representation theory of ﬁnite dimensional algebras anton cox · representation theory of ﬁnite...

Representation theory of finite dimensional algebras

Anton Cox

Notes for the London Taught Course Centre

Autumn 2008

Centre for Mathematical ScienceCity University

Northampton SquareLondon EC1V 0HB England

1

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 5Recommended reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 6

Chapter 1. Algebras and modules . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 71.1. Associative algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 71.2. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91.3. Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 111.4. Representations of quivers . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 131.5. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 15

Chapter 2. Semisimplicity and some basic structure theorems . . . . . . . . . . . . . . . . 172.1. Simple modules and semisimplicity . . . . . . . . . . . . . . . . .. . . . . . . . . 172.2. Schur’s lemma and the Artin-Wedderburn theorem . . . . . .. . . . . . . . . . . . 192.3. The Jacobson radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 212.4. The Krull-Schmidt theorem . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 232.5. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 25

Chapter 3. Projective and injective modules . . . . . . . . . . . . .. . . . . . . . . . . . 273.1. Projective and injective modules . . . . . . . . . . . . . . . . . .. . . . . . . . . 273.2. Idempotents and direct sum decompositions . . . . . . . . . .. . . . . . . . . . . 313.3. Simple and projective modules for bound quiver algebras . . . . . . . . . . . . . . 343.4. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 35

Chapter 4. Representation type and Gabriel’s theorem . . . . .. . . . . . . . . . . . . . . 374.1. Representation type . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 374.2. Representation type of quiver algebras . . . . . . . . . . . . .. . . . . . . . . . . 404.3. Dimension vectors and Cartan matrices . . . . . . . . . . . . . .. . . . . . . . . . 414.4. Reflection functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 434.5. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 46

Chapter 5. Further directions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 475.1. Ring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 475.2. Almost split sequences and the geometry of representations . . . . . . . . . . . . . 475.3. Local representation theory . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 495.4. Representations of other algebraic objects . . . . . . . . .. . . . . . . . . . . . . 495.5. Quantum groups and the Ringel-Hall algebra . . . . . . . . . .. . . . . . . . . . . 50

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 53

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

3

Introduction

This course will provide a basic introduction to the representation theory of algebras, concen-trating mainly on the finite dimensional case. Representation theory is concerned with the studyof how various algebraic objects act on vector spaces, in a manner which respects the originalalgebraic structure. Finite dimensional algebras, while of interest in their own right, provide a (rel-atively) elementary setting in which to develop some of the basic language, while still exhibitingmost of the key features that can arise.

It is common for a first course in representation theory to concentrate on the character theory offinite groups over the complex numbers. This has a number of advantages, not least that charactersare much easier to construct than the corresponding representations. However, the theory is ratherunrepresentative in certain crucial respects.

Most important of these is that such representations are always semisimple. This means that itis enough to classify those representations which have no sub-representations, which (for a givengroup) is a finite number. All other representations can thenbe constructed from these via directsums. In general one cannot hope to construct all representations of an algebra. In fact, we willsee that such an aim is provably impossible to achieve (except in certain special cases). Instead wewill develop various tools to analyse representations in general.

In this course we will focus not on group algebras (although these will play a role), but ratheron certain algebras associated to quivers. These have many advantages (even over group algebras)in terms of ease of computation and of constructing examples, but are rich enough to give a betterflavour of general aspects of representation theory. Indeed, it will turn out that to understandthe representation theory ofany finite dimensional algebra over an algebraically closed field itis enough to understand the representation theory associated to quivers and quotients of quivers.This includes group algebras as a special case (and over any algebraically closed field, not just thecomplex numbers).

In Chapter 1 we will begin with various basic definitions and examples. First we will lookat algebras and modules, and then at quivers and their representations. We will then see that thequiver setting gives rise to examples in the algebra setting.

Chapter 2 covers the core classical representation theory of algebras. We begin with an analysisof the relation between simple representations and representations in general, and then consider forwhich algebras we can reduce to the study of simples alone. Such algebras are called semisimple,and the Artin-Wedderburn Theorem will give a complete classification in this case.

If an algebra is not semisimple, then the Jacobson radical ofthe algebra can be regarded as ameasure of its non-semisimplicity. We will develop the basic properties of this. The Krull-SchmidtTheorem then tells us that it is enough to determine the indecomposable modules (a class whichcontains the simple modules but is in general much larger).

5

6 INTRODUCTION

In non-semisimple settings projective and injective modules play a key role. In this coursewe will only be able to touch on the basic definitions, and willsay nothing of the vital role theyplay in cohomology. This is in part because we will not have the time to develop the necessarybackground in category theory which is an important part of modern day algebra.

Chapter 3 gives the basic definitions of projective and injectives, before going on to a studyof the role of idempotents in representation theory. Using these is what allows us to reduce to thestudy of quivers, although we will only give an outline of thereduction method here. We thenshow how simple, projective, and injective modules can be easily constructed for quivers.

Chapter 4 introduces the notion of representation type. This is a measure of how hard it isto fully understand the representation theory of an algebra. The fundamental theorem of Drozdsays that every algebra falls into one of three types; the first two being (in principle) completelyunderstandable, while the third is provably impossible to fully understand.

We will sketch how the classification by type can be carried out in two special cases: groupalgebras, and for representations of quivers without relations. The latter case will allow us tointroduce some more ideas from the representation theory ofquivers which are used in the proof.

Finally in Chapter 5 we will indicate some further topics which the reader may wish to inves-tigate.

Recommended reading

Due to the limited number of lectures available, the lectures will consist of an outline of themain theory together with some examples. These notes will fill in more of the details, but withonly sketches of the proofs in places. Each Chapter ends witha brief selection of exercises forthe reader. For a far more comprehensive treatment of this material (together with many moreexamples and exercises) the reader is recommended to look at[ASS06, Chapters I-III].

For simplicity, [ASS06] only considers algebras over algebraically closed fields.An excellent(if rapid) introduction which considers more general ringscan be found in [Ben91, Chapters 1 and4]. Other notes available on the web which cover similar material are [Bru03] and [Bar06]. Ourexposition draws on all of these sources, as well as unpublished lecture notes of Erdmann.

Two books which go into more advanced topics than this course(and which are not entirelysuitable for the beginner) are [ARS94] and [GR97].

CHAPTER 1

Algebras and modules

In this course we will be interested in the representation theory of finite dimensional algebrasdefined over a field. We begin by recalling certain basic definitions concerning fields.

DEFINITION 1.0.1. A field k isalgebraically closedif every non-constant polynomial withcoefficients in k has a root in k. A field hascharacteristicp if p is the smallest positive integer suchthat

p

∑i=1

1 = 0.

If there is no such p then the field is said to havecharacteristic 0. A field isinfinite if it containsinfinitely many elements.

Henceforthk will denote some field.

1.1. Associative algebras

DEFINITION 1.1.1. Analgebra overk, or k-algebrais a k-vector space A with a bilinear map

A×A −→ A(x,y) 7−→ xy.

We say that the algebra isassociativeif for all x,y,z∈ A we have

x(yz) = (xy)z.

An algebra A isunital if there exists an element1∈ A such that1x = x1 = x for all x∈ A. Such anelement is called theidentity in A. (Note that such an element is necessarily unique.) We say thatan algebra isfinite dimensionalif the underlying vector space is finite dimensional. An algebra Ais commutativeif xy = yx for all x,y∈ A.

It is common to abuse terminology and take algebra to mean an associative unital algebra, andwe will follow this convention. There are several importantclasses of non-associative algebras (forexample Lie algebras) but we shall not consider them here.Thus all algebras we consider willbe associative and unital.

EXAMPLE 1.1.2. (a) Let k[x1, . . . ,xn] denote the vector space of polynomials in the (commut-ing) variables x1, . . . ,xn. This is an infinite dimensional commutative algebra with multiplicationgiven by the usual multiplication of polynomials, and identity given by the trivial polynomial1.

(b) Let k〈x1, . . . ,xn〉 denote the vector space of polynomials in thenon-commutingvariablesx1, . . . ,xn. A general element is of the form∑n

i=1λiwi for some n where for each i,λi ∈ k and

7

8 1. ALGEBRAS AND MODULES

wi = xa1i1

xa2i2

. . .xatit for some t. Given two elements∑n

i=1λiwi and∑mi=1λ ′

i w′i the product is defined to

be the elementn

∑i=1

m

∑j=1

λiλ ′i wiw

′j

where wiw j denotes the element obtained from wi and wj by concatenation. This is an infinitedimensional associative algebra with identity given by thetrivial polynomial1. If n > 1 then thealgebra is non-commutative.

(c) Given a group G, we denote by kG thegroup algebraobtained by considering the vectorspace of formal linear combinations of group elements. Given two elements∑n

i=1 λigi and∑mi=1 µihi

with λi,µi ∈ k and gi ,hi ∈ G we define the product to be the elementn

∑i=1

m

∑j=1

λiµ jgih j .

The identity element is the identity element e∈ G regarded as an element of kG. The algebra kG isfinite dimensional if and only if G is a finite group, and is commutative if and only if G is abelian.

(d) The set Mn(k) of n×n matrices with entries in k is a finite dimensional algebra, thema-trix algebra, with the usual matrix multiplication, and identity element the matrix I. If n> 1 itis non-commutative. Equivalently, let V be an n-dimensional k-vector space, and consider theendomorphism algebra

Endk(V) = { f : V −→V | f is k-linear}.

This is an algebra with multiplication given by compositionof functions. Fixing a basis for V theelements ofEndk(V) can be written in terms of matrices with respect to this basis, and in this waywe can identifyEndk(V) with Mn(k).

(e) If A is an algebra then so is Aop, theopposite algebra, which equals A as a vector space,but with multiplication map(x,y) 7−→ yx.

As usual in Algebra, we are not just interested in objects (inthis case algebras), but also infunctions between them which respect the underlying structures.

DEFINITION 1.1.3. A homomorphismbetween k-algebras A and B is a linear mapφ : A−→Bsuch thatφ(1) = 1 andφ(xy) = φ(x)φ(y) for all x,y∈ A. This is anisomorphismprecisely whenthe linear map is a bijection.

DEFINITION 1.1.4. Given an algebra A, asubalgebraof A is a subspace S of A containing1,such that for all x,y∈ S we have xy∈ S. Aleft (respectively right) idealin A is a subspace I of Asuch that for all x∈ I and a∈ A we have ax∈ I (respectively xa∈ I). If I is a left and a right idealthen we say that I is anideal in A.

EXAMPLE 1.1.5. (a) If H is a subgroup of a group G, then kH is a subalgebra of kG.

(b) Given two algebras A and B, and a homomorphismφ : A−→ B, the setim(φ) is a subal-gebra of B, whileker(φ) is an ideal in A.

Idempotents play a crucial role in the analysis of algebras.

DEFINITION 1.1.6. An element e∈ A is anidempotentif e2 = e. Two idempotents e1 and e2 inA areorthogonalif

e1e2 = e2e1 = 0.

1.2. MODULES 9

An idempotent e is calledprimitive if it cannot be written in the form e= e1 +e2 where e1 and e2are non-zero orthogonal idempotents. An idempotent e iscentralif ea= ae for all a∈ A.

1.2. Modules

Representation theory is concerned with the study of the wayin which certain algebraic objects(in our case, algebras) act on vector spaces. There are two ways to express this concept; in termsof representations or (in more modern language) in terms of modules.

DEFINITION 1.2.1. Given an algebra A over k, arepresentationof A is an algebra homomor-phism

φ : A−→ Endk(M)

for some vector space M. Aleft A-moduleis a k-vector space M together with a bilinear mapA×M −→M, which we will denote by(a,m) 7−→ am, such that for all m∈ M and x,y∈ A we have1m= m and(xy)m= x(ym). Similarly, aright A-moduleis a k-vector space M and a bilinear mapφ : M×A−→ M such that m1 = m and m(xy) = (mx)y for all m∈ M and x,y∈ A. We will adoptthe convention that all modules are left modules unless stated otherwise.

DEFINITION 1.2.2. An A-module isfinite dimensionalif it is finite dimensional as a vectorspace. An A-module M isgeneratedby a set{m1 : i ∈ I} (where I is some index set) if everyelement m of M can be written in the form

m= ∑i∈I

aimi

for some ai ∈ A. We say that M isfinitely generatedif it is generated by a finite set of elements. IfA is a finite dimensional algebra then M is finitely generated if and only if M is finite dimensional.

LEMMA 1.2.3. (a) There is a natural equivalence between left (respectively right) A-modulesand right (respectively left) Aop-modules.(b) There is a natural equivalence between representationsof A and left A-modules.

PROOF. We give the correspondence in each case; details are left tothe reader. Given a leftmoduleM for A with bilinear mapφ : A×M −→ M, define a rightAop-module structure onMvia the mapφ ′ : M ×A −→ M given byφ ′(m,x) = φ(x,m). It is easy to verify thatφ is anAop-homomorphism.

Given a representationφ : A −→ Endk(M) of A we define anA-module structure onM bysetting

am= φ(a)(m)

for all a∈ A andm∈ M. Conversely, given anA-moduleM, the mapM −→ M given bym 7−→ rmis linear, and gives the desired representationφ : A−→ Endk(M). �

DEFINITION 1.2.4. A homomorphismbetween A-modules M and N is a linear mapφ : M −→N such thatφ(am) = aφ(m) for all a ∈ A and m∈ M. This is anisomorphismprecisely when thelinear map is a bijection.

DEFINITION 1.2.5. Given an A-module M, asubmoduleof M is a subspace N of M such thatfor all n∈N and a∈A we have an∈N. (Note that N is an A-module in its own right.) The quotientspace

M/N = {m+N : m∈ M}


(under the relation m+N = m′ +N if and only if m−m′ ∈ N) has an A-module structure given bya(m+N) = am+N, and is called thequotientof M by N.

EXAMPLE 1.2.6. (a) The algebra A is a (left or right) A-module, with respect to the usualmultiplication map on A. If I is a left ideal of A then I is a submodule of the left module A.

(b) If A = k then A-modules are just k-vector spaces.

(c) If A= k[x1, . . . ,xn] then an A-module is a k-vector space M together with commuting lineartransformationsαi : M −→ M (whereαi describes the action of xi).

(d) Every A-module M has M and the empty vector space0 as submodules.

LEMMA 1.2.7 (Isomorphism Theorem).If M and N are A-modules andφ : M −→ N is ahomomorphism of A-modules then

im(φ) ∼= M/ker(φ)

as A-modules.

PROOF. Copy the proof for linear maps between vector spaces, noting that the additional struc-ture of a module is preserved. �

DEFINITION 1.2.8. If an A-module M has submodules L and N such that M= L⊕N as avector space then we say that M is thedirect sumof L and N. A module M isindecomposableif itis not the direct sum of two non-zero submodules (and isdecomposableotherwise). A module M issimple(or irreducible) if M has no submodules except M and0.

For vector spaces, the notions of indecomposability and irreducibility coincide. However, thisis not the case for modules in general.

EXAMPLE 1.2.9. Let C2 denote the cyclic group with elements{1,g}, and consider the two-dimensional kC2-module M with basis{m1,m2} where gm1 = m2 and gm2 = m1. If M = N1⊕N2with N1 and N2 non-zero then each Ni is the span of a vector of the formλ1m1 + λ2m2 for someλ1,λ2 ∈ k. Applying g we deduce thatλ1 = ±λ2, and hence Ni must be the span of m1−m2 orm1+m2. But N1 = N2 if k has characteristic2, which contradicts our assumption. Thus M is neverirreducible, but is indecomposable if and only if the characteristic of k is2. We will see that thisexample generalises to arbitrary group algebras when we consider Maschke’s Theorem.

There is a close relationship between the representation theory ofA andAop.

DEFINITION 1.2.10. Let M be a finite dimensional (left) A-module. Then thedual moduleM∗ is the dual vector spaceHomk(M,k) with a right A-module action given by(φa)(m) = φ(am)for all a ∈ A, m∈ M andφ ∈ Homk(M,k). By Lemma 1.2.3 this gives M∗ the structure of a leftAop-module.

Taking the dual of anAop-module gives anA-module, and it is easy to verify (as for vectorspaces) that

LEMMA 1.2.11.For any finite dimensional A-module M we have M∗∗ ∼= M.

1.3. QUIVERS 11

1.3. Quivers

DEFINITION 1.3.1. A quiverQ is a directed graph. We will denote the set of vertices by Q0,and the set of edges (which we callarrows) by Q1. If Q0 and Q1 are both finite then Q is afinitequiver. Theunderlying graphQ of a quiver Q is the graph obtained from Q by forgetting allorientations of edges.

A path of lengthn in Q is a sequence p= α1α2 . . .αn where eachαi is an arrow andαi startsat the vertex whereαi+1 ends. For each vertex i, there is a path of length0, which we denote byεi . A quiver isacyclic if the only paths which start and end at the same vertex have length0, andconnectedif Q is a connected graph.

EXAMPLE 1.3.2. (a) For the quiver Q given by

•1

α��•2

γ//

δ//β 55 •3 •ρ

oo

the set of paths of length greater than1 is given by

{β n+2,β n+1α,γβ n+1,δβ n+1,γβ nα,δβ nα : n≥ 0}.

(b) For the quiver Q given by•1α 55 βii

the set of paths corresponds to words inα andβ (along with the trivial word).

(c) For the quiver Q given by

•1α // •2

β// •3 •4

γoo

the set of paths is

{ε1,ε2,ε3,ε4,α,β ,γ,βα}.

We would like to associate an algebra to a quiver; however, weneed to take a little care.

DEFINITION 1.3.3. Thepath algebrakQ of a quiver Q is the k-vector space with basis the setof paths in Q. Multiplication is via concatenation of paths:if p = α1α2 . . .αn and q= β1β2 . . .βmthen

pq= α1α2 . . .αnβ1β2 . . .βm

if αn starts at the vertex whereβ1 ends, and is0 otherwise.

We have not yet checked that the above definition does in fact define an algebra structure onkQ.

LEMMA 1.3.4. Let Q be a quiver. Then kQ is an associative algebra. Further kQ has anidentity element if and only if Q0 is finite, and is finite dimensional if and only if Q is finite andacyclic.


PROOF. The associativity of multiplication inkQ is straightforward. Next note that the ele-mentsεi satisfy

εiε j = δi j εi

and hence form a set of orthogonal idempotents. Further, forany pathp∈ kQ we haveεi p = p ifp ends at vertexi and 0 otherwise. Hence ifQ0 is finite then

∑i∈Q0

εi p = p.

Similarly

∑i∈Q0

pεi = p

and hence1 = ∑

i∈Q0

εi

is the unit inkQ.

Conversely, suppose thatQ0 is infinite and 1∈ kQ. Then 1= ∑λi pi for some (finite) set ofpathspi and scalarsλi . Pick a vertexj such that for alli the pathpi does not end atj. Thenε j1= 0,which gives a contradiction.

Finally, if Q0 or Q1 is not finite thenkQ is clearly not finite dimensional. Given a finite setof vertices with finitely many edges, there are only finitely many paths between them unless thequiver contains a cycle. �

EXAMPLE 1.3.5. Each of the quivers in Example 1.3.2 is finite, and so the corresponding kQcontains a unit. However, the path algebras corresponding to 1.3.2(a) and 1.3.2(b) are not finitedimensional. Indeed, it is easy to see that the path algebra for (b) is isomorphic to k〈x,y〉, underthe map takingα to x andβ to y. The path algebra for 1.3.2(c) is an8-dimensional algebra.

Because of Lemma 1.3.4 we will only consider finite quiversQ, so that the corresponding pathalgebras are unital.

DEFINITION 1.3.6. Given a finite quiver Q, the ideal RQ of kQ generated by the arrows in Qis called thearrow idealof kQ. Then RmQ is the ideal generated by all paths of length m in Q. Anideal I in kQ is calledadmissibleif there exists m≥ 2 such that

RmQ ⊆ I ⊆ R2

Q.

If I is admissible then(Q, I) is called abound quiver, and kQ/I is a bound quiver algebra.

Note that ifQ is finite and acyclic then any ideal contained inR2Q is admissible, asRm

Q = 0 if mis greater than the maximal path length inQ.

EXAMPLE 1.3.7. Let Q be as in Example 1.3.2(b), and let I= 〈βα,β 2〉. This is not an ad-missible ideal in kQ as it does not containαm for any m≥ 1, and so does not contain Rm

Q for anym≥ 2.

PROPOSITION1.3.8. Let Q be a finite quiver with admissible ideal I in kQ. Then kQ/I is finitedimensional.

1.4. REPRESENTATIONS OF QUIVERS 13

PROOF. As I is admissible there existsm≥ 2 such thatRmQ ⊆ I . Hence there is a surjective al-

gebra homomorphism fromkQ/RmQ ontokQ/I . But the former algebra is clearly finite dimensional

as there are only finitely many paths of length less thanm. �

DEFINITION 1.3.9. A relation in kQ is a finite linear combination of paths of length at leasttwo in Q such that all paths have the same start vertex and the same end vertex. If{ρ j : j ∈ J} is aset of relations in kQ such that the ideal generated by the setis admissible then we say that kQ isbound by the relations.

EXAMPLE 1.3.10.Consider the quiver in Example 1.3.2(a) and the relations

{γβ 2α −δα,γβ +δβ ,β 5}.

Any path of length at least7 must containβ 5, and so Q is bound by this set of relations.

In fact the above example generalises: it is easy to see that any idealI in R2Q is admissible if it

contains each cycle inQ to some power. Further, we have

PROPOSITION1.3.11.Let Q be a finite quiver. Every admissible ideal in kQ is generated by afinite sequence of relations in kQ.

PROOF. (Sketch) It is easy to check that every admissible idealI is finitely generated by someset{a1, . . . ,an} (asRm

Q and I/RmQ are finitely generated). However, in general a set of generators

for I will not be a set of relations, as the paths in eachai may not all have the same start vertex andend vertex. However, the non-zero elements in the set

{εxaiεy : 1≤ i ≤ n,x,y∈ Q0}

are all relations, and this set generatesI . �

1.4. Representations of quivers

DEFINITION 1.4.1. Let Q be a finite quiver. ArepresentationM of Q over k is a collection of k-vector spaces{Ma : a∈ Q0} together with a linear mapφα : Ma −→Mb for each arrowα : a−→ bin Q1. The representation M isfinite dimensionalif all the Ma are finite dimensional.

DEFINITION 1.4.2.Given two representations M and M′ of a finite quiver Q, ahomomorphismfrom M to N is a collection of linear maps fi : Mi −→ M′

i such that for each arrowα : i −→ j wehaveφ ′

α fi = f jφα .

When giving examples of representations of quivers we will usually fix bases of each of thevector spaces, and represent the maps between them by matrices with respect to column vectors inthese bases.

EXAMPLE 1.4.3. Consider the quiver

•1α //

δ BBB

BBBB

B•2

β// •3 •4

γoo

•5

ρ

>>||||||||

.


This has a representation

k(1

0) //

(01) ��=

====

=== k2

(1 11 0) // k2 k3

(1 2 10 1 0)oo

k2(0 1

1 1)

??��

Notice how easy it was to give a representation: there are no compatibility relations to bechecked (apart from that the linear maps go between the appropriate dimension) so examples canbe easily generated for any path algebra. This is very different from writing down explicit modulesfor an algebra (in general).

Definition 1.4.1 looks rather different from that for an algebra. However, the next lemmashows that representations ofQ correspond tokQ-modules in a natural way.

LEMMA 1.4.4. Let M be a representation of a finite acyclic quiver Q. Consider the vectorspace

M′ =⊕

a∈Q0

Ma.

This can be given the structure of a kQ-module by defining for eachα : i −→ j a mapφ ′α : M −→M

byφ ′

α(m1, . . . ,mn) = (0, . . . ,0,φα(mi),0, . . .0)

where the non-zero entry is in position j, and for each i∈ Q0 a mapεi : M −→ M by

εi(m1, . . . ,mn) = (0, . . . ,0,mi,0, . . . ,0)

where the non-zero entry is in position i. Conversely, suppose that N is a kQ-module. Then weobtain a representation of Q by setting Na = εaN and definingφα for α : a−→ b to be the restrictionof the action ofα ∈ kQ to Na.

PROOF. Checking that the above definitions give akQ-module and a representation ofQ re-spectively is routine. �

We also need the notion of a representation of a bound quiver.Note that we do not need toassume thatQ is acyclic here, as admissible ideals guarantee that the associated quotient algebra isfinite dimensional.

DEFINITION 1.4.5. Given a path p= α1α2 . . .αn in a finite quiver Q from a to b and a repre-sentation M of Q we define the linear mapφp from Ma to Mb by

φp = φαnφαn−1 . . .φα1.

If ρ is a linear combination of paths pi with the same start vertex and the same end vertex thenφρis defined to be the corresponding linear combination of theφpi . Given an admissible ideal I in kQwe say that M isbound byI if φρ = 0 for all relationsρ ∈ I.

EXAMPLE 1.4.6. Consider the representation in Example 1.4.3. Let p= βα and q= ρδ .Then

φp =

(

1 11 0

)(

10

)

=

(

11

)

φq =

(

0 11 1

)(

01

)

=

(

11

)

and so this representation is bound by the ideal〈βα −ρδ 〉.

1.5. EXERCISES 15

It is easy to verify that the correspondence between representation of finite acyclicQ andkQ-modules given in Lemma 1.4.4 extends to a correspondence between representations of finiteQbound byI andkQ/I -modules.

In this course we are avoiding the language of category theory. This is mainly due to lackof time: the language of categories and functors is a very powerful one, and many results inrepresentation theory are best stated in this way. Roughly,a category is a collection ofobjects(e.g.kQ-modules) andmorphisms(e.g. kQ-homomorphisms), and the idea is to study the category asa whole rather than just the objects or morphisms separately. A functor is then a map from onecategory to another which transports both objects and morphisms in a suitably compatible way.In this language the above result relating bound representations ofQ andkQ/I -modules gives anequivalence between the corresponding categories.

1.5. Exercises

(1) Suppose thatI is an ideal in an algebraA.

(a) Show thatA/I has an algebra structure such that there is a surjective homomorphismfrom A to A/I .

(b) Suppose thatA is an algebra with idealI , and thatM is anA/I -module. Show thatMcan be given the structure of anA-module.

(c) If M is anA-module, what condition must it satisfy to be anA/I -module?

(2) Suppose that(P,≤) is a partially ordered set of cardinalityn, and definekP to be thesubset ofMn(k) given by

kP= {M = (mi j ) : mi j = 0 if i 6≤ j}.

(a) Show thatkP is a subalgebra ofMn(k) (this is called theincidence algebraof (P,≤)).(b) Show thatP can be identified with the set{1, . . .n} in such a way thatkP can be

identified with a subalgebra of the algebraLTn(k) of lower triangular matrices inMn(k).

(c) Deduce that ifQ is a finite acyclic quiver with at most one arrow between each pairof vertices, thenkQ is a subalgebra ofLTn(k) for somen.

(d) Illustrate your last construction in the case of the quiver in Example 1.3.2(c).(e) Which quiver correspond to the whole ofLTn(k)?

(3) Suppose thatQ is a quiver, and letQop be the quiver obtained by reversing all the arrows.Show that there is an isomorphism of algebrask(Qop) ∼= (kQ)op.

(4) Suppose thatG is a group. Show thatkG∼= (kG)op.

(5) Classify the simple modules for the cyclic groupCn over an algebraically closed field ofcharacteristicp≥ 0.

(6) Suppose thatM = (Ma,φa) is a representation of some finite quiverQ.


(a) Given vector spacesNa ≤ Ma, what conditions must be satisfied for(Na,φa) to be asubrepresentationN of Q?

(b) Suppose thatM is a representation ofQ bound by an admissible idealI . Show thatthe representationN is also bound byI .

(c) If Q hasn vertices, given non-isomorphic simple representations ofkQ, and also ofkQ/I . (Hint: what condition on the dimensions of theNa guarantees the absence of aproper subrepresentation?)

(d) If Q is acyclic then we will see in Chapter 2 that these examples form a complete setof simple representations. However, it is also possible to show this directly. Supposethat M is a representation of an acyclicQ such that more than oneMa is non-zero.Show thatM has a proper subrepresentation.

(e) Suppose thatQ is finite but contains some cycle. Show thatQ now has infinitelymany non-isomorphic simple representations overC.

(7) In this exercise we will classify the indecomposable representations of the quiverQ givenby

•1α1 // •2

α2 // •3α3 // . . .

αn−2 // •n−1αn−1 // •n .

Let M = (Mi,φi) be an indecomposable representation ofQ.(a) Show that ifφi is not injective thenM j = 0 for j > i.(b) Similarly show that ifφi is not surjective thenM j = 0 for j ≤ i.(c) Deduce thatM is isomorphic to a representation of the form

0 // . . . // 0 // kid // . . . id // k // 0 // . . . // 0 .

(d) Show that then(n+1)2 such modules are pairwise non-isomorphic.

We will see in Chapter 4 that this example is part of a more general picture.

(8) Let S3 denote the symmetric group on three symbols. Decompose the group algebraCS3into a direct sum of simple representations forS3. (You may find it convenient to identifyCS3 with a space of permutation matrices.)

CHAPTER 2

Semisimplicity and some basic structure theorems

In this chapter we will review some of the classical structure theorems for finite dimensionalalgebras. In most cases results will be stated with only a sketch of the proof. Henceforth we willrestrict our attention to finite dimensional modules.

2.1. Simple modules and semisimplicity

Recall that a simple module is a moduleSsuch that the only submodules areSand 0. Theseform the building blocks out of which all other modules are made:

LEMMA 2.1.1. If M is a finite dimensional A-module then there exists a sequence of submod-ules

0 = M0 ⊂ M1 ⊂ ·· · ⊂ Mn = M

such that Mi/Mi−1 is simple for each1≤ i ≤ n. Such a series is called acomposition seriesfor M.

PROOF. Proceed by induction on the dimension ofM. If M is not simple, pick a submoduleM1of minimal dimension, which is necessarily simple. Now dim(M/M1) < dimM, and so the resultfollows by induction. �

Moreover, we have

THEOREM 2.1.2 (Jordan-Hölder).Suppose that M has two composition series

0 = M0 ⊂ M1 ⊂ ·· · ⊂ Mm = M, 0 = N0 ⊂ N1 ⊂ ·· · ⊂ Nn = M.

Then n= m and there exists a permutationσ of {1, . . .n} such that

Mi/Mi+1∼= Nσ(i)/Nσ(i)+1.

PROOF. The proof is similar to that for groups. �

Life would be (relatively) straightforward if every modulewas a direct sum of simple modules.

DEFINITION 2.1.3. A module M issemisimple(or completely reducible) if it can be written asa direct sum of simple modules. An algebra A issemisimpleif every finite dimensional A-moduleis semisimple.

LEMMA 2.1.4. If M is a finite dimensional A-module then the following are equivalent:(a) If N is a submodule of M then there exists L a submodule of M such that M= L⊕N.(b) M is semisimple.(c) M is a (not necessarily direct) sum of simple submodules.

17

18 2. SEMISIMPLICITY AND SOME BASIC STRUCTURE THEOREMS

PROOF. (Sketch) Note that (a) implies (b) and (b) implies (c) are clear. For (c) implies (a)consider the set of submodules ofA whose intersection withN is 0. Pick one such,L say, ofmaximal dimension; ifN⊕L 6= M then there is some simpleS in M not in N⊕L. But this wouldimply thatS+L has intersection 0 withA, contradicting the maximality ofL. �

LEMMA 2.1.5. If M is a semisimple A-module then so is every submodule and quotient moduleof M.

PROOF. (Sketch) IfN is a submodule thenM = N⊕L for someL by the preceding Lemma.But thenM/L ∼= N, and so it is enough to prove the result for quotient modules.

If M/L is a quotient module consider the projection homomorphismπ from M to M/L. WriteM as a sum of simple modulesSi and verify thatπ(S) is either simple or 0. This proves thatM/Lis a sum of simple modules, and so the result follows from the preceding lemma. �

To show that an algebra is semisimple, we do not want to have tocheck the condition for everypossible module. Fortunately we have

PROPOSITION2.1.6. Every finite dimensional A-module is isomorphic to a quotient of An forsome n. Hence an algebra A is semisimple if and only if A is semisimple as an A-module.

PROOF. (Sketch) Suppose thatM is a finite dimensionalA-module, spanned by some elementsm1, . . . ,mn. We define a map

φ : ⊕ni=1A−→ M

by

φ((a1, . . . ,an)) =n

∑i=1

aimi .

It is easy to check that this is a homomorphism ofA-modules, and so by the isomorphism theoremwe have that

M ∼= ⊕ni=1A/kerφ .

The result now follows from the preceding lemma. �

For finite groups we can say exactly whenkG is semisimple:

THEOREM 2.1.7 (Maschke).Let G be a finite group. Then the group algebra kG is semisimpleif and only if the characteristic of k does not divide|G|, the order of the group.

PROOF. (Sketch) First suppose that the characteristic ofk does not divide|G|. We must showthat everykG-submoduleM of kG has a complement as a module. Clearly as vector spaces wecan findN such thatM⊕N = kG. Let π : kG−→ M be the projection mapπ(m+n) = m for allm∈ M andn∈ N. We want to modifyπ so that it is a module homomorphism, and then show thatthe kernel is the desired complement.

Define a mapTπ : kG−→ M by

Tπ(m) =1|G| ∑

g∈G

g(π(g−1m)).

Note that this is possible as|G|−1 exists ink. It is then routine to check thatTπ is a kG-modulemap.

2.2. SCHUR’S LEMMA AND THE ARTIN-WEDDERBURN THEOREM 19

Now let K = ker(Tπ), which is a submodule ofkG. We want to show thatkG= M⊕K. Firstshow thatTπ acts as the identity onM, which implies thatM ∩K = 0. Next note that by therank-nullity theorem for linear maps,kG = M + K. Combining these two facts we deduce thatkG= M⊕K as required.

For the reverse implication, considerw = ∑g∈Gg∈ kG. It is easy to check that every elementof g fixesw, and hencew spans a one-dimensional submoduleM of kG. Now suppose that there isa complementary submoduleN of kG, and decompose 1= e+ f whereeand f are the idempotentscorresponding toM andN respectively. We havee= λw for someλ ∈ k, ande2 = e= λ 2w2. Itis easy to check thatw2 = |G|w and henceλw = λ 2|G|w which implies that 1= λ |G|. But thiscontradicts the fact that|G| = 0 in k. �

The next result will be important in the following section.

LEMMA 2.1.8. The algebra Mn(k) is semisimple.

PROOF. Let Ei j denote the matrix inA = Mn(k) consisting of zeros everywhere except for the(i, j)th entry, which is 1. We first note that

1 = E11+E22+ · · ·+Enn

is an orthogonal idempotent decomposition of 1, and henceA decomposes as a direct sum ofmodules of the formAEii . We will show that these summands are simple.

First observe thatAEii is just the set of matrices which are zero except possibly in column i.Pick x∈ AEii non-zero; we must show thatAx= AEii . As x is non-zero there is some entryxmi inthe matrixx which is non-zero. But then

E jmx = xmiE ji ∈ Ax

and henceE ji ∈ Ax for all 1≤ j ≤ n. But this implies thatAx= AEii as required. �

2.2. Schur’s lemma and the Artin-Wedderburn theorem

We begin with Schur’s lemma, which tells us about automorphisms of simple modules.

LEMMA 2.2.1 (Schur).Let S be a simple A-module andφ : S−→S a non-zero homomorphism.Thenφ is invertible.

PROOF. Let M = kerφ andN = imφ ; these are both submodules ofS. But S is simple andφ 6= 0, soM = 0 andφ is injective. Similarly we see thatN = S, soφ is surjective, and henceφ isinvertible. �

LEMMA 2.2.2. If k is algebraically closed and S is finite dimensional with non-zero endomor-phismφ , thenφ = λ . idS, for some non-zeroλ ∈ k.

PROOF. As k is algebraically closed and dimS< ∞ the mapφ has an eigenvalueλ ∈ k. Thenφ −λ idS is an endomorphism ofSwith non-zero kernel (containing all eigenvectors with eigen-value λ ). Arguing as in the preceding lemma we deduce that ker(φ − λ idS) = S, and henceφ = λ idS. �


Given anA-moduleM we set

EndA(M) = {φ : M −→ M | φ is anA-homomorphism}.

This is a subalgebra of Endk(M). More generally, ifM andN areA-modules we set

HomA(M,N) = {φ : M −→ N | φ is anA-homomorphism}.

Arguing as in the proof of Lemma 2.2.1 above we obtain

LEMMA 2.2.3 (Schur).If k is algebraically closed and S and T are simple A-modules then

HomA(S,T) ∼=

{

k if S∼= T0 otherwise.

We can now give a complete classification of the finite dimensional semisimple algebras.

THEOREM 2.2.4 (Artin-Wedderburn).Let A be a finite dimensional algebra over an alge-braically closed field k. Then A is semisimple if and only if

A∼= Mn1(k)⊕Mn2(k)⊕·· ·⊕Mnt (k)

for some t∈ N and n1, . . . ,nt ∈ N.

PROOF. (Sketch) We saw in Lemma 2.1.8 thatMn(k) is a semisimple algebra, and ifA andBare semisimple algebras, then it is easy to verify thatA⊕B is semisimple.

For the reverse implication suppose thatM andN areA-modules, withM = ⊕ni=1Mi andN =

⊕mi=1Ni . The first claim is that HomA(M,N) can be identified with the space of matrices

{(φi j )1≤i≤n,1≤ j≤m | φi, j : M j −→ Ni anA-homomorphism}

and that ifM = N with Mi = Ni for all i then this space of matrices is an algebra by matrix multi-plication, isomorphic to EndA(M). This follows by an elementary calculation.

Now apply this to the special case whereM = N = A, and

A = (S1⊕S2⊕·· ·⊕Sn1)⊕ (Sn1+1⊕·· ·⊕Sn1+n2)⊕·· ·⊕ (Sn1+n2+···+nt−1+1⊕·· ·⊕Sn1+n2+···+nt )

is a decomposition ofA into simples such that two simples are isomorphic if and onlyif they occurin the same bracketed term. By Schur’s Lemma above we see thatφi j in this special case is 0 ifSi andSj are in different bracketed terms, and is someλi j ∈ k otherwise. There is then an obviousisomorphism of HomA(A,A) with Mn1(k)⊕·· ·⊕Mnt (k). Finally, we note that for any algebraAwe have

EndA(A,A) ∼= Aop

and hence

A = (Aop)op∼= Mn1(k)op⊕·· ·⊕Mnt (k)

op.

But it is easy to see thatMn(k) ∼= Mn(k)op via the map taking a matrixX to its transpose, and sowe are done. �

We can also describe all the simple modules for such an algebra.

2.3. THE JACOBSON RADICAL 21

COROLLARY 2.2.5. Suppose that

A∼= Mn1(k)⊕Mn2(k)⊕·· ·⊕Mnt (k).

Then A has exactly t isomorphism classes of simple modules, one for each matrix algebra. IfSi is the simple corresponding to Mni(k) then dimSi = ni and Si occurs precisely ni times in adecomposition of A into simple modules.

PROOF. (Sketch) Choose a basis forA such that for each elementa ∈ A the mapx 7−→ ax isgiven by a block matrix

A1 0 0 · · · 00 A2 0 · · · 0...

...0 · · · 0 0 At

whereAi ∈ Mni (k). ThenA is the direct sum of the spaces given by the columns of this matrix,each of dimensionni . Arguing as in Lemma 2.1.8 we see that each of these column spaces is asimpleA-module. Swapping rows in a given block gives isomorphic modules. Thus there are atmostt non-isomorphic simples in a decomposition ofA (and hence by Proposition 2.1.6 at mostt isomorphism classes). Two simples from different blocks cannot be isomorphic (by consideringthe action of the matrix which is the identity in blockAi and zero elsewhere). �

REMARK 2.2.6. If k is not algebraically closed then the proofs of Lemmas 2.2.2 and 2.2.3 nolonger hold. Instead one deduces that for a simple module S the spaceEndA(S,S) is a divisionring over k. (Adivision ring is a non-commutative version of a field.) There is then a version ofthe Artin-Wedderburn theorem, but where each Mn(k) is replaced by some Mn(Di) with Di somedivision ring containing k.

2.3. The Jacobson radical

Suppose thatA is not a semisimple algebra. One way to measure how far from semisimple itis would be to find an idealI in A such thatA/I is semisimple andI is minimal with this property.

DEFINITION 2.3.1. TheJacobson radical(or just radical) of an algebra A, denotedJ (A) (orjustJ ), is the set of elements a∈ A such that aS= 0 for all simple modules S. It is easy to verifythat this is an ideal in A.

DEFINITION 2.3.2. An ideal isnilpotentif there exists n such that In = 0. A maximal submod-ule in a module M is a module L⊂ M which is maximal by inclusion. Theannihilator Ann(M) ofa module M is the set of a∈ A such that aM= 0. This is easily seen to be a submodule of A.

When discussing the Jacobson radical, the following resultis useful.

LEMMA 2.3.3. Let A be a finite dimensional algebra. Then A has a largest nilpotent ideal.

PROOF. Consider the set of nilpotent ideals inA, and chose one,I , of maximal dimension.If J is another nilpotent ideal then the idealI + J is also nilpotent. (IfIn = 0 andJm = 0 then(I + J)m+n = 0, as the expansion of any expression(a+ b)n+m with a ∈ I andb ∈ J contains atleastn copies ofa or m copies ofb.) But then dim(I +J) = dimI and henceJ ⊆ I . �


THEOREM 2.3.4 (Jacobson).Let A be a finite dimensional algebra. The idealJ (A) is(a) the largest nilpotent ideal N in A.(b) the intersection D of all maximal submodules of A.(c) the smallest submodule R of A such that A/R is semisimple.

PROOF. (a) First suppose thatS is simple. ThenNS is a submodule ofS. If NS= S then byinductionNmS= S for all m≥ 1. But this contradicts the nilpotency ofN, and soN ⊆ J . For thereverse inclusion, consider a composition series forA

0 = An ⊂ An−1 ⊂ ·· · ⊂ A0 = A.

As Ai/Ai+1 is simple we havea(Ai/Ai+1) = 0 for all a∈ J . But this implies thatJ Ai ⊆ Ai+1,and hence

J n ⊂ J nA⊂ An = 0.

(b) Suppose thata ∈ J andM is a maximal submodule ofA. ThenA/M is simple and soa(A/M) = 0. In other words,a(1+M) = 0+M and soa∈ M. ThusJ ⊂ M for every maximalsubmodule ofA.

For the reverse inclusion, suppose thatJ 6⊆ D. Then there exists some simpleS ands∈ Swith Ds 6= 0. NowDs is a submodule ofS, and henceDs= S. Thus there existsd ∈ D with ds= s;sod−1∈ Ann(S) 6⊆ A, and there exists a maximal submoduleM of A with Ann(S)⊆ M. But thend ∈ D ⊆ M and 1−d ∈ M implies that 1∈ M, which contradictsM ⊂ A.

(c) (Sketch) First we claim thatD can be expressed as the intersection of finitely many maximalsubmodules ofA. To see this pick some submoduleL which is the intersection of finitely manymaximal submodules, such that dimL is minimal. ClearlyD ⊆ L. For any maximalM in A wemust have thatL = L∩M, and henceL ⊆ D.

ThusD = M1∩M2∩ . . .∩Mn for some maximal submodulesM1, . . .Mn. There is a homomor-phism

φ : A/D −→ A/M1⊕·· ·A/Mn

given byφ(a) = (a+M1, . . . ,a+Mn). It is easy to see this is injective. As eachMi is maximal wehave embeddedA/D into a semisimple module, and henceA/D is semisimple by Lemma 2.1.5.

Now suppose thatA/X is semisimple. It remains to show thatD ⊆ X. Write A/X as a directsum of simplesSi = Li/X. Then it is easy to check that the submoduleMi = ∑i 6= j Li is a maximalsubmodule ofA, and that the intersection of theMi equalsX. By definition this intersection containsD, as required. �

The Jacobson radical can be used to understand the structureof A-modules:

LEMMA 2.3.5 (Nakayama).If M is a finite dimensional A-module such thatJ M = M thenM = 0.

PROOF. (Sketch, for the caseA is finite dimensional) Suppose thatM 6= 0 and choose a minimalset of generatorsm1, . . . ,mt of M as anA-module. Nowmt ∈ M = J M implies that

mt =t

∑i=1

aimi

2.4. THE KRULL-SCHMIDT THEOREM 23

for someai ∈ J , and so

(1−at)mt =t−1

∑i=1

aimi .

Now at ∈ J implies thatat is nilpotent, and then it is easy to check that 1−at must be invert-ible. But this implies thatmt can be expressed in terms of the remainingmi , which contradictsminimality. �

We have the following generalisation of Nakayama’s Lemma.

PROPOSITION 2.3.6. If A is a finite dimensional algebra and M is a finite dimensional A-module thenJ M equals(a) the intersection D of all maximal submodules of M.(b) the smallest submodule R of M such that M/R is semisimple.

PROOF. (Sketch) Suppose thatMi is a maximal submodule ofM. ThenM/Mi is simple, andhence by Nakayama’s lemmaJ (M/Mi) = 0. ThereforeJ M ⊆ J Mi ⊆ Mi and soJ M ⊆ D.

By Theorem 2.3.4 the moduleM/J M is semisimple, as it is a module forA/J . Nowsuppose thatL is a submodule ofM such thatM/L is semisimple. LetM/L = M1/L⊕·· ·⊕Mt/Lwhere eachMi/L is simple. Then the modulesNj = ∑i 6= j Mi are maximal submodules ofM andLis the intersection of theNj . HenceJ M is a submodule ofL asJ M is a submodule ofD. TakingL = J M we see thatD is a submodule ofJ M which completes the proof. �

Motivated by the last result, we have

DEFINITION 2.3.7. Theradicalof a module M is defined to be the moduleJ M. Note thatwhen M= A this agrees with the earlier definition of the radical of an algebra. Theheador topofM, denotedhd(M) or top(M), is the module M/J M. By the last proposition the sequence

M ⊃ J M ⊃ J 2M ⊃ ·· · ⊃ J tM ⊃ J t+1M = 0

is such that each successive quotient is the largest semisimple quotient possible. This is called theLoewy seriesfor M, and t+1 is theLoewy lengthof M.

The head of a moduleM is the largest semisimple quotient ofM. It can be shown that thesubmodule ofM generated by all simple submodules is the largest semisimple submodule ofM;we call this thesocleof M, and denote it by soc(M).

2.4. The Krull-Schmidt theorem

Given a finite dimensionalA-moduleM, it is clear that we can decomposeM as a direct sum ofindecomposable modules. The Krull-Schmidt theorem says that this decomposition is essentiallyunique, and so it is enough to classify the indecomposable modules for an algebra.

THEOREM 2.4.1 (Krull-Schmidt).Let A be a finite dimensional algebra and M be a finitedimensional A-module. If

M = M1⊕M2⊕·· ·⊕Mn = N1⊕N2⊕·· ·⊕Nm

are two decompositions of M into indecomposables then n= m and there exists a permutationσof {1, . . .n} such that Ni ∼= Mσ(i).


PROOF. The idea is to proceed by induction onn, at each stage cancelling out summands whichare known to be isomorphic. The details are slightly technical, and so will be omitted here. Insteadwe will review below some of the ideas used in the proof. �

A key idea in the proof of the Krull-Schmidt theorem is the notion of a local algebra.

DEFINITION 2.4.2. An algebra A islocal if it has a unique maximal right (or left) ideal.

There are various characterisations of a local algebra.

LEMMA 2.4.3. Suppose that A is a finite dimensional algebra over an algebraically closedfield. Then the following are equivalent:(a) A is a local algebra.(b) The set of non-invertible elements of A form an ideal.(c) The only idempotents in A are0 and1.(d) The quotient A/J is isomorphic to k.

PROOF. This is not difficult, but is omitted as it requires a few preparatory results. �

REMARK 2.4.4. In fact (a) and (b) are equivalent for any algebra A. However,there existexamples of infinite dimensional algebras with only0 and1 as idempotents which are not local,for example k[x]. Also, if the field is not algebraically closed then A/J will only be a divisionring in general.

LEMMA 2.4.5 (Fitting).Let M be a finite dimensional A-module, andφ ∈ EndA(M). Then forlarge enough n we have

M = im(φn)⊕ker(φn).

In particular, if M is indecomposable then any non-invertible endomorphism of M must be nilpo-tent.

PROOF. Note thatφ i+1(M) ⊆ φ i(M) for all i. As M is finite dimensional there must exist ann such thatφn+t(M) = φn(M), for all t ≥ 1 and soφn is an isomorphism fromφn(M) to φ2n(M).Form∈ M let x be an element such thatφn(m) = φ2n(x). Now

m= φn(x)+(m−φn(x)) ∈ im(φn)+ker(φn)

and soM = im(φn)+ker(φn). If φn(m) ∈ im(φn)∩ker(φn) thenφ2n(m) = 0, and soφn(m) = 0.Thus the sum is direct, as required. �

Local algebras are useful as they allow us to detect indecomposable modules.

LEMMA 2.4.6. Let M be a finite dimensional A-module. Then M is indecomposable if andonly if EndA(M) is a local algebra.

PROOF. First suppose thatM = M1 ⊕M2, and for i = 1,2 let ei be the map fromM to Mwhich mapsm1 +m2 to mi. Thenei ∈ EndA(M) is non-invertible (as it has non-zero kernel). Bute1 +e2 = 1, which is invertible, which implies that EndA(M) is not local by Lemma 2.4.3.

Now suppose thatM is indecomposable. LetI be a maximal right ideal in EndA(M), andpick φ ∈ EndA(M)\I . By maximality we have EndA(M) = EndA(M)φ + I . Thus we can write1 = θφ + µ whereθ ∈ EndA(M) andµ ∈ I . Note that any element inI cannot be an isomorphism

2.5. EXERCISES 25

of M (as it would then be invertible), and hence by Fitting’s Lemma we have thatµn = 0 for somen >> 0. But then

(1+ µ + µ2 + . . .+ µn−1)θφ = (1+ µ + µ2 + . . .+ µn−1)(1−µ) = 1−µn = 1

and soφ is an isomorphism. But thenI consists precisely of the non invertible elements inEndA(M), and the result follows by Lemma 2.4.3. �

2.5. Exercises

(1) Let A = k[x] andM be the 2-dimensionalA module wherex acts via the matrix(

0 10 0

)

with respect to some basis ofM. Prove thatM is not a semisimple module.

(2) Prove the assertion in the proof of the Artin-Wedderburntheorem that

EndA(A,A) ∼= Aop.

(3) Thecentreof an algebraA, denotedZ(A), is the set ofz∈ A such thatza= azfor all a∈A.This is a subalgebra ofA. If k is algebraically closed andS is a simpleA-module showthat for allz∈ Z(A) there existsλ ∈ k such thatzm= λm for all m∈ S.

(4) Show thatk[x]/(xn) is a local algebra.

(5) Let G be a finite group of orderpn, andk be a field of characteristicp.(a) Prove that the idealI generated by the set

{1−g : g∈ Z(G)}

is nilpotent inkG.(b) Show thatI is the kernel of some map fromkG to k(G/Z(G)).(c) Deduce thatkG is local. You may use the fact that for all suchG we haveZ(G) 6= 1.

(6) Show thatk[x,y] is not a local algebra, but only contains the two idempotents0 and 1.This demonstrates the need for finite dimensionality in Lemma 2.4.3.

CHAPTER 3

Projective and injective modules

3.1. Projective and injective modules

DEFINITION 3.1.1. A short exact sequenceof A-modules is a sequence of the form

0−→ Lφ

−→ Mψ

−→ N → 0

such that the mapφ is injective, the mapψ is surjective, andimφ = kerψ. More generally, asequence

· · · −→ Lφ

−→ Mψ

−→ N −→ ·· ·

is exact atM if imφ = kerψ. If a sequence is exact at every module then it is calledexact. (Thusa short exact sequence is exact.)

Note that in a short exact sequence as above we have that

M/L ∼= N

by the isomorphism theorem, and dimM = dimL+dimN. When a sequence starts or ends in a 0it is common to assume that it is exact (as we will do in what follows).

LEMMA 3.1.2. Given a short exact sequence of A-modules

0−→ Lφ

−→ Mψ

−→ N → 0

the following are equivalent:(a) There exists a homomorphismθ : N −→ M such thatψθ = idN.(b) There exists a homomorphismτ : M −→ L such thatτφ = idL.(c) There is a module X with M= X⊕ker(ψ).

PROOF. (Sketch) We will show that (a) is equivalent to (c); that (b)is equivalent to (c) issimilar. First suppose thatθ as in (a) exists. Thenθ must be an injective map. LetX = im(θ),a submodule ofM isomorphic toN. It is easy to check thatX ∩ ker(ψ) = 0 and that dim(X ⊕ker(ψ)) = dimM by exactness atM. ThereforeM = X⊕ker(ψ).

Now suppose thatM = X⊕ker(ψ). Consider the restriction ofψ to X; it is clearly an isomor-phism and soθ can be taken to be an inverse toψ. �

DEFINITION 3.1.3.An A-module P isprojectiveif for all surjective A-module homomorphismsθ : M −→ N and for allφ : P−→ N there existsψ : P−→ M such that gψ = φ .

27

28 3. PROJECTIVE AND INJECTIVE MODULES

Thus a moduleP is projective if there always existsψ such that the following diagram com-mutes

P

φ��

ψ

~~M

θ // N // 0.

(Note that here we are using our convention about exactness for the bottom row in the diagram.)

There is a dual definition, obtained by reversing all the arrows and swapping surjective andinjective.

DEFINITION 3.1.4. An A-module I isinjective if for all injective A-module homomorphismsθ : N −→ M and for allφ : N −→ I there existsψ : M −→ I such thatψg = φ .

Thus a moduleI is injective if there always existsψ such that the following diagram commutes

0 // M

φ��

θ // N

ψ~~

I

EXAMPLE 3.1.5. For m≥ 1 the module Am is projective. To see this, denote the ith coordinatevector(0, . . . ,0,1,0. . . ,0) by vi , and suppose thatφ(vi) = ni ∈ N. As g is surjective, there existsmi ∈ M such that g(mi) = ni . Given a general element(a1, . . . ,am) ∈ Am define

ψ(a1, . . . ,am) =m

∑i=1

aimi .

It is easy to verify that this gives the desired A-module homomorphism.

We would like a means to recognise projective modulesP without having to consider all pos-sible surjections and morphisms fromP. The following lemma provides this, and shows that theabove example is typical.

LEMMA 3.1.6. For an algebra A the following are equivalent.(a) P is projective.(b) Wheneverθ : M −→ P is a surjection then M∼= P⊕ker(θ).(c) P is isomorphic to a direct summand of Am for some m.

PROOF. First suppose thatP is projective. We have by definition a commutative diagram

P

idP��

ψ

��M

θ // P // 0.

and by Lemma 3.1.2 this implies thatM ∼= P⊕ker(θ).

Now suppose that (b) holds. Given anyA-moduleM with generatorsmi , i ∈ I , there is asurjection from⊕i∈I A ontoM given by the map taking 1 in theith copy ofA to mi . TakingM = Pa projective we deduce that (c) holds.

3.1. PROJECTIVE AND INJECTIVE MODULES 29

Finally suppose thatAm∼= P⊕X for someP andX. Let π be the projection map fromAn ontoP, andι be the inclusion map fromP into Am. Given modulesM andN and a surjection fromM toN we have the commutative diagram

An

π��ψ1

��

P

φ��

ιOO

ψ1ι

~~M

θ // P // 0.

It is easy to check thatψ1ι gives the desired mapψ for P in the definition of a projective module.�

Suppose thatA is finite dimensional and

A = P(1)⊕·· ·⊕P(n) (1)

is a decomposition ofA into indecomposable direct summands. By the last result these summandsare indecomposable projective modules.

LEMMA 3.1.7. Suppose that A is a finite dimensional algebra. Let P be a projective A-modulewith submodule N, and suppose that every homomorphismφ : P−→ P maps N to N. Then there isa surjection fromEndA(P) ontoEndA(P/N) and if P is indecomposable then so is P/N.

PROOF. (Sketch) Givenφ : P−→ P let φ be the obvious map fromP/N to P/N. Check thisis well-defined; it is clearly a homomorphism. The mapφ −→ φ gives an algebra homomorphismfrom EndA(P) to EndA(P/N); givenψ ∈ EndA(P/N) use the projective property ofP to constructa mapφ so thatφ = ψ.

If P is indecomposable then EndA(P) is a local algebra by Lemma 2.4.6. Therefore there isa unique maximal right ideal in EndA(P), and hence a unique maximal ideal in EndA(P/N) (aswe have shown that this is a quotient of EndA(P)). Thus EndA(P/N) is local, and henceP/N isindecomposable. �

THEOREM 3.1.8. Let A be a finite dimensional algebra, and decompose A as in (1). SettingS(i) = P(i)/J P(i) we have(a) The module S(i) is simple, and every simple A-module is isomorphic to some S(i).(b) We have S(i) ∼= S( j) if and only if P(i) ∼= P( j).

PROOF. (Sketch) (a) The modulesS(i) is semisimple, so it is enough to check it is indecom-posable. Note thatP(i) andJ P(i) satisfy the assumptions in Lemma 3.1.7, and soS(i) is inde-composable.

Let S be a simple module and choosex 6= 0 in S. As 1x = x there is someP(i) such thatP(i)x 6= 0 (asAx 6= 0 andA is the direct sum of theP(i)). Define a homomorphism fromP(i) to S,soS is a simple quotient ofP(i). But J P(i) is the unique maximal submodule ofP(i), and so asS is simple we haveS∼= P(i)/J P(i).

(b) If P(i) ∼= P( j) via φ it is easy to see thatφ(J P(i)) ⊆ J P( j)). Henceφ induces ahomomorphism fromS(i) to Sj . As φ is invertible this has an inverse, and soS(i) ∼= S( j) bySchur’s Lemma.


If S(i) ∼= S( j) then use the projective property to construct a homomorphism ψ from P(i)to P( j). Show that the image of this map cannot be insideJ P( j), so asJ P( j) is a maximalsubmoduleψ must have image all ofP( j). By Lemma 3.1.6 we deduce thatP(i)∼= P( j)⊕ker(ψ),and so asP(i) is indecomposable we have ker(ψ) = 0. Thusψ is an isomorphism. �

By Krull-Schmidt, this implies that a finite dimensional algebraA has only finitely many iso-morphism classes of simple modules.

DEFINITION 3.1.9. Let M be a finite dimensional A-module. Aprojective coverfor M is aprojective module P such that

P/J P∼= M/J Mand there exists a surjectionπ : P−→ M.

LEMMA 3.1.10.Let A be a finite dimensional algebra. Every finite dimensional A-module hasa projective cover, which is unique up to isomorphism. In particular, suppose that

M/J M ∼= S(1)n1 ⊕S(2)n2 ⊕·· ·⊕S(t)nt .

ThenP = P(1)n1 ⊕P(2)n2 ⊕·· ·⊕P(t)nt

is a projective cover of M via the canonical surjection on each component.

PROOF. (Sketch) It is clear that the givenP satisfiesP/J P ∼= M/J M. Use the projectiveproperty to construct a homomorphismπ from P to M; it is easy to see that im(π)+J M = Mby the commutativity of the related diagram. But thenJ (M/ im(π)) = (J M + im(π))/ im(π) =M/ im(π) and so by Nakayama’s Lemma we haveM/N = 0. Thusπ is surjective as required.�

DEFINITION 3.1.11.A projective resolutionof a module M is an exact sequence

· · · −→ P3 −→ P2 −→ P1 −→ M −→ 0

such that all the Pi are projective.

By induction using Lemma 3.1.10 we deduce

PROPOSITION 3.1.12. If A is a finite dimensional algebra then every finite dimensional A-module has a projective resolution.

There is a similar theory for injective modules, but insteadof developing this separately wewill instead use dual modules to relate the two.

The injective analogue of a projective cover is called theinjective envelopeof M. An injectiveresolutionof M is an exact sequence

0−→ M −→ I1 −→ I2 −→ I3 −→ ·· ·

such that all theIi are injective.

THEOREM 3.1.13.Suppose that A is a finite dimensional algebra and M a finite dimensionalA-module.(a) M is simple if and only if M∗ is a simple Aop-module.(b) M is projective if and only if M∗ is an injective Aop-module.(c) M is injective if and only if M∗ is a projective Aop-module.(d) The injective envelope of M is I if and only if the projective cover of the Aop-module M∗ is I∗.

3.2. IDEMPOTENTS AND DIRECT SUM DECOMPOSITIONS 31

PROOF. This is a straightforward application of duality. �

Projective and injective modules play a crucial role in the study of the cohomology of rep-resentations. In a non-semisimple representation theory there are certain spaces associated toHomA(M,N) calledextension groupsExtiA(M,N). To introduce these properly, we would needto work with the category of modules, and introduce the notion of a derived functor. Unfortunatelythis is beyond the scope of the current course.

3.2. Idempotents and direct sum decompositions

Every algebra has at least two idempotents, 0 and 1. IfA is not local then there exists anotheridempotente∈A andeand 1−eare two non-zero orthogonal idempotents, giving a decompositionof A-modules

A = Ae⊕A(1−e).

If e is a central idempotent then so is 1−e, and the above decomposition becomes a direct sumof algebras. Conversely, ifA = M1⊕M2 as anA-module, then the corresponding decomposition1 = e1 + e2 is an orthogonal idempotent decomposition of 1. If the decomposition ofA is as adirect sum of algebras, then the corresponding idempotentsare central.

DEFINITION 3.2.1. We say that an algebra isconnectedor indecomposableif 0 and1 are theonly central idempotents in A.

Note that ifA is not connected, sayA= A1⊕A2, then anyA-moduleM decomposes as a directsumM1⊕M2 whereMi is anAi-module fori = 1,2. (This follows by decomposing 1∈ A andapplying it toM.) Thus we can reduce the study of the representations of an algebra to the casewhere the algebra is connected.

Suppose thatA is a finite dimensional algebra. By repeatedly decomposingA as anA-modulewe can write

A = P1⊕·· ·⊕Pn

where thePi are indecomposable left ideals inA. (The sum is finite asA is finite dimensional.)There is a corresponding decomposition of 1 as a sum of primitive orthogonal idempotents. Con-versely any such decomposition of 1 gives rise to a decomposition of A into indecomposable leftideals. Note that we can identify primitive idempotents by the following application of Lemma2.4.6

COROLLARY 3.2.2. Suppose that A is a finite dimensional algebra. Then an idempotent e∈ Ais primitive if and only if eAe is local.

DEFINITION 3.2.3.Suppose that A is a finite dimensional algebra with a completeset{e1, . . .en}of primitive orthogonal idempotents. Then A isbasicif Aei

∼= Aej implies that i= j.

Basic algebras have the following nice properties.

PROPOSITION3.2.4. Suppose that k is algebraically closed.(a) A finite dimensional k-algebra A is basic if and only if

A/J ∼= k×k×·· ·×k.

(b) Every simple module over a basic algebra is one dimensional.


PROOF. (Sketch) (a) Suppose thatA is basic and consider a complete set of primitive idempo-tentse1, . . . ,en for A. By Theorem 3.1.8 the modulesSi = (A/J )ei are simpleA/J -modules.Also, asA is basic these simples are non-isomorphic. Then Schur’s lemma implies that Hom-spaces between such simples are isomorphic to 0 ork, and one can define an injective homomor-phism

φ : A/J −→ EndA/J (S1⊕·· ·⊕Sn) ∼= k×·· ·×k.

By dimensions this is an isomorphism.

If A/J is basic then theSi above are all non-isomorphic (as the primitive idempotentsareeven central inA/J ), and the same argument as above implies thatA is basic.

(b) Any simpleA-module is also anA/J -module by Nakayama’s Lemma. But by part (a)this is isomorphic tok×·· ·×k, which implies the result. �

Given an arbitrary finite dimensional algebraA, we can associate a basic algebra to it in thefollowing manner.

DEFINITION 3.2.5. Suppose that A is finite dimensional and has a complete set{e1, . . .en} ofprimitive orthogonal idempotents. Pick idempotents ei1, . . .eit from this set such that Aeia

∼= Aeibimplies that a= b, and so that the collection is maximal with this property. Then define

eA =t

∑a=1

eia

and set Ab = eAAeA, thebasic algebra associated toA. (It is easy to see that this is indeed a basicalgebra, and is independent of the choice of idempotents.)

As we have not given a precise definition of a category, we shall state the next result withoutproof.

THEOREM 3.2.6. Suppose that A is a finite dimensional algebra. Then the category of finitedimensional A-modules is equivalent to the category of finite dimensional Ab-modules.

This means that to understand the representation theory of afinite dimensional algebra it isenough to consider representations of the corresponding basic algebra.

Let us now consider the special case of the path algebra of a quiver.

LEMMA 3.2.7. Let Q be a finite quiver. Then the sum

1 = ∑i∈Q0

εi

is a decomposition into a complete set of primitive orthogonal idempotents for kQ.

PROOF. All that remains to prove is that theεi are primitive, for which it is enough to showthatεikQεi is local. Suppose thate∈ εikQεi is an idempotent. Thene= λεi +w whereλ ∈ k andw is a sum of paths froma to a. But then

0 = e2−e= (λ 2−λ )εi +w2 +(2λ −1)w

implies thatw = 0 andλ = 0 or λ = 1. �

We can now characterise the connected path algebras of quivers.

3.2. IDEMPOTENTS AND DIRECT SUM DECOMPOSITIONS 33

LEMMA 3.2.8. Let Q be a finite quiver. Then kQ is connected if and only if Q is aconnectedquiver.

PROOF. (Sketch) It can be shown thatkQ is connected if and only if there does not exist apartitionQ0 = X∪Y of the set of vertices such that for allx∈ X andy∈Y we haveεxkQεy = 0 =εykQεx.

Clearly if Q is not connected then there exists a partitionQ0 = X∪Y so that there is no pathfrom a vertex inX to a vertex inY (or vice versa). Thus in this caseεxkQεy = 0 = εykQεx, andkQis not connected.

If kQ is not connected butQ is connected, there exists a partitionQ0 = X ∪Y as above, andelementsx∈ X andy∈Y with an arrowα : x−→ y in Q. But thenα ∈ εxkQεy which contradictsour assumption onkQ. �

THEOREM 3.2.9. Let Q be a finite, connected, acyclic quiver. Then kQ is a basicconnectedalgebra with radical given by the arrow ideal.

PROOF. (Sketch) By Lemma 3.2.7 we have a decomposition

kQ/RQ =⊕

a,b∈Qo

εa(kQ/RQ)εb.

As Rcontains all non-trivial paths each summand is non-zero only whena = b, in which case it isisomorphic tok. Thus we will be done by Proposition 3.2.4 and Lemma 3.2.8 if we can show thatRQ = J . But asQ is acyclic there exists a maximal path length inQ. HenceRn

Q = 0 for n >> 0,and soR⊆ J by Theorem 2.3.4. It is not too hard to show that in fact any nilpotent idealI suchthatA/I is a product of copies ofk must equalJ (A). �

Now we consider the case of bound quiver algebras.

PROPOSITION3.2.10.Let Q be a finite quiver with admissible ideal I in kQ. Then(a) The set

{ei = εi + I : i ∈ Q0}

is a complete set of primitive orthogonal idempotents in kQ/I.(b) The algebra kQ/I is connected if and only if Q is a connected quiver.(c) The algebra kQ/I is basic, with radical RQ/I.

PROOF. (Sketch) The proofs of (a) and (b) are similar to those forkQ. Part (c) is almostimmediate from the corresponding result forkQ. �

We have seen that the representation theory of finite dimensional algebras reduces to the studyof connected basic algebras. The last result says that boundquiver algebras for connected quiversare such algebras. We conclude this section with

THEOREM 3.2.11. Let A be a basic, connected, finite dimensional k-algebra over an alge-braically closed field. Then there is a connected quiver Q associated to A and an admissible idealI in kQ such that

A∼= kQ/I .


Thus over algebraically closed fields the study of finite dimensional algebras can be reducedto the study of bound quiver algebras.

PROOF. (Sketch) Rather than give a detailed proof, we will sketch how to construct the quiverassociated toA.

Let {e1, . . . ,en} be a complete set of primitive orthogonal idempotents inA. ThenQ has vertexset{1, . . . ,n}. Given 1≤ i, j ≤ n, the number of arrows fromi to j equals the dimension of thevector spaceei(J /J 2)ej .

One then checks that this quiver is independent of the choiceof idempotents and is connected.Then one defines a homomorphism fromkQ to A, and show that this is (i) surjective, and (ii) haskernel which is an admissible ideal inkQ. The result then follows from the first isomorphismtheorem. �

For a discussion of what happens whenk is not algebraically closed see [Ben91, Section 4.1].

3.3. Simple and projective modules for bound quiver algebras

In general it is hard to determine explicitly the simple modules for an algebra. Indeed, some ofthe most important open questions in representation theoryrelate to determining simple modules.However, in the case of a bound quiver algebra the simple modules can be written down entirelyexplicitly.

We will also see that the indecomposable projectives can also be easily constructed. The sameis true for indecomposable injectives, but we will not consider these in detail here.

Let kQ/I be a bound quiver algebra. We know by Proposition 3.2.10 and Theorem 3.1.8 thatthe simple modules are parameterised by the vertices ofQ, and are all one dimensional (as thealgebras are basic). Given this, the following result is almost clear.

PROPOSITION3.3.1. Let kQ/I be a bound quiver algebra. For a∈ Q0, let S(a) be the repre-sentation of Q such that

S(a)b =

{

k a= b0 a 6= b

and for all arrowsα the mapφα = 0. Then

{S(a) : a∈ Q0}

is a complete set of non-isomorphic simple modules for kQ/I.

PROOF. The only thing that remains to check is that the various simples are not isomorphic,but this is straightforward. �

The description of the projective modulesP(a) is slightly more complicated.

PROPOSITION3.3.2.Let kQ/I be a bound quiver algebra, and P(a) the projective correspond-ing to εa. Then P(a) can be realised in the following manner.For b∈ Q0 let P(a)b be the k-vector space with basis the set of all elements of theform w+ I wherew is a path from a to b. Given an arrowα : b−→ c, the mapφα : P(a)b −→ P(a)c is given by leftmultiplication byα + I.

3.4. EXERCISES 35

PROOF. This is a straightforward consequence of the explicit identification of quiver represen-tations withkQ/I -modules given earlier. �

The description of injective modules for a bound quiver algebra is similar, using Theorem3.1.13.

3.4. Exercises

(1) Let A be an algebra containingean idempotent, and letM be a leftA-module.

(a) Show thateAeis an algebra, andeM is a lefteAe-module.(b) Show that HomA(Ae,M) is a lefteAe-module where the action ofa∈ eAeon a mor-

phismφ ∈ HomA(Ae,M) is given by

aφ(−) = φ(a−).

(c) Show that there is an isomorphism of lefteAe-moduleseM∼= HomA(Ae,M).

(2) A first course on representation theory often considers only representations of finite groupsoverC. In this case much can be learnt from the study ofcharacters. Given a finite groupG, a representationV of dimensionn can be described by giving a group homomorphismρ : G−→ End(V). By choosing a basis ofV we obtain a map fromG into GLn(C). Wedefine thecharacterof V to be the mapχV : G −→ C given byχV(g) = Tr(ρ(g)), thetrace of the matrixρ(g). This looks like it throws away a lot of information; howeverthisexercise will show that it is still a powerful tool.

(a) Show that the character ofV does not depend on the chosen basis.(b) Show that ifV andW are two isomorphic representations ofG thenχV = χW. Hint:

Let φ be an isomorphism fromV to W. Pick a basis forV and consider the corre-sponding basis ofW obtained viaφ . Now compare the actions ofg ∈ G on eachbasis.

(c) Suppose thatV andW are two simple non-isomorphic representations ofG. By theArtin-Wedderburn TheoremCG is isomorphic to a direct sum of matrix algebras,and there is a corresponding idempotent decomposition 1= ∑ei . Show that thereexistsi such thatei acts as the identity onV and as 0 onW. (You may wish to recallCorollary 2.2.5.)

(d) Deduce from the above that ifV andW are two simple representations ofG thenV ∼= W if and only if χV = χW.

(3) Determine the indecomposable projectives and their radicals for the following boundquivers.(a)

•1α1 // •2

α2 // •3α3 // •4

(b) The same quiver as in (a) but with the relationα3α2 = 0.


(c)

•1α1 //

α3��

•2

α2��

βuu

•3α4 // •4

with the relationβ 2 = 0.

(d) The same quiver as in (c) with the relations

β 2 = 0 α2α1 = 2α4α3.

(4) Suppose thatQ is a finite acyclic quiver. Show that all the linear maps in an indecompos-able projective representation ofQ must be injective.

CHAPTER 4

Representation type and Gabriel’s theorem

4.1. Representation type

We have seen that a finite dimensional algebra has only finitely many isomorphism classes ofsimple modules. It is natural to ask if the same is also true ofindecomposables. However, this isnot generally the case.

DEFINITION 4.1.1. An algebra hasfinite representation typeif there are only finitely manyisomorphism classes of finite dimensional indecomposable modules. Otherwise the algebra hasinfinite representation type.

By Krull-Schmidt it is clear that for a representation finitealgebra we have complete knowl-edge of its representation theory once we have constructed acomplete (finite) set of indecompos-able modules (although that is not necessarily easy!). Semisimple algebras are clearly of finiterepresentation type.

EXAMPLE 4.1.2. Suppose that k is algebraically closed. Then the algebra A= k[x]/(xn) hasfinite representation type. Any A-module M is a vector space together with a linear mapφ : M −→M such thatφn = 0. Considerφ as a matrix with respect to some basis. Then the correspondingJordan canonical form forφ is a block diagonal matrix where each block is a t× t matrix of theform

Jt(0) =

0 1 0 0 · · · 00 0 1 0 · · · 0...

. . ....

.... . .

...0 · · · 0 0 0 10 · · · 0 0 0 0

.

for some t≤ n (as no larger block satisfiesφn = 0). But if M is indecomposable then there is onlyone such block. Therefore there are precisely n isomorphismclasses of indecomposable modules:one each of dimension1,2, . . . ,n.

EXAMPLE 4.1.3. The algebra A= k[x,y]/(x2,y2) has infinite representation type. Let M= k2n

for some n≥ 1 and choseλ ∈ k. Then let x and y act respectively by

X =

(

0 In0 0

)

Y =

(

0 Jn(λ )0 0

)

where In is the n×n identity matrix, and Jn(λ ) = Jn(0)+λ In. It is easy to verify that X2 = Y2 = 0and XY=YX, so this defines an A-module. One can also check it is indecomposable. Clearly thesemodules are non-isomorphic for different values of n (and infact they are also non-isomorphic fordifferent values ofλ ).

37

38 4. REPRESENTATION TYPE AND GABRIEL’S THEOREM

EXAMPLE 4.1.4. Let Q be a quiver such that there exist two vertices i and j suchthat thereare (at least) two arrowsα,β : i −→ j. Then as in the last example we find infinitely many non-isomorphic indecomposables by setting Mi = M j = kn and representingα by the matrix In andβby the matrix Jn(λ ).

LEMMA 4.1.5. If A has finite representation type and I is an ideal in A then A/I has finiterepresentation type.

PROOF. Suppose thatM is anA/I -module. Then we can define anA-module structure onMby settingam= (a+ I)m for all a∈ A andm∈ M. FurtherM is indecomposable forA if and onlyif it is for A/I , and two modules are isomorphic asA/I -modules if and only they are isomorphic asA-modules. �

If an algebra is not representation finite, is there any hope to classify the finite dimensionalindecomposable modules?

DEFINITION 4.1.6. Suppose that k is an infinite field. An algebra over k hastame represen-tation typeif it is of infinite type and for all n∈ N, all but finitely many isomorphism classes ofn-dimensional indecomposables occur in a finite number of one-parameter families.

Thus there is some hope that one can classify all indecomposable representations for algebrasof tame representation type.

REMARK 4.1.7. (a) We could make precise what we mean by a one-parameter family of rep-resentations; for our purposes however the above definitionwill be good enough. The idea of aone-parameter family is illustrated in the variation withλ of the representations defined in Exam-ple 4.1.3.(b) Some authors define tame representation type to include finite representation type.

DEFINITION 4.1.8. A k-algebra A haswild representation typeif for all finite dimensionalk-algebras B, the representation theory of B can be embeddedinto that of A.

REMARK 4.1.9.Again, we could give a more precise definition of what we mean by embeddingone representation theory inside another, but this would require the language of categories.

This means that understanding all indecomposable representations ofA implies an understand-ing of all representations ofeveryfinite dimensional algebra, which should sound like a hopelesstask. That it is can be seem from

REMARK 4.1.10. It follows from an alternative definition of wild representation type that therepresentation theory of k〈x,y〉 can be embedded into that of any wild algebra. But the wordproblem for finitely presented groups can be embedded into the representation theory of k〈x,y〉,and this problem has been proved to beundecidable.

The following fundamental theorem is due to Drozd.

THEOREM 4.1.11 (Trichotomy theorem).Over an algebraically closed field, every finite di-mensional algebra is either of finite, tame, or wild representation type.

PROOF. A proof of this theorem is beyond the scope of this course. �

4.1. REPRESENTATION TYPE 39

In general we do not have a complete classification of algebras of finite (or tame) representationtype. However in the special case of a quiver algebra or a group algebra we can give such aclassification. We will conclude this section by considering the group case. As one would expectfrom Maschke’s theorem, this now depends on the field as well as the group. We begin with aspecial case.

PROPOSITION4.1.12.Let G be a finite group of order pn, and k be a field of characteristic p.Then kG has finite representation type if and only if G is cyclic.

PROOF. (Sketch) First suppose thatG is cyclic. Then by Example 4.1.2 it is enough to showthatkG∼= k[x]/(xpn

). Letgbe a generator forG, and define a mapφ : k[x]−→ kGby f 7−→ f (1−g).

We claim that this is a surjective algebra homomorphism, with kernel containing(xpn). From

this it follows by comparing dimensions and Lemma 1.2.7 thatφ induces the desired algebraisomorphism. To see the claim, note that(1−g)pi

= 1−gpiin characteristicp, as all other binomial

coefficients vanish, and henceφ(xpn) = 0. Then verify that 1,(1−g),(1−g)2, . . . ,(1−g)pn−1 form

a basis forkG.

For the reverse implication, basic group theory implies that there existsN⊳G such thatG/N∼=Cp×Cp. It is then enough by Lemma 4.1.5 to show thatk(G/N) has infinite representation type.By a similar argument to the preceding paragraph, one can show that

k(G/N) ∼= k[x,y]/(xp,yp).

As (xp,yp) ⊆ (x2,y2), it is enough to show thatk[x,y]/(x2,y2) has infinite type. But this was donein Example 4.1.3. �

Using this it is possible to prove

THEOREM 4.1.13 (Higman).Let G be a finite group and k a field. Then kG has finite repre-sentation type if and only if either(a) k has characteristic zero, or(b) k has characteristic p> 0 and G has a cyclic Sylow p-subgroup.

PROOF. (Sketch) Ifk has characteristic zero thenkG is semisimple by Maschke’s theorem,and we are done. Ifk has positive characteristic then we would like to argue thatkG of finite typeif and only if kH is of finite type whereH is a Sylowp-subgroup ofG, as then we are done byProposition 4.1.12.

As H is a Sylowp-subgroup ofG the index ofH in G is coprime top, and so is non-zero ink.The reduction to the case ofkH now proceeds by a Maschke-type averaging argument. �

The tame cases can also be classified.

THEOREM 4.1.14. Let G be a finite group, and k be an infinite field of characteristic p > 0.Then kG has tame representation type if and only if p= 2 and the Sylow2-subgroups of G aredihedral, semidihedral, or generalised quaternion.


4.2. Representation type of quiver algebras

In the special case of a quiver algebra we have a complete classification of those of finite andof tame representation types. We will begin by consider the finite type case, for which we willneed to introduce certain Dynkin diagrams. These are illustrated in Figure 4.1.

......

......

6

7

8

1 2 3 4 nA

n:

Dn

:

1

3

2 4 5 n

7

6

6

1 2 3 4 5E7 :

1 2 3 4 5E8 :

1 2 3 4 5E6 :

FIGURE 4.1. The Dynkin diagrams of typesAn, Dn, E6, E7, andE8

THEOREM 4.2.1 (Gabriel).Suppose that Q is a finite quiver. Then kQ has finite representationtype if and only ifQ is a disjoint union of Dynkin diagrams of types A, D, or E as inFigure 4.1.

If you know any of the theory of Lie algebras then you may recognise Dynkin diagrams asbeing associated with a root system. (This explains the strange labelling scheme: there are alsoroot systems of typesBn andCn, as well asF4 andG2.)

There is a similar classification of tame quiver algebras, this time in terms of certainextendedDynkin diagrams(also known asEuclidean diagrams).

THEOREM 4.2.2. Suppose that Q is a finite quiver and k an infinite field. Then kQ has tamerepresentation type if and only ifQ is a disjoint union of extended Dynkin diagrams as in Figure4.2 possibly together with Dynkin diagrams of types A, D, E asin Figure 4.1.

In the next two sections we will introduce some of the main ideas used in the proof of Gabriel’stheorem. First we will introduce some combinatorics associated to representations which for sim-ples and projectives only depends on the underlying graph. This provides the link with the languageof Lie theory (although a knowledge of this is not necessary here). In the final section of this chap-ter we will outline how this combinatorics, together with reflection functors, can be used to proveone implication of Gabriel’s theorem.

4.3. DIMENSION VECTORS AND CARTAN MATRICES 41

......

......

6

7

8

An

:

Dn

:

1 2 3 4 5E6 :

1 2 3 4 n

1

2

3

4

n

654321E7 :

761 2 3 4 5E8 :

FIGURE 4.2. The extended Dynkin diagrams of typesAn, Dn, E6, E7, andE8

4.3. Dimension vectors and Cartan matrices

In this section we will assume for convenience that the vertex setQ0 of a finite quiverQ hasbeen identified with{1, . . . ,n} for somen.

DEFINITION 4.3.1. Suppose that M= (Mi,φi) is a representation of a finite quiver Q withvertices1, . . .n. Then thedimension vectorof M is the n-tuple

dimM = (dimM1, . . . ,dimMn).

EXAMPLE 4.3.2. (a) The dimension vector of the representation considered in Example 1.4.3is (1,2,2,3,2).

(b) Clearly the simple representations of Q have dimension vectors with1 in the ith position(for some i) and0 elsewhere. We will denote this vector by e(i).

(c) By Proposition 3.3.2 we have that p(i) = dimP(i) is the vector whose jth coordinate is thenumber of paths from i to j.

We can now define a matrix related tokQ which will play an important role in what follows.

DEFINITION 4.3.3. TheCartan matrixC of kQ is the n× n matrix whose ith column is thevector p(i)T .


EXAMPLE 4.3.4. Let Q be the quiver

•1α // •2

β// •3

γ// •4 .

This has Cartan matrix

1 0 0 01 1 0 01 1 1 01 1 1 1

.

LEMMA 4.3.5. For all i ∈ Q0 we have

e(i) = p(i)−n

∑i=1

a(i, j)p( j) (2)

where a(i, j) is the number of arrows from i to j.

PROOF. (Sketch) LetA = kQ, and setSi = Aei/J ei . Then we have thate(i) = dim(Si) andp(i) = dim(Aei) and so

e(i) = dim(Aei)−dim(J ei).

Thus it is enough to show that

dim(J ei) =n

∑i=1

a(i, j)p( j).

Now J ei is the span of all paths of positive length starting ati, which equals the direct sum of allAα whereα is an arrow starting ati. It is easy to see thatAα ∼= Aε j whereα : i −→ j via the mapxα 7−→ xε j . �

COROLLARY 4.3.6. The Cartan matrix of Q is invertible overZ.

PROOF. Transposing the vectors in (2) we obtain

e(i)T = p(i)T −n

∑i=1

a(i, j)p( j)T .

The left-hand side is the columns of the identity matrix, while the right-hand side involves thecolumns ofC. ThusC has a left inverseI +(−a(i, j)). �

EXAMPLE 4.3.7. Returning to the quiver in Example 4.3.4 we see that

C−1 =

1 0 0 0−1 1 0 0

0 −1 1 00 0 −1 1

.

We can now use the Cartan matrix to define various forms onZn. We will write C−T for(C−1)T .

DEFINITION 4.3.8. We define theEuler characteristic, a (not in general symmetric) bilinearform onZ

n by〈x,y〉 = xC−TyT

4.4. REFLECTION FUNCTORS 43

and an associated symmetric form by

(x,y) = 〈x,y〉+ 〈y,x〉.

It is an elementary exercise to show that

LEMMA 4.3.9. For all i and j in Q0 we have

〈p(i),e( j)〉= δi j .

DEFINITION 4.3.10.For i ∈ Q0 define a map si : Zn −→ Zn by

si(x) = x− (x,e(i))e(i).

This is a linear map and it is easy to verify that s2i = id. We define W to be the subgroup ofGLn(Z)

generated by the si . We say thatx ∈ Zn is positiveif xi ≥ 0 for all i, with strict inequality for atleast one i, and writex > 0. Then the set ofpositive rootsfor Q is the set

{w(e(i)) : w(e(i)) > 0, w∈W, 1≤ i ≤ n}.

Root systems arise in a variety of places, such as Lie theory,and are well understood. Thefollowing fact is not hard to prove.

LEMMA 4.3.11.Suppose thatQ is of type A, D, or E. Then the set

{w(e(i)) : w∈W, 1≤ i ≤ n}

(and hence the set of positive roots) is finite.

The relevance of the above combinatorial framework to representation theory is the followingresult.

THEOREM 4.3.12 (Gabriel).Suppose that Q is a finite quiver such thatQ is of type A, D orE. Then the map V7−→ dimV gives a bijection between isomorphism classes of finite dimensionalindecomposable representations and the positive roots of Q.

(Combining this with Lemma 4.3.11 proves that ADE type quivers have finite representationtype.)

One way to prove Theorem 4.3.12 is using reflection functors.

4.4. Reflection functors

DEFINITION 4.4.1. Let Q be a finite quiver, and suppose that i is a vertex such thatthere areno arrows starting from i. Then we say that i is asink in Q. Similarly, if there are no arrows endingat i then we say that i is asourcein Q.

Suppose thati is a sink (or source) ofQ. We wish to define a new quiversiQ and a functorfrom kQ-modules toksiQ-modules. (This just means that it should mapkQ-modules toksiQ-modules and also map morphisms betweenkQ-modules to corresponding morphisms forksiQ in acompatible manner.) We begin withsiQ.

DEFINITION 4.4.2. Suppose that i is a sink (or source) of Q. Then siQ is the quiver obtainedby reversing the direction of all arrows ending at i.


Now suppose thatM = (Mi,φi) is a representation ofQ. We next wish to define a representationof siQ when i is a sink. Suppose for concreteness that the arrows enteringi are labelledα j withα j : i j −→ i for 1≤ j ≤ t.

DEFINITION 4.4.3. Let C+i (M) be the siQ-representation with C+(M) j = M j for all j 6= i. The

space C+i (M)i is defined by the exact sequence

0−→C+i (M)i

θ−→

t⊕

j=1

Mi j

ψ−→ Mi (3)

whereψ = ∑tj=1 φi j . The linear maps in C+i (M) are unchanged if the arrow has not been reversed,

and areθ followed by projection onto the relevant summand if the arrow has been reversed. Givena morphismφ between two representations of Q a corresponding morphism C+

i (φ) can be definedbetween siQ-modules, which makes C+

i into a functor. We call this areflection functor.

As the notation suggests, there is a relation between reflection functors and the combinatoricsof the preceding section. This follows from

PROPOSITION 4.4.4. Suppose that M is a finite dimensional indecomposable representationof a finite quiver Q. Then C+i (M) is 0 if M is a simple representation, and is indecomposableotherwise. In the latter case we have that

dimC+i (M) = si(dimM).

PROOF. (Sketch) It is clear thatC+i (M) = 0 if M is simple. Next one shows: (i) thatM is

indecomposable only ifM is simple or the mapψ in (3) is surjective, and (ii) that ifN = C+i (M)

then there is a homomorphismEndkQ(M) −→ EndksiQ(N)

which is surjective if (3) is surjective.

Now supposeM is indecomposable but not simple. Then EndkQ(M) is local by Lemma 2.4.6and we have a surjection onto EndksiQ(N). Arguing as in the proof of 3.1.7 we see that this latteralgebra is also local, and soN is indecomposable.

The dimension claim follows from elementary linear algebra, together with a comparison withthe corresponding combinatorics for dimension vectors. �

Now suppose thati is a source inQ. There is a similar definition of a reflection functorC−i in

this case were we reverse the direction of all the arrows in (3). Again one can show thatC−i takes

simple representations to 0 and non-simple indecomposableto indecomposables as in Proposition4.4.4. From the definitions it is easy to verify that

C−i C+

i (M) ∼=

{

M M 6∼= Si0 M ∼= Si

and similarly

C+i C−

i (M) ∼=

{

M M 6∼= Si0 M ∼= Si .

From this follows

4.4. REFLECTION FUNCTORS 45

COROLLARY 4.4.5. Suppose that i is a sink in Q. Then there is a bijection betweennon-simplefinite dimensional indecomposable kQ-modules and non-simple indecomposable ksiQ-modulesgiven by C+i . Hence kQ and ksiQ have the same representation type.

Any finite acyclic quiver has a sink and a source. Thus we can label the vertices ofQ startingwith the sinks, then taking the sinks in the quiver without these vertices, and so on. Thus we mayassume that if there is an edge fromi to j theni < j. We will call such a labelling anadmissiblelabelling.

DEFINITION 4.4.6. Suppose that Q has an admissible labelling. Then the functor

C+ = C+n C+

n−1 . . .C+1

is defined. We call this theCoxeter functorwith respect to this ordering. Note that every arrowin Q is reversed precisely twice in the construction of C+, and so C+ takes representations of Qto representations of Q. Similarly there is a functor C− = C−

1 . . .C−n . There are corresponding

elements s+ and s− in W.

Using the finiteness of the set of positive roots from Lemma 4.3.11 it is now possible to prove

LEMMA 4.4.7. If y∈Zn satisfies s+y = y theny = 0. Also, ifx∈Zn with x > 0 then(s+)nx = 0for n >> 0.

Now we can sketch the proof of Theorem 4.3.12.

PROOF. (Sketch) First suppose thatQ is of typeADE, and thatM is a finite dimensional in-decomposable representation ofQ. Then forn >> 0 we have(C+)nM = 0. This follows fromLemma 4.4.7 as dim(C+)nM ≥ 0 for all n, but equals(s+)ndimM by Proposition 4.4.4.

Thus there existsn such thatX = (C+)nM 6= 0 but (C+)X = 0. Therefore there is anisuch thatC+

i−1 . . .C+1 (X) 6= 0 but C+

i C+i−1 . . .C+

1 (X) = 0. By Proposition 4.4.4 this implies thatC+

i . . .C+1 (X) ∼= Si (for the relevant quiver). We can reverse our steps and reconstructM from Si

usingC−j functors, which also gives the dimension vector forM in terms of the action ofW on

e(i). It is easy to see that this gives the desired bijection between dimension vectors and finitedimensional indecomposable modules. �

This gives one half of Gabriel’s Theorem 4.2.1. To prove thatall other quivers have infiniterepresentation type, one proceeds case by case. Show that various simple quivers have infinite typeby hand, such as a quiver with multiple arrows (see Example 4.1.4), or a quiver with four arrowsfrom distinct vertices meeting at a single vertex. Then showthat every quiver contains one of theseexamples as a subquiver (and hence is of infinite type) exceptthe quivers of type ADE. Finally byusing reflection functors we see that the representation type depends only on the underlying graph.

To conclude, a word or two about infinite dimensional representations. As one might expectthese are considerably more complicated. Here are two general theorems for the finite and infinitetype cases.

THEOREM 4.4.8 (Auslander).If A is a finite dimensional algebra of finite representation typethen every indecomposable A-module is finite dimensional, and every module is a direct sum ofindecomposables.


THEOREM 4.4.9 (Roiter).If A is a finite dimensional algebra of infinite representation typethen there are indecomposable A-modules with arbitrarily many composition factors.

4.5. Exercises

(1) Let Q be the quiver

•1α // •2 •3

βoo .

(a) Show that this has six isomorphism classes of indecomposable modules with dimen-sion vectors(1,0,0), (0,1,0), (0,0,1), (1,1,0), (0,1,1) and(1,1,1).

(b) Determine the Cartan matrix forQ.(c) Verify that the dimension vectors of projective and simple representations are orthog-

onal with respect to the Euler characteristic.(d) Determine the groupW as a subgroup of GL3(Z), and hence verify that the dimension

vectors of indecomposable modules are in bijection with thepositive roots ofQ.(e) Consider the reflection functor forQ corresponding to the unique sink inQ. Deter-

mine the effect of this functor on each of the indecomposablerepresentations ofQ,verifying that in each case the new representation is indecomposable.

(2) Consider the 3-Kronecker quiver Qgiven by

•1

α

""β//

γ<<•2 .

For λ ,µ ∈ k let M(λ ,µ) be the representation ofQ such thatM1 = k2, M2 = k, and themaps corresponding toα, β and γ are given by the matrices(1,0), (0,1), and(λ ,µ)respectively. Show that the representationsM(λ ,µ) are indecomposable and pairwisenon-isomorphic (and hence that this quiver has a two-parameter family of indecompos-ables).

These are not the only indecomposables (this quiver has wildrepresentation type!). In[Bar06, Proposition 2.1] it is shown that classifying the indecomposables of this quiverwould allow one to classify the indecomposables foranyquiver.

(3) Investigate what happens if you apply the theory of reflection functors to the 3-Kroneckerquiver and its representationsM(λ ,µ) described above.

CHAPTER 5

Further directions

In this Chapter we will briefly review some of the many ways in which the material in thiscourse can be extended. Given the time available we can only sketch an indication of the kind oftopics that can be covered: more detailed surveys can be found in the references.

5.1. Ring theory

Much of the classical material developed in Chapters 1-3 canalso be considered when the fieldk is replaced by a (commutative) ring. However this can introduce considerable complications —particularly when we consider representations over the integers. Good basic introductions can befound in [Mat86] and the (194 page!) introduction to [CR81]. The latter also gives an extensiveexposition of the integral representation theory of finite groups. A shorter discussion more in thespirit of the later part of these notes can be found in [Ben91].

5.2. Almost split sequences and the geometry of representations

We have only begun the study of representations of finite dimensional algebras. There areseveral important ideas which we have not had time to touch on, and we will give a brief sketch ofa few of them in this section.

Consider a short exact sequence ofA-modules

0→ Lφ→ M

ψ→ N → 0.

If M is the direct sum ofL andN then we call the sequencesplit. Recall from Lemma 3.1.2 thatthis is equvalent to the existence of a left inverse toφ and to the existence of a right inverse toψ.We call a morphism with a left inverse asection, and with a right inverse aretraction.

Clearly if our sequence is split we understandM completely if we understandL andN. How-ever, we would like to be able to deal with non-split sequences. Almost split sequences turn out toplay an important role.

We say that a homomorphismφ : L → M is left minimalif every elementsθ ∈ EndA(M) withθφ = φ is an automorphism. (There is a similar definition forright minimal.) The mapφ as aboveis calledleft almost splitif φ is not a section, and for every morphismτ : L→U that is not a sectionthere existsτ ′ : M →U such thatτ ′φ = τ. This definition is similar to that for an injective module;the corresponding ‘projective’ version is calledright almost split.

Now we can give the main definition. A sequence

0→ Lφ→ M

ψ→ N → 0

47

48 5. FURTHER DIRECTIONS

is almost splitif φ is left minimal and left almost split, andψ is right minimal and right almostsplit. It is clear that an almost split sequence is not split.However, it is not immediately clear thatthere exist any such sequences.

First one shows that ifφ : L → M is left minimal and left almost split, thenM is unique up toisomorphism. Ifφ is merely left almost split thenL must be an indecomposable. (And of coursethere are similar righthand versions of these results.) Thus if there is an almost split sequence asabove thenL andN must be indecomposable, andM is uniquely determined. Further,L cannot beinjective andN cannot be projective.

The Auslander-Reiten translateis an explicit functor which takes anA-moduleM to an A-moduleτM. (The precise definition is a little too involved for the timeavailable to us.) Using this,Auslander and Reiten were able to prove

THEOREM 5.2.1 (Auslander-Reiten).If M is indecomposable and not projective then there isexists an almost split sequence

0→ τM → E → M → 0.

There is a similar result for indecomposable noninjectivesusing the inverse translateτ−1.

Auslander and Reiten also introduced theAuslander-Reiten quiverassociated to a finite dimen-sional algebraA. This is a quiver whose vertices are the isomorphism classesof indecomposablerepresentations ofA, and whose arrows correspond to bases for the spaces of certain irreduciblemorphismsbetween indecomposables. Studying this, together with theeffect of the Auslander-Reiten translate upon it, is an important aspect of the modern theory.

For example, using this one can prove the following conjectures of Brauer and Thrall:

THEOREM 5.2.2. If A is not representation finite then A has indecomposable modules of arbi-trarily large dimension.

THEOREM 5.2.3. If k is algebraically closed and A is not representation finite then there areinfinitely many positive integers n such that there are infinitely many non-isomorphic n-dimensionalindecomposable A-modules.

The theory of almost split sequences and AR-quivers is developed in [ARS94] and [ASS06].

Another direction of study is inspired by the reflection functors used in the proof of Gabriel’sTheorem. This leads to a general area oftilting theory, which tries to replace the algebraA be-ing studied by a simpler algebra which is closely related. Again, an extensive theory has beendeveloped — see for example [ASS06].

Finally in this section, we should note that there is an important approach to representationsof finite dimensional algebras which we have completely ignored in these notes, which relies ongeometric techniques.

If we fix a dimension vectorα, then the space of representations of a given quiver with thatdimension forms an algebraic variety. Thus we may use the methods of algebraic geometry tostudy this variety. This is a very powerful technique, but does require more geometry than wehave time to introduce in these notes. For an indication of how the results in this course (such asGabriel’s theorem) can be approached in this manner, see [Bru03]. There is also a more generalsurvey focussing on the geometric aspects in [CB93].

5.4. REPRESENTATIONS OF OTHER ALGEBRAIC OBJECTS 49

5.3. Local representation theory

We have not looked in detail at the special case of group algebras of finite groups in character-istic p > 0. We did see in the discussion of representation type that the Sylowp-subgroups play akey role. There is a general approach to studying group representations which proceeds by relatingthe representation theory of a groupG to that of certain normalisers ofp-groups inG.

Given H ≤ G, we can generalise the notion of projective modules forG to relatively H-projectivemodules. One way to define this is to copy the definition we havegiven, but add therequirement that the desired homomorphism must exist as a morphism ofkH-modules. Using thisthe Green and Brauer correspondences can be defined which reduce to the study of the representa-tion theory of normalisers ofp-groups inG.

This leads to an extensive and well-developed theory. An excellent introduction, which startsin the spirit of these notes, can be found in [Alp86].

5.4. Representations of other algebraic objects

In this series of lecures we have concentrated on representations of (mainly finite dimensional)associative algebras. But there are other algebraic structures we could have studied. We willintroduce a few of the most important examples.

A Lie algebrais an example of a non-associative algebra. The bilinear mapof two elementsxandy is traditionally denoted by[x,y]. To give a Lie algebra structure this map must beantisym-metric:

[x,x] = 0

and satisfy theJacobi identity:

[x, [y,z]]+ [y, [z,x]]+ [z, [x,y]] = 0.

Given two Lie algebrasg andh, ahomomorphismfrom g to h is a linear map which respects theLie algebra structures, i.e. such that

φ([x,y]) = [φ(x),φ(y)].

Note that any associative algebraA can be given a Lie algebra structure by using the standardmultiplication to define

[x,y] = xy−yx.

In particular, given a vector spaceM the algebra Endk(M) has a Lie algebra structure, and wedefine arepresentationof g to be a vector spaceM together with a Lie algebra homomorphismfrom g to Endk(M).

In a similar way we can define representations of various other classes of algebraic objectsby showing that Endk(M) or Autk(M) (the space of invertible linear maps) lies in that class, andrequiring that the linear map is a homomorphism in that class.

For example, ifk= C or R and we start with aLie group G(a group that is also a differentiablemanifold, such that the group operations are smooth maps) then if M is a finite dimensional vectorspace then Autk(M) is also a Lie group. (For infinite dimensionalM more care is needed.) Thus arepresentationof G is a vector spaceM and a homomorphism of Lie groups fromG to Autk(M).This situation can be generalised to arbitrary algebraically closed fields by consideringalgebraic

50 5. FURTHER DIRECTIONS

groupsand their representations. Instead of being a differentialmanifold we require that the groupis an algebraic variety with group operations which are morphisms of varieties.

The representation theory of Lie algebras and of Lie or algebraic groups is closely related,and all three theories have been very well developed. An introduction to the basics of Lie algebrarepresentation theory can be found in [Car05] or [Hum72]. For Lie groups it is necessary to knowsome basic manifold theory, while for algebraic groups there is a fair amount of algebraic geometryrequired. See [FH91] (or the more advanced [Bum04]) for Lie groups and [Jan03] for algebraicgroups — although the latter presumes a good knowledge of thebasic structure of such groups asdescribed in [Spr98] or [Hum75].

Given a Lie algebrag, there exists a correspondinguniversal enveloping algebras U(g). Thisis an infinite dimensional associative algebra which (via the usual Lie algebra structure on an as-sociative algebra) preserves the representation theory ofg. A classical introduction is [Dix96]; seealso [Hum08] for a guide to the relatedcategoryO of certain infinite dimensional representationsof g overC.

The special class ofsemisimple algebraic groups(or the associated Lie algebras) can be clas-sified; the classification is based around Dynkin diagrams. There are correspondingfinite groupsof Lie type, and one way of studying these is via a reduction from the corresponding algebraicgroup. An introduction to this approach can be found in [DM91], while [Hum06] gives a moreelementary and up-to-date survey of the field.

5.5. Quantum groups and the Ringel-Hall algebra

To define an algebra we needed a multiplication map: a bilinear map fromA×A to A. We candefine an analogous structure called acoalgebraby defining every map in the opposite direction,and consideringcomultiplication: a bilinear map fromA to A×A. (There are various conditions inthe definition which we will not describe here.) Algebras that are also coalgebras in a compatibleway are calledbialgebras, and if they have one additional property (corresponding tothe inversionof elements in a group) we obtain aHopf algebra. There are plenty of interesting examples ofHopf algebras — including group algebras and the universal enveloping algebra of a Lie algebra.

Quantum groupshave been defined in a number of different ways. In each case, the basic ideais to take some Hopf algebra related to a Lie algebrag and introduce an extra parameterq∈ k. Thestructure of these algebras will depend onq, but whenq tends to 1 we should recover the originalHopf algebra in the limit. The standard construction is realised as a deformation of the universalenveloping algebra ofg.

Quantum groups have been studied for many reasons. They arose in the mathematical physicsliterature (which is a rich source of interesting representation theories), and have since provedvery useful in the study of representations of algebraic groups in positive characteristic. (The bestresults to date on the structure of simple modules for algebraic groups proceed via a comparisonwith the associated quantum group.) They have also shed new light on the classical theory; certainremarkable bases called crystal (or canonical) bases were found in the quantum world which werenot previously known in the classical case.

5.5. QUANTUM GROUPS AND THE RINGEL-HALL ALGEBRA 51

There are many different approaches to quantum groups, reflecting their varied applications.Two good examples are [Jan96] and [Kas95]. There is also a nice introduction to the theory ofcrystal bases in [HK02].

Why have we made a detour into Lie theory in the last two sections? Well, it turns outthat quantum groups are closely related to representationsof finite dimensional algebras. Ringel(generalising work of Hall) defined certain algebras, theRingel-Hall algebrasassociated to a fi-nite dimensional algebraA. These have basis the set of isomorphism classes of indecomposableA-modules, and multiplication is defined in terms of possibleextensions of one module by an-other. Ringel then proved that for a quiver algebra this algebra is isomorphic to a quantum groupassociated to the corresponding Lie algebra. Thus the theory of finite dimensional algebras isclosely related to that of Lie algebras. The relationship between these two theories is described in[DDPW08].

Bibliography

[Alp86] J. Alperin,Local representation theory, Cambridge studies in advanced mathematics, vol. 11, Cambridge,1986.

[ARS94] M. Auslander, I. Reiten, and S. Smalø,Representation theory of Artin algebras, Cambridge studies inadvanced mathematics, vol. 36, Cambridge, 1994.

[ASS06] I. Assem, D. Simson, and A. Skowronski,Elements of the representation theory of associative algebrasI, LMS student texts, vol. 65, Cambridge, 2006.

[Bar06] M. Barot,Representations of quivers, 2006, Notes for an Advanced Summer School on RepresentationTheory and Related Topics at the ICTP,http://www.matem.unam.mx/barot/resear h.html .

[Ben91] D. J. Benson,Representations and cohomology I, Cambridge studies in advanced mathematics, vol. 30,Cambridge, 1991.

[Bru03] J. Brundan,Topics in representation theory: Chapter 2, finite dimensional algebras, 2003, U. of Oregonlecture notes,http://darkwing.uoregon.edu/�brundan/math607winter03/index.html .

[Bum04] D. Bump,Lie groups, Graduate texts in mathematics, vol. 225, Springer, 2004.[Car05] R. Carter,Lie algebras of finite and affine type, Cambridge studies in advanced mathematics, vol. 96,

Cambridge, 2005.[CB93] W. Crawley-Boevey,Geometry of representations of algebras, 1993, notes from a graduate course at

Oxford University,http://www.maths.leeds.a .uk/�pmtw /geomreps.pdf .[CR81] C. W. Curtis and I. Reiner,Methods of representation theory, vol. 1, Wiley, 1981.[DDPW08] B. Deng, J. Du, B. Parshall, and J. Wang,Finite dimensional algebras and quantum groups, Mathematical

surveys and monographs, vol. 150, AMS, 2008.[Dix96] J. Dixmier, Enveloping algebras, Graduate studies in mathematics, vol. 11, AMS, 1996, reprinting of

1977 English translation.[DM91] F. Digne and J. Michel,Representations of finite groups of Lie type, LMS student texts, vol. 21, Cam-

bridge, 1991.[FH91] W. Fulton and J. Harris,Representation theory, Graduate texts in mathematics, vol. 129, Springer, 1991.[GR97] P. Gabriel and A. V. Roiter,Representations of finite-dimensional algebras, Springer, 1997.[HK02] J. Hong and S. Kang,Introduction to quantum groups and crystal bases, Graduate studies in mathematics,

vol. 42, AMS, 2002.[Hum72] J. E. Humphreys,Introduction to Lie algebras and representation theory, Graduate texts in mathematics,

vol. 9, Springer, 1972.[Hum75] , Linear algebraic groups, Graduate texts in mathematics, vol. 21, Springer, 1975.[Hum06] , Modular representations of finite groups of Lie type, LMS lecture notes, vol. 326, Cambridge,

2006.[Hum08] , Representations of sesmisimple Lie algebras in the BGG categoryO, Graduate studies in math-

ematics, vol. 94, AMS, 2008.[Jan96] J. C. Jantzen,Lectures on quantum groups, Graduate studies in mathematics, vol. 6, AMS, 1996.[Jan03] , Representations of algebraic groups, second ed., Mathematical Surveys and Monographs, vol.

107, AMS, 2003.[Kas95] C. Kassel,Quantum groups, Graduate texts in mathematics, vol. 155, Springer, 1995.[Mat86] H. Matsumura,Commutative ring theory, Cambridge studies in advanced mathematics, vol. 8, Cam-

bridge, 1986.[Spr98] T. A. Springer,Linear algebraic groups, second ed., Progress in mathematics, vol. 9, Birkhäuser, 1998.

53

Index

admissible labelling, 45algebra, 7

associative, 7basic, 31commutative, 7connected, 31finite dimensional, 7incidence, 15indecomposable, 31Lie, 49local, 24opposite, 8semisimple, 17unital, 7universal enveloping algebra, 50

annihilator, 21antisymmetric, 49arrow, 11arrow ideal, 12Artin-Wedderburn theorem, 20Auslander-Reiten translate, 48Auslander-Reiten quiver, 48

bound quiver algebra, 12

Cartan matrix, 41categoryO, 50centre, 25character, 35completely reducible, 17composition series, 17Coxeter functor, 45

dimension vector, 41direct sum, 10division ring, 21Dynkin diagram, 40

endomorphism algebra, 8Euclidean diagram, 40Euler characteristic, 42exact, 27extended Dynkin diagram, 40

fieldalgebraically closed, 7characteristic of, 7infinite, 7

Fitting’s lemma, 24

Gabriel’s theorem, 40group

algebraic, 50Lie, 49of Lie type, 50

group algebra, 8

head, 23Higman’s theorem, 39homomorphism

of quiver representations, 13left (or right) almost split, 47left (or right) minimal, 47of algebras, 8of Lie algebras, 49of modules, 9

ideal, 8admissible, 12left/right, 8nilpotent, 21

idempotent, 8central, 9orthogonal, 8primitive, 9

identity element, 7injective envelope, 30isomorphism

of algebras, 8of modules, 9

isomorphism theorem, 10

Jacobi identity, 49Jacobson radical, 21Jordan-Hölder theorem, 17

Krull-Schmidt theorem, 23

55

56 INDEX

Lie algebra, 49universal enveloping algebra of, 50

Lie group, 49Loewy length, 23Loewy series, 23

Maschke’s theorem, 18matrix algebra, 8module, 9

decomposable, 10dual, 10finite dimensional, 9finitely generated, 9generated by, 9indecomposable, 10injective, 28irreducible, 10projective, 27semisimple, 17simple, 10

Nakayama’s lemma, 22

path, 11path algebra, 11positive roots, 43projective cover, 30

quiver, 11acyclic, 11bound, 12connected, 11finite, 11underlying graph, 11

quotient module, 10

radicalof an algebra, 21of a module, 23

reflection functor, 44relation in a quiver, 13representation

of algebraic groups, 50of algebras, 9of Lie algebras, 49of Lie groups, 49of quivers, 13

representation typefinite, 37infinite, 37tame, 38wild, 38

resolutioninjective, 30projective, 30

retraction, 47Ringel-Hall algebras, 51

Schur’s lemma, 19, 20section, 47sequence

almost split, 48short exact sequence, 27

split, 47sink, 43socle, 23source, 43subalgebra, 8submodule, 9

maximal, 21

top, 23Trichotomy theorem, 38

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