quivers and three-dimensional lie algebras · 2017-01-31 · chapter 1 introduction in the late...

Quivers and Three-Dimensional Lie Algebras

Jeffrey Pike

Thesis submitted to the Faculty of Graduate and Postdoctoral Studies

in partial fulfillment of the requirements for the degree of Master of Science in

Mathematics 1

Department of Mathematics and Statistics

Faculty of Science

University of Ottawa

c© Jeffrey Pike, Ottawa, Canada, 2015

1The M.Sc. program is a joint program with Carleton University, administered by the Ottawa-Carleton Institute of Mathematics and Statistics

Abstract

We study a family of three-dimensional Lie algebras Lµ that depend on a continuous

parameter µ. We introduce certain quivers, which we denote by Qm,n (m,n ∈ Z)

and Q∞×∞, and prove that idempotented versions of the enveloping algebras of the

Lie algebras Lµ are isomorphic to the path algebras of these quivers modulo certain

ideals in the case that µ is rational and non-rational, respectively. We then show

how the representation theory of the quivers Qm,n and Q∞×∞ can be related to the

representation theory of quivers of affine type A, and use this relationship to study

representations of the Lie algebras Lµ. In particular, though it is known that the

Lie algebras Lµ are of wild representation type, we show that if we impose certain

restrictions on weight decompositions, we obtain full subcategories of the category of

representations of Lµ that are of finite or tame representation type.

ii

Acknowledgements

The research for this thesis began in the Summer of 2012 at the University of Ottawa

under the supervision of Alistair Savage. We would like to thank him not only for

introducing us to the topics of this project, but also for all the interesting discussions

and for his insightful guidance over the course of this project.

iii

Contents

List of Figures vi

1 Introduction 1

2 Quivers 5

2.1 Quivers and the Path Algebra . . . . . . . . . . . . . . . . . . . 5

2.2 Quiver Representations . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Quiver Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Quivers with Relations . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Graded Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Lie Algebras 29

3.1 Elementary Notions . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Lie Algebras of Low Dimension . . . . . . . . . . . . . . . . . . . 30

3.3 The Universal Enveloping Algebra of Lµ and its Representations 33

3.3.1 Universal Enveloping Algebras . . . . . . . . . . . . . . . . . 33

3.3.2 Rational Case . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.3 Non-Rational Case . . . . . . . . . . . . . . . . . . . . . . . 35

4 The Quivers Qm,n and Q∞×∞ 37

4.1 Relation to the Lie Algebras Lµ . . . . . . . . . . . . . . . . . . 37

4.1.1 Rational Case . . . . . . . . . . . . . . . . . . . . . . . . . . 38

iv

CONTENTS v


4.2 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Rational Case . . . . . . . . . . . . . . . . . . . . . . . . . . 42


A Category Theory 52

A.1 Categories and Functors . . . . . . . . . . . . . . . . . . . . . . . 52

A.2 Abelian Categories and Adjunctions . . . . . . . . . . . . . . . . 54

A.3 Kan Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B Affine Varieties 61

Bibliography 66

List of Figures

2.1 The quiver Qm,n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The quiver Q∞×∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 The quiver morphism g2,1 : Q2,1 → Q3 . . . . . . . . . . . . . . . . . 11

2.4 The quiver morphism f : Q∞×∞ → Q∞ . . . . . . . . . . . . . . . . 11

2.5 A quiver morphism that is not a covering morphism . . . . . . . . . 12

4.1 The functor gRm+n

! is not full . . . . . . . . . . . . . . . . . . . . . . 43

4.2 The functor gRm+n

! is not essentially surjective . . . . . . . . . . . . . 44

4.3 The functor fR∞! is not full . . . . . . . . . . . . . . . . . . . . . . . 47

4.4 The functor fR∞! is not essentially surjective . . . . . . . . . . . . . 47

4.5 The category Rep(Q∞×∞, R∞×∞) is at least tame . . . . . . . . . . 48

vi

Chapter 1

Introduction

In the late 20th century, the mathematician Pierre Gabriel discovered a beautiful re-

lationship between the root systems of Lie algebras and the representations of quivers

[5], which are directed graphs. This result fuelled further research of the connection

between Lie theory and quivers, as the correspondence between the two allows for a

more intuitive and geometric means of studying Lie algebras. In the present work,

we study the representation theory of a particular collection of Lie algebras by first

relating them to certain quivers.

When working over the complex numbers, one can completely classify all Lie

algebras of dimension three, up to isomorphism. The possibilities are (see for example

[3, Chapter 3]):

1. the three-dimensional abelian Lie algebra,

2. the direct sum of the unique nonabelian two-dimensional Lie algebra with the

one-dimensional Lie algebra,

3. the Heisenberg algebra,

4. the Lie algebra sl2(C),

1

1. Introduction 2

5. the Lie algebra with basis x, y, z and commutation relations [x, y] = y, [x, z] =

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

6. the Lie algebras Lµ, µ ∈ C∗ = C \ 0 (see Section 3.2).

The representation theory of the first four Lie algebras in the above list is well un-

derstood, while the representation theory of the last two is less so. In particular, for

generic µ, very little is known about the representation theory of the Lie algebras

Lµ, and so in order to obtain a better understanding of the representation theory of

three-dimensional Lie algebras over the complex numbers, we first require a better

understanding of this particular family. Due to a result of Makedonskyi [10, Theorem

3], it is known that the Lie algebras Lµ are of wild representation type. However, in

the current paper, we will exploit the relationship between quivers and Lie algebras

to draw several conclusions about the representations of these Lie algebras.

The Euclidean group is the group of isometries of R2 having determinant 1, and

the Euclidean algebra is the complexification of its Lie algebra. The Euclidean group

is one of the oldest and most studied examples of a group: it was studied implicitly

even before the notion of a group was formalized, and it has applications not only

throughout mathematics but in quantum mechanics, relativity, and other areas of

physics as well. In [12, Theorem 4.1], it is shown that the category of representations

of the Euclidean algebra admitting weight space decompositions is equivalent to the

category of representations of the preprojective algebras of quivers of type A∞. In

the current paper, we show that the category of representations of Lmn

(m,n ∈ Z,

n 6= 0) admitting weight space decompositions can be embedded inside the category

of representations of the preprojective algebra of the affine quiver of type A(1)m+n, where

by convention A(1)0 denotes the quiver of type A∞. It can be shown that the Euclidean

algebra is isomorphic to the Lie algebra L−1, and so if µ = −1 then m = −n so that

this agrees with what is presented in [12]. Thus the current work can be thought of

as a generalization of some of the results of that paper. Analogous to the rational

1. Introduction 3

case, we also show how the representations with weight space decompositions of Lµ,

µ ∈ C \Q, form a subcategory of the category of representations of the preprojective

algebra of the quiver of type A∞.

We begin by defining the modified enveloping algebras of the Lie algebras Lµ,

denoted Uµ, and we note that category of representations of Uµ is equivalent to the

category of representations of Lµ admitting weight space decompositions. We then

introduce certain quivers, denoted Qm,n (m,n ∈ Z) and Q∞×∞, and show that the

algebras Uµ are isomorphic to the path algebras of Qm,n and Q∞×∞ modulo certain

ideals in the case that µ is rational and non-rational, respectively (see Proposition

4.1.2). We use the theory of quiver morphisms to relate the representation theory of

the quivers Qm,n and Q∞×∞ to the representation theory of affine quivers of type A,

which is well understood. In the case that µ ∈ Q, the main result is the following

(Theorems 4.2.3 and 4.2.4):

Theorem. Let m,n ∈ Z, gcd(m,n) = 1, and let a, b ∈ Z be such that 0 ≤ b− a < m.

Let Cm,na,b denote the full subcategory of Uµ -Mod consisting of modules V such that

Vk = 0 whenever k < a or k > b, where Vk denotes the kth weight space of V . Then

Cm,na,b is of finite representation type when n 6= 1, and Cm,1a,b is of tame representation

type.

When µ ∈ C \Q, we introduce a Z action on Uµ -Mod and our main result is the

following (Corollary 4.2.11):

Theorem. Let A be a finite subset of Z with the property that A does not contain any

five consecutive integers. Then there are a finite number of Z-orbits of isomorphism

classes of indecomposable Uµ-modules V such that Vij = 0 whenever i− j /∈ A, where

Vij denotes the (i, j) weight space of V .

The organization of the paper is as follows. In Chapter 2 we recall some basic

notions from the theory of quivers and their representations, and we develop the

1. Introduction 4

theory of morphisms between quivers. In Chapter 3 we introduce the family Lµ of

3-dimensional Lie algebras that will be studied in the rest of the paper. We then

describe the (modified) universal enveloping algebras Um,n and Uµ associated with

the Lie algebras Lµ. Finally, in Chapter 4, we establish a relationship between the

representation theory of the Lie algebras Lµ and the representation theory of the

quivers Qm,n and Q∞×∞, and use some of the results of Chapter 2 to study these

quivers.

In Appendix A we cover some of the basics of category theory which we use

throughout. We also include a brief discussion of the theory of Kan extensions, the

use of which simplifies much of the discussion in Section 2.3, though is not strictly

necessary. In Appendix B, we review some elementary definitions and results in the

theory of affine varieties, which are used in Section 2.5.

Notation. Throughout this work, all vector spaces and linear maps will be over

C. Given a C-algebra A, we will take the term module over A to mean left module

over A. The category of (left) modules over A is denoted A -Mod. By the usual abuse

of language, we will use the terms module and representation interchangeably. Below

is a table of important notation we will use throughout.

Index of Notation

Qm,n, Q∞×∞ Page 5 CQ, P(Q) Page 6

Qs, Q∞ ϕ∗, ϕ∗, ϕ! Definition 2.3.7

ϕ∗R′ , ϕR′∗ , ϕR

′

! Page 22 EQα , ER

α , ΛQ∗α , N (ER

α ) Page 24

Um,n, Um,n Page 34 Uµ, Uµ Page 36

Chapter 2

Quivers

This chapter introduces the theory of quivers and their representations. We put

special emphasis on quiver morphisms and certain functors associated with them.

2.1 Quivers and the Path Algebra

A quiver Q is a 4-tuple (X,A, t, h), where X and A are sets, and t and h are functions

from A to X. The sets X and A are called the vertex and arrow sets respectively.

If ρ ∈ A, we call t(ρ) the tail of ρ, and h(ρ) the head. We can think of an element

ρ ∈ A as an arrow from the vertex t(ρ) to the vertex h(ρ). We will often denote a

quiver simply by Q = (X,A), or even more simply by a picture, leaving the maps t

and h implied.

Example 2.1.1 (The quiver Qm,n). Let m,n ∈ Z be nonzero integers and consider

the quiver Qm,n = (Z, Am,n) where Am,n = ρkj | k ∈ Z, j = 1, 2 is the arrow set. We

define a map σ : 1, 2 → m,n such that σ(1) = m and σ(2) = n. Then ρkj is an

arrow whose tail, t(ρkj ), is the vertex k and whose head, h(ρkj ), is the vertex k+ σ(j).

If gcd(m,n) 6= 1, then the quiver Qm,n decomposes into a disjoint union of quivers of

the form Qm′,n′ , where gcd(m′, n′) = 1. Thus we may assume when dealing with the

5

2. Quivers 6

quivers Qm,n that gcd(m,n) = 1.

CC CC

CC CC

0 1 m m+ 1 n n+ 1−m 1−m1− n−n· · · · · ·· · ·· · · · · ·· · ·

Figure 2.1: The quiver Qm,n

Example 2.1.2 (The quiver Q∞×∞). We now consider the quiver Q∞×∞ = (Z ×

Z, A∞×∞) where the set of arrows is A∞×∞ = ρkd | d ∈ 1, 2, k ∈ Z×Z. We define

the map θ : 1, 2 → (1, 0), (0, 1) by θ(1) = (1, 0) and θ(2) = (0, 1). Then ρkd is the

arrow whose tail, t(ρkd), is the vertex k = (i, j) and whose head, h(ρkd), is the vertex

(i, j) + θ(d).

(−1,−1) (0,−1) (1,−1)

(−1,0) (0, 0) (1, 0)

(−1,1) (0, 1) (1, 1). . .

. . .

. . .

. . .

. . .

. . .

.

.. ...

.

..

..

....

..

.

Figure 2.2: The quiver Q∞×∞

A path in a quiver Q is a sequence τ = ρnρn−1 · · · ρ1 of arrows such that h(ρi) =

t(ρi+1) for each 1 ≤ i ≤ n− 1. We define t(τ) = t(ρ1) and h(τ) = h(ρn).

Definition 2.1.3 (Path algebra). Let Q be a quiver. The path algebra of Q is the

C-algebra whose underlying vector space has for basis the set of paths in Q and with

2. Quivers 7

multiplication given by:

τ2 · τ1 =

τ2τ1, if h(τ1) = t(τ2),

0, otherwise,

where τ2τ1 denotes the concatenation of the paths τ1 and τ2. We denote the path

algebra of Q by CQ.

For any vertex x ∈ X we let ex denote the trivial path starting and ending at x

and with multiplication given by exτ = δh(τ)xτ and τex = δt(τ)xτ for any path τ .

For example, any nontrivial path in the quiver Qm,n of Example 2.1.1 above

is of the form ρk+σ(j1)+···+σ(js−1)js

· · · ρk+σ(j1)j2

ρkj1 with k ∈ Z, s ∈ N and ji ∈ 1, 2

for i = 1, . . . , s. Elements of the path algebra CQm,n are linear combinations of

these paths and the trivial paths at each vertex. Similarly, elements of the form

ρk+θ(d1)+···+θ(ds−1)ds

· · · ρk+θ(d1)d2

ρkd1 along with the trivial paths at each vertex constitute

a basis for the path algebra CQ∞×∞, where k ∈ Z × Z, s ∈ N, and dn ∈ 1, 2 for

n = 1, ..., s.

For every arrow ρ ∈ A in a quiver Q = (X,A), define an arrow ρ by t(ρ) = h(ρ)

and h(ρ) = t(ρ), and let A denote the set of all such ρ. Then we call←→Q = (X,A∪ A)

the double quiver of Q.

Definition 2.1.4 (Preprojective algebra). Let Q be a quiver,←→Q its double quiver,

and let I be the two-sided ideal of C←→Q generated by elements of the form∑

ρ∈A,h(ρ)=i

ρρ−∑

ρ∈A,t(ρ)=i

ρρ. (2.1.1)

Then the algebra C←→Q /I is called the preprojective algebra of Q, and is denoted P(Q).

The relations (2.1.1) are called the Gelfand-Ponomarev relations in←→Q .

Now that we have introduced the concept of paths of a quiver Q = (X,A), we

can think of Q as being a small category as follows:

2. Quivers 8

(i) the objects of Q are given by Ob(Q) = X, and

(ii) for any B,B′ ∈ Ob(Q), HomQ(B,B′) is the set of all paths τ in Q with t(τ) = B

and h(τ) = B′.

It is not difficult to see that this defines a category. Indeed, composition is given by

concatenation of paths and for any B ∈ Ob(Q) the identity morphism on B is given

by the trivial path eB. We will often pass between this categorical definition of a

quiver and our original definition.

2.2 Quiver Representations

Definition 2.2.1 (Quiver Representation). Let Q be a quiver. A representation of

Q is a covariant functor V : Q→ VectC.

More explicitly, a representation of a quiver Q = (X,A) is a collection of vector

spaces V (x) | x ∈ X along with a collection of linear maps V (ρ) : V (t(ρ)) →

V (h(ρ)) | ρ ∈ A. For any path τ = ρn · · · ρ1 in Q we set V (τ) = V (ρn) · · ·V (ρ1).

Let V,W be two representations of a quiver Q = (X,A). A morphism from V to

W , σ ∈ HomRep(Q)(V,W ), is a natural transformation σ : V → W . A morphism σ is

specified by a collection of linear maps σ(x) : V (x)→ W (x) | x ∈ X) such that for

every ρ ∈ A the following diagram commutes:

W (t(ρ))W (ρ)−−−→ W (h(ρ))

σ(t(ρ))

x xσ(h(ρ))

V (t(ρ)) −−−→V (ρ)

V (h(ρ)).

(2.2.1)

Thus we may consider the category Rep(Q) having as objects representations of

Q, and with morphisms as described above. The category of representations of the

quiver Q = (X,A) is equivalent to the category of representations of the path algebra

CQ (see for example [1, Theorem 1.6]).

2. Quivers 9

Remark 2.2.2. When viewed as a category, the morphisms (paths) in a quiver Q

are generated by the paths of length 1 (arrows). Thus the action of a representation

V on any path in Q is uniquely determined by the action of V on the arrows of Q.

Definition 2.2.3 (Representation Type). Let Q be a quiver and let C be a subcategory

of Rep(Q). We say C is of finite type if there are only finitely many isomorphism

classes of indecomposable representations in C. We say C is of tame type if for every

n ∈ N all but finitely many isomorphism classes of indecomposable representations

of dimension n in C occur in a finite number of families that are parametrized by a

single, complex parameter. We say C is of wild type if it is neither of finite type nor

tame type.

In [5], Gabriel was able to prove that the notion of representation type of Rep(Q)

is closely related to the study of Dynkin diagrams. This beautiful result was one of

the main motivating factors for studying the representation theory of quivers and its

relationship to Lie theory.

Theorem 2.2.4 (Gabriel’s Theorem). Let Q be a quiver.

1. Rep(Q) is of finite type if and only if the underlying graph of Q is a union of

Dynkin diagrams of type A,D, or E.

2. Rep(Q) is of tame type if and only if the underlying graph of Q is a union of

Dynkin diagrams of type A,D, or E and of extended Dynkin diagrams of type

A, D, or E (with at least one extended Dynkin diagram).

Let Q = (X,A) be some quiver. If V, U ∈ Rep(Q) and ϕ ∈ HomRep(Q)(V, U),

then the kernel of ϕ, kerϕ, is the representation of Q with vector spaces given by

(kerϕ)(x) = ker(ϕ(x)) for every x ∈ X, and maps given by the restriction of the

maps in V to these subspaces. The cokernel of ϕ is the representation of Q given by

(cokerϕ)(x) = U(x)/ imϕ(x) and with maps induced on these spaces by the maps of

U .

2. Quivers 10

2.3 Quiver Morphisms

Definition 2.3.1. Let Q = (X,A, t, h) and Q′ = (X ′, A′, t, h) be two quivers. Then

a quiver morphism ϕ ∈ Hom(Q,Q′) consists of a pair of maps ϕx : X → X ′ and

ϕa : A→ A′ such that ϕxt = tϕa and ϕah = hϕx.

Remark 2.3.2. Throughout this chapter we have moved back and forth between

thinking of quivers as directed graphs and thinking of quivers as the free category

generated by the directed graph. While this can often be a useful identification,

we should mention that the natural functor Quiv → Cat, where Quiv denotes the

category of directed graphs, is a faithful embedding that is not full. That is, when

thinking of two quivers Q,Q′ as small categories, there can exist functors between Q

and Q′ that are not quiver morphisms.

Example 2.3.3. For all s ∈ N, define the quiver Qs = (Z/sZ, ρi, ρi), where t(ρi) =

h(ρi+1) = i and h(ρi) = t(ρi+1) = i + 1. Let m,n ∈ Z with gcd(m,n) = 1.

Then we have gcd(m,m + n) = 1 and so for all k ∈ Z there is a unique inte-

ger 0 ≤ jk < m + n such that k ≡ jkm mod (m + n). Consider the morphism

gm,n ∈ Hom(Qm,n, Qm+n), where Qm,n denotes the quiver of Example 2.1.1, given by

(gm,n)x(k) = jk, (gm,n)a(ρk1) = ρjk , and (gm,n)a(ρ

k2) = ρjk for all k ∈ Z. It is not

difficult to see that this does indeed define a quiver morphism. The morphism g2,1 is

pictured below, where the coloured vertices and arrows of Q2,1 are mapped under g

to the vertices and arrows of the same colour. We will usually just write g, and let it

be understood that the morphism g depends on m and n.

Example 2.3.4. Let Q∞ denote the quiver (Z, ρi, ρi), where t(ρi) = i = h(ρi+1), and

h(ρi) = i + 1 = t(ρi+1). If Q∞×∞ denotes the quiver from Example 2.1.2, consider

the morphism f ∈ Hom(Q∞×∞, Q∞) given by fx(i, j) = i − j, fa(ρ(i,j)1 ) = ρi−j, and

fa(ρ(i,j)2 ) = ρi−j for all (i, j) ∈ Z × Z. Once again it is easy to check that f is

indeed a quiver morphism. The morphism f is displayed in the figure below, where

2. Quivers 11

wwg2,1

Figure 2.3: The quiver morphism g2,1 : Q2,1 → Q3

the coloured vertices and arrows of Q∞×∞ are mapped under f to the vertices and

arrows of the same colour.

. . .

. . .

. . .

. . .

. . .

. . .

.

.. ...

.

..

..

....

..

.

=⇒f

. . . . . .

Figure 2.4: The quiver morphism f : Q∞×∞ → Q∞

For the most part, we will be interested in quiver morphisms satisfying a certain

property, known as covering morphisms, which we introduce now.

Definition 2.3.5 (Covering Morphism). Let Q = (X,A) and Q′ = (X ′, A′) be quivers

and let ϕ ∈ Hom(Q,Q′). Then ϕ will be said to be a covering morphism if for every

x ∈ X and every path τ ′ in Q′ such that h(τ ′) = ϕ(x) there exists a unique path τ in

Q such that h(τ) = x and ϕ(τ) = τ ′.

2. Quivers 12

For example, the morphisms g and f of Examples 2.3.3 and 2.3.4 are covering

morphisms. On the other hand, the following example provides a quiver morphism

which is not a covering morphism.

Example 2.3.6. Consider the quiver morphism ϕ pictured below. This morphism

is not a covering morphism, since the path of length one ending at ϕ(3) has two

preimages under ϕ.

=⇒ϕ

21

3

ϕ(1), ϕ(2)

ϕ(3)

Figure 2.5: A quiver morphism that is not a covering morphism

Given a quiver morphism ϕ : Q → Q′, there are three important functors one

can use to study the relationship between the representations of the two quivers.

Definition 2.3.7. Let Q = (X,A) and Q′ = (X ′, A′) be two quivers and let ϕ ∈

Hom(Q,Q′).

(i) The restriction functor of ϕ is the functor ϕ∗ : Rep(Q′) → Rep(Q) defined for

all V ∈ Rep(Q′) by

(a) For any x ∈ X, ϕ∗(V )(x) = V (ϕ(x)).

(b) For any ρ ∈ A, ϕ∗(V )(ρ) = V (ϕ(ρ)).

If V, U ∈ Rep(Q) and f ∈ HomQ(V, U) then the morphism ϕ∗(f) is defined by

ϕ∗(f)(x) = f(ϕ(x)) for every x ∈ X.

(ii) If ϕ is a covering morphism, then the right extension functor of ϕ is the functor

ϕ∗ : Rep(Q)→ Rep(Q′) defined for all V ∈ Rep(Q) by:

2. Quivers 13

(a) For any x′ ∈ X ′, ϕ∗(V )(x′) =∏

x∈ϕ−1(x′) V (x), where by convention we take

the empty product to be the zero vector space.

(b) For any ρ′ ∈ A′, ϕ∗(V )(ρ′) =∏

ρ∈ϕ−1(ρ′) V (ρ).

If V, U ∈ Rep(Q′) and f ∈ HomQ′(V, U) then the morphism ϕ∗(f) is defined by

ϕ∗(f)(x′) =∏

x∈ϕ−1(x′) f(x).

(iii) If ϕ is a covering morphism, then the left extension functor of ϕ is the functor

ϕ! : Rep(Q)→ Rep(Q′) defined for all V ∈ Rep(Q) by:

(a) For any x′ ∈ X ′, ϕ!(V )(x′) =⊕

x∈ϕ−1(x′) V (x), where by convention we take

the empty coproduct to be the zero vector space.

(b) For any ρ′ ∈ A′, ϕ!(V )(ρ′) =⊕

ρ∈ϕ−1(ρ′) V (ρ).

If V, U ∈ Rep(Q′) and f ∈ HomQ′(V, U) then the morphism ϕ!(f) is defined by

ϕ!(f)(x′) =⊕

x∈ϕ−1(x′) f(x).

Remark 2.3.8. If we view a representation of Q′ as a functor V : Q′ → VectC then

the restriction functor ϕ∗ is described by the map V 7→ V ϕ. The (left and right)

extension functors have similar categorical definitions in terms of what are known as

(left and right) Kan extensions, and these definitions allow one to discuss the extension

functors even in the case where ϕ is not a covering morphism. More precisely, in the

language of Section A.3, the category Rep(Q) is written as the functor category

VectQC , and then the restriction functor ϕ∗ is written VectϕC : VectQ′

C → VectQC . Since

Q is small and VectC is complete, then for any functor V : Q → VectC (that is, for

any representation V ∈ Rep(Q)) there exists a right Kan extension of V along ϕ by

Theorem A.3.2, which we denote by Ranϕ V ∈ VectQ′

C . Moreover, for any vertex x in

Q′, we have

Ranϕ V (x) = lim←−((x ↓ ϕ)Qx−→ Q

V−→ VectC).

2. Quivers 14

The comma category x ↓ ϕ is given by pairs (y, ρ : x → y) of vertices y in Q and

arrows ρ : x → ϕ(y) in Q′. The functor Qx takes pairs (y, ρ) to y. Thus Ranϕ V (x)

can be described as taking the limit of the vector spaces V (y) over all vertices y in Q

and all arrows ρ : x → ϕ(y) in Q′. In order to generalize Definition 2.3.7 above, we

set ϕ∗ to be the functor V 7→ Ranϕ V . In the case that ϕ is a covering morphism, for

any y in Q and any arrow ρ : x→ ϕ(y) in Q′, there is precisely one arrow mapped to

ρ by ϕ and we retrieve the definition of ϕ∗ given above. Similarly, one can generalize

the definition of ϕ! in terms of left Kan extensions. Since in this work we deal only

with covering morphisms, we will work with Definition 2.3.7 directly in order to keep

the technical prerequisites to a minimum. However, we would like to point out that

using these generalized definitions of the left and right extension functors, one can

apply Theorem A.3.4 to deduce that ϕ∗ is right adjoint to ϕ∗, and ϕ! is left adjoint

to ϕ∗. This is a fact that was proven directly in [2, Theorem 4.1] for the case when

ϕ is a covering morphism (we include the proof below in Theorem 2.3.9).

In Section 4.2 we will use the above functors to study the representation theory

of the quivers Qm,n and Q∞×∞ of Examples 2.1.1 and 2.1.2. It turns out that the

functors in Definition 2.3.7 are closely related.

Theorem 2.3.9. [2, Theorem 4.1] Let Q,Q′ be two quivers and let ϕ ∈ Hom(Q,Q′)

be a covering morphism. Then ϕ∗ is left adjoint to ϕ∗ and right adjoint to ϕ!.

Proof: In [2, Theorem 4.1] it is shown explicitly that ϕ∗ is left adjoint to ϕ∗.

The proof that ϕ∗ is right adjoint to ϕ! is completely dual to the argument presented

there. We include the proof here for completenes.

For any V ∈ Rep(Q), U ∈ Rep(Q′) we need to exhibit an isomorphism

HomRep(Q′)(ϕ!(V ), U) ∼= HomRep(Q)(V, ϕ∗(U))

that is natural in the arguments V and U . First, we note that for any x′ ∈ X ′ we

2. Quivers 15

have

HomC(ϕ!(V )(x′), U(x′)) ∼=⊕

x:x′=ϕ(x)

HomC(V (x), U(x′))

∼= HomC(V (x), ϕ∗(U)(x)). (2.3.1)

Now for any σ ∈ HomRep(Q′)(ϕ!(V ), U), define σ ∈ HomRep(Q)(V, ϕ∗(U)) by σ(x0) =

σ(x′)ix0 for every x0 ∈ X with ϕ(x0) = x′. Here ix0 denotes the inclusion map

ix0 : V (x0) →⊕

x∈ϕ−1(x′) V (x) = ϕ!(V )(x′). Then the assignment σ 7→ σ defines a

morphism HomRep(Q′)(ϕ!(V ), U) → HomRep(Q)(V, ϕ∗(U)). Indeed, let x0, y0 ∈ X and

ρ0 : x0 → y0. Then if x′ = ϕ(x0), y′ = ϕ(y0), and ρ′ = ϕ(ρ0), the diagram⊕x∈ϕ−1(x′) V (x)

ϕ!(V )(ρ′)−−−−−→⊕

y∈ϕ−1(y′) V (y)

σ(x′)

y yσ(y′)

ϕ∗(U)(x0) −−−−−→ϕ∗(U)(ρ0)

ϕ∗(U)(y0)

(2.3.2)

is commutative since σ is a morphism. But by the definition of ϕ!, the diagram

V (x0)V (ρ0)−−−→ V (y0)

ix0

y yiy0⊕x∈ϕ−1(x′) V (x) −−−−−→

ϕ!(V )(ρ′)

⊕y∈ϕ−1(y′) V (y)

(2.3.3)

is also commutative. Placing diagram (2.3.3) on top of diagram (2.3.2) shows that

σ ∈ HomRep(Q)(V, ϕ∗(U)).

It is not difficult to see that if we fix either V or U then the assignment σ 7→ σ de-

fines a natural transformation. Suppose we fix V . Then for any β ∈ HomRep(Q′)(U1, U2)

and any σ ∈ HomRep(Q′)(ϕ!(V ), U1) we have

βσ(x0) = βσ(ϕ(x0))ix0 = β(ϕ(x0))σ(ϕ(x0))ix0 .

On the other hand,

(ϕ∗(β)σ)(x0) = ϕ∗(β)(x0)σ(x0) = β(ϕ(x0))σ(ϕ(x0))ix0 ,

2. Quivers 16

and so the assignment is natural in the argument U , as claimed. One shows similarly

that the assignment is natural in V .

If is left to show that for any representations V and U , the map σ 7→ σ

defined above is an isomorphism. To do so, we construct its inverse. For any

τ ∈ HomRep(Q)(V, ϕ∗(U)), define τ ∈ HomRep(Q′)(ϕ!(V ), U) by τ(x′) =

⊕x∈ϕ−1(x′) τ(x)

for every x′ ∈ X ′, where we have considered τ as an element of the coproduct (2.3.1)

above. Clearly the assignments σ → σ and τ → τ are mutual inverses, so if we can

show that τ defines a morphism of representations, then we are done. Let ρ′ : x′ → y′

be an arrow in Q′. For every y ∈ X such that ϕ(y) = y′, let ρy be the unique arrow

in Q with ϕ(ρy) = ρ′ and h(ρy) = y. Then any morphism τ : V → ϕ∗(U) gives a

commutative diagram

V (t(ρy))V (ρy)−−−→ V (y)

τ(t(ρy))

y yτ(y)

U(x′) −−−→U(ρ′)

U(y′)

for each such y. Taking the direct sum gives the following commutative diagram⊕x∈ϕ−1(x′) V (x)

⊕V (ρy)−−−−−→

⊕y∈ϕ−1(y′) V (y)

τ(x′)

y yτ(y′)

U(x′) −−−→U(ρ′)

U(y′),

which shows that τ is a morphism of representations.

Remark 2.3.10. It is clear from Definition 2.3.7 that if we restrict the functors ϕ∗

and ϕ! to the subcategory of Rep(Q) consisting of finite dimensional representations,

then ϕ∗ and ϕ! are naturally isomorphic. In that case, ϕ∗ and ϕ! are both left and

right adjoint to each other when ϕ is a covering morphism. Such a pair is often called

a biadjoint pair.

The following corollary will prove useful.

2. Quivers 17

Corollary 2.3.11. The functors ϕ∗, ϕ∗, and ϕ! are additive. Moreover, ϕ∗ is exact,

ϕ∗ is left exact, and ϕ! is right exact.

Proof: This follows immediately from Theorem 2.3.9 as all adjoint functors are

additive, and left (resp. right) adjoint functors are right (resp. left) exact (see Section

A.2).

We will now focus on other properties of these functors.

Lemma 2.3.12. Let Q,Q′ be quivers and let ϕ ∈ Hom(Q,Q′) be a covering morphism.

Then both ϕ∗ and ϕ! are faithful.

Proof: Let V, U ∈ Rep(Q) and consider the map (ϕ!)V U : HomRep(Q)(V, U) →

HomRep(Q′)(ϕ!(V ), ϕ!(U)). We wish to show that this map is injective. Since it

is a group homomorphism, it is enough to consider the preimage of the morphism

0: ϕ!(V ) → ϕ!(U). This morphism is defined by the linear maps 0(x′) : ϕ!(V )(x′) →

ϕ!(U)(x′) for every x′ ∈ X ′. So if ϕ!(σ) = 0, then∐

x∈ϕ−1(x′) σ(x) = 0 for every x′,

and it follows that σ(x) = 0 for every x ∈ X. Hence σ = 0. To show that ϕ∗ is

faithful is similar.

Remark 2.3.13. We have seen that if we use the generalized definitions of the left

and right extension functors in terms of Kan extensions, Theorem 2.3.9 is still true.

However, in this generalized setting, Lemma 2.3.12 does not hold.

Lemma 2.3.14. Let Q,Q′ be quivers and let ϕ ∈ Hom(Q,Q′) be a covering morphism.

Then both ϕ∗ and ϕ! are exact.

Proof: By Corollary 2.3.11, we need only show that ϕ∗ is right exact and ϕ! is

left exact. Since the categories Rep(Q) and Rep(Q′) are abelian, it is enough to show

that ϕ∗ preserves cokernels and ϕ! preserves kernels.

2. Quivers 18

Let V, U ∈ Rep(Q) and σ ∈ HomRep(Q)(V, U). The vector spaces of the represen-

tation ϕ∗(cokerσ) are defined by ϕ∗(cokerσ)(x′) =∏

x∈ϕ−1(x′) U(x)/ imσ(x). How-

ever, the map ϕ∗(σ)(x′) : ϕ∗(V )(x′)→ ϕ∗(U)(x′) is given by∏

x∈ϕ−1(x′) σ(x). Hence

coker(ϕ∗(σ))(x′) = ϕ∗(U)(x′)/ imϕ∗(σ)(x′)

=∏

x∈ϕ−1(x′)

U(x)/∏

x∈ϕ−1(x′)

imσ(x)

∼=∏

x∈ϕ−1(x′)

U(x)/ imσ(x)

= ϕ∗(cokerσ)(x′).

It is now clear that for any arrow ρ′ ∈ A′ the maps ϕ∗(cokerσ)(ρ′) and coker(ϕ∗(σ))(ρ′)

will be equal, and hence ϕ∗ preserves cokernels.

The representation ϕ!(kerσ) ∈ Rep(Q) has vector spaces given by

ϕ!(kerσ)(x) =⊕

x∈ϕ−1(x′)

(kerσ)(x) =⊕

x∈ϕ−1(x′)

ker(σ(x))

for all x ∈ X. On the other hand, the morphism ϕ!(σ) ∈ HomRep(Q)(ϕ!(V ), ϕ!(U)) is

defined by ϕ!(σ)(x′) = ⊕x∈ϕ−1(x′)σ(x) and hence

ker(ϕ!(σ))(x′) = ker

⊕x∈ϕ−1(x′)

σ(x)

=⊕

x∈ϕ−1(x′)

ker(σ(x)).

Thus ϕ!(kerσ)(x′) = ker(ϕ!(σ))(x′) for all x′ ∈ X ′. Again, it is easy to see that for

any ρ′ ∈ A′ the maps ϕ!(kerσ)(ρ′) and ker(ϕ!(σ))(ρ′) will also coincide, and so ϕ!

preserves kernels.

2.4 Quivers with Relations

Definition 2.4.1 (Relation). A relation in a quiver Q is an element r ∈ CQ. More

explicitly, a relation in Q is an expression of the form∑k

j=1 ajτj, where for every

2. Quivers 19

j ∈ 1, . . . , k we have aj ∈ C and τj is a path in Q.

We say that a representation V ∈ Rep(Q) satisfies the relation∑k

j=1 ajτj if∑kj=1 ajV (τj) = 0. If R is a set of relations, we denote by Rep(Q,R) the category of

representations of Q which satisfy the relations R. If I is the two-sided ideal in CQ

generated by R, then there is an equivalence of categories Rep(CQ/I) ∼= Rep(Q,R)

(see for example [1, Theorem 1.6]).

Remark 2.4.2. If A is an associative algebra then there is a natural equivalence of

categories Rep(A) ∼= A -Mod. Thus the equivalences described above imply that the

categories Rep(Q) and Rep(Q,R) are abelian. For more information on categories of

representations of quivers, see [1].

Any quiver morphism ϕ : Q → Q′ induces an algebra homomorphism ϕ : CQ →

CQ′, and so for any set of relations R′ in Q′ we can consider the preimage

ϕ−1(R′) = k∑j=1

ajτj |k∑j=1

ajϕ(τj) ∈ R′.

We will now show that if ϕ is a covering morphism andR′ is a set of relations inQ′ then

the functors ϕ∗,ϕ∗, and ϕ! restrict naturally to the subcategories of representations

satisfying the relations R′ and ϕ−1(R′). First we recall that if ϕ : Q→ Q′ is a covering

morphism and τ ′ is a path in Q′, then for any y ∈ X with ϕ(y) = h(τ ′) there is a

unique path τ in Q ending at y with ϕ(τ) = τ ′. We will denote this unique path τ

by ϕ−1(τ ′)y.

Lemma 2.4.3. Let ϕ : Q→ Q′ be a covering morphism. Then for any path τ ′ in Q′

and any representation V ∈ Rep(Q) we have

ϕ!(V )(τ ′) =⊕

y:h(τ ′)=ϕ(y)

V (ϕ−1(τ ′)y), and (2.4.1)

ϕ∗(V )(τ ′) =∏

y:h(τ ′)=ϕ(y)

V (ϕ−1(τ ′)y). (2.4.2)

2. Quivers 20

Proof: We begin by noting that since ϕ is a covering morphism, for any path τ ′

in Q′ there is a bijection between paths τ in Q with ϕ(τ) = τ ′ and vertices x ∈ X

with ϕ(x) = h(τ ′) given by x↔ ϕ−1(τ ′)x. Let τ ′ be a path in Q′. We will proceed by

induction on the length of the path, which we denote by n. If n = 0, then the path

τ ′ is a trivial path at some vertex y′ in Q′. Then for any vertex y ∈ ϕ−1(y′), ϕ−1(τ ′)y

is the trivial path at y. Hence we have

ϕ!(V )(τ ′) = idϕ!(V )(y′) =⊕

y∈ϕ−1(y′)

idV (y) =⊕

y:h(ρ′)=ϕ(y)

V (ϕ−1(ρ′)y).

Now let n ≥ 1 and write τ ′ = α′ρ′, where α′ is a path in Q′ of length n − 1 and

ρ′ ∈ A′. Then if we assume the result is true for the path α′, we have

ϕ!(V )(α′)ϕ!(V )(ρ′) =⊕

y:h(α′)=ϕ(y)

V (ϕ−1(α′)y)⊕

x:h(ρ′)=ϕ(x)

V (ϕ−1(ρ′)x). (2.4.3)

Since ϕ is a covering morphism, |α | ϕ(α) = α′| = |τ | ϕ(τ) = τ ′| as both

sets are in bijection with the set of vertices in X that get mapped to h(α′) = h(τ ′).

Moreover, for any y ∈ X with ϕ(y) = h(τ ′), if we let x = t(ϕ−1(α′)y) then we have

ϕ−1(τ ′)y = ϕ−1(α′)yϕ−1(ρ′)x. It follows that each ϕ−1(τ ′)y shows up exactly once

in the decomposition (2.4.3) above. On the other hand, if x ∈ X is a vertex such

that the arrow ϕ−1(ρ′)x cannot be extended to a path which maps to τ ′ under ϕ,

then in the decomposition (2.4.3) V (x) is mapped to 0. We can therefore ignore such

components, and we conclude that

ϕ!(τ′) =

⊕y:h(τ ′)=ϕ(y)

V (ϕ−1(τ ′)y),

from which the result follows by induction. To show that

ϕ∗(τ′) =

∏y:h(τ ′)=ϕ(y)

V (ϕ−1(τ ′)y)

is similar.

2. Quivers 21

Proposition 2.4.4. Let Q,Q′ be quivers, R′ a set of relations in Q′, and ϕ : Q→ Q′

a covering morphism.

(i) If V ∈ Rep(Q′, R′), then ϕ∗(V ) ∈ Rep(Q,ϕ−1(R′)).

(ii) If V ∈ Rep(Q,ϕ−1(R′)) then ϕ!(V ) ∈ Rep(Q′, R′) and ϕ∗(V ) ∈ Rep(Q′, R′).

Proof:

(i) Let∑k

j=1 ajτ′j ∈ R′. Then since V ∈ Rep(Q′, R′), we have

∑kj=1 ajV (τ ′j) = 0.

Hence for any y ∈ X with ϕ(y) = h(τ ′j) we have

k∑j=1

ajϕ∗(V )(ϕ−1(τ ′j)y) =

k∑j=1

ajV (τ ′j) = 0.

It follows that ϕ∗(V ) ∈ Rep(Q,ϕ−1(R′)).

(ii) Once again, suppose∑k

j=1 ajτ′j ∈ R′. Let V ∈ Rep(Q,ϕ−1(R′)) and let h(τ ′j) =

y′ for every j. Then using Lemma 2.4.3 we get:

k∑j=1

ajϕ!(V )(τ ′j) =k∑j=1

aj⊕

y:y′=ϕ(y)

V (ϕ−1(τ ′j)y)

=⊕

y:y′=ϕ(y)

k∑j=1

ajV (ϕ−1(τ ′j)y)

= 0.

It follows that ϕ!(V ) ∈ Rep(Q′, R′). To show that ϕ∗(V ) ∈ Rep(Q′, R′) is

similar.

We will denote by ϕ∗R′ the restriction of the functor ϕ∗ to the subcategory

Rep(Q′, R′). Similarly, we denote by ϕR′

! and ϕR′∗ the restrictions of the functors

2. Quivers 22

ϕ! and ϕ∗ to the subcategory Rep(Q,ϕ−1(R′)). By the results of Section 2.3, for

any set of relations R′ in Q′ the functors ϕ∗R′ , ϕR′

! , and ϕR′∗ are all additive, exact,

and both ϕR′

! and ϕR′∗ are faithful. Moreover, Theorem 2.3.9 implies that ϕ∗R′ is left

adjoint to ϕR′∗ and right adjoint to ϕR

′

! .

2.5 Graded Dimension

In [8], Lusztig introduced certain varieties associated to quivers, now called Lusztig’s

nilpotent quiver varieties. These varieties were used to introduce a canonical basis

of the lower half of the universal enveloping algebra associated to a Kac-Moody Lie

algebra. These quiver varieties have now become an important tool in the study of

Lie algebras and their representations, and so a natural question to ask is how the

results of this chapter translate into the language of these varieties. While a thorough

investigation of this relationship is outside the scope of the current work, we provide

a starting point for such questions here. More precisely, we show that the restriction

functor and the left and right extension functors associated to a quiver morphism

induce a morphism between the associated quiver varieties, and that the image of

these morphisms are subvarieties of their codomains. It would be fruitful to better

understand the images of these maps, and this provides a potential avenue for further

study. For example, it would be useful to know when the images of these maps are

isomorphic to irreducible components of a Lusztig quiver variety, as these components

correspond to the elements of the canonical basis alluded to earlier. We begin with

some definitions which lead to the description of the Lusztig quiver varieties, and

then we examine how the functors described in Section 2.4 behave with respect to

this construction.

Definition 2.5.1 (Graded Dimension). Let Q = (X,A) and let V ∈ Rep(Q) be

such that dimV (x) < ∞ for all x ∈ X. The graded dimension of V is the function

2. Quivers 23

α : X → N defined by α(x) = dimV (x). If α is the graded dimension of V , we write

dimV = α. We say α is of finite type if∑

x∈X α(x) < ∞, and we say V is finite

dimensional if dimV is of finite type.

We will denote the set of all functions α : N → X by NX . Let α ∈ NX and let

V ∈ Rep(Q) be such that dimV = α. Then for any x ∈ X, we have V (x) ∼= Cα(x),

and so by fixing bases for each V (x) we can identify V with an element of the space

EQα :=

⊕ρ∈A

Hom(Cα(t(ρ)),Cα(h(ρ))).

We call EQα the representation space of dimension α of Q. Of course, EQ

α is isomorphic

to the space⊕

ρ∈A Matα(t(ρ))×α(h(ρ))(C), where Matn×m(C) denotes the space of n×m

matrices with complex entries.

For any α ∈ NX , we define the group

GL(α) :=∏x∈X

GLα(x)(C),

where GLn(C) denotes the group of automorphisms of Cn. Then GL(α) acts on the

space EQα via (g · T )ρ = gh(ρ)Tρg

−1t(ρ) for all ρ ∈ A, T ∈ EQ

α . This action is illustrated

by the commutativity of the following diagram

Cα(t(ρ)) Tρ−−−→ Cα(h(ρ))

gt(ρ)

y ygh(ρ)Cα(t(ρ)) −−−→

(g·T )ρCα(h(ρ)).

Clearly, two elements T, S ∈ EQα are isomorphic when thought of as representations

of Q if and only if they lie in the same GL(α)-orbit.

Now suppose ϕ : Q→ Q′ is a quiver morphism, and let V be a representation of

Q′ having graded dimension α. Then for any x ∈ X we have

dimϕ∗(V )(x) = dimV (ϕ(x)) = α(ϕ(x)),

2. Quivers 24

and hence dimϕ∗(V ) = αϕ. In other words, ϕ∗ induces a map EQ′α → EQ

αϕ which is

obviously linear.

For any α ∈ NX of finite type, define ϕ!(α) ∈ NX′ by ϕ!(α)(x′) =∑

x∈ϕ−1(x′) α(x).

Then if T ∈ EQα , we have dimϕ!(T )(x′) = ϕ!(α)(x′), and hence ϕ! induces a map

EQα → EQ′

ϕ!(α).

Due to the results of Section 2.4, this discussion extends to the case of quivers

with relations. That is, if R is a set of relations in a quiver Q, then we can identify

representations V ∈ Rep(Q,R) with elements of the space

ERα := T ∈ EQ

α | T satisfies the relations R.

Then if R′ is a set of relations in Q′, ϕ∗R′ gives a linear map ER′α → E

ϕ−1(R′)αϕ , and ϕR

′

!

gives a linear map Eϕ−1(R′)α → ER′

ϕ!(α).

We now examine a specific example of this. We start with a quiver morphism

ϕ : Q → Q′, and suppose Q′ =←→Q∗ for some quiver Q∗. That is, Q′ is the double

quiver of some other quiver. We will consider certain subsets of the spaces ER′α .

Definition 2.5.2 (Nilpotent Representation). A representation V of a quiver Q =

(X,A) is said to be nilpotent if there exists some N ∈ N such that for any path of

the form ρN · · · ρ1 in Q, where ρi ∈ A for all i, we have V (ρN · · · ρ1) = 0.

Now let R′ be the set of Gelfand-Ponomarev relations (2.1.1) in Q′. We will

study the sets

ΛQ∗

α := T ∈ ER′

α | T is nilpotent.

These objects were introduced by Lusztig [8], and are known as Lusztig’s (nilpotent)

quiver varieties. Later on we will show that if ϕ is a covering morphism, then ϕ!(T )

is nilpotent if T is nilpotent. Hence ϕ! gives a map from the set of nilpotent elements

of Eϕ−1(R′)α , which we denote by N (E

ϕ−1(R′)α ), into the Lusztig quiver variety ΛQ∗

ϕ!(α).

Let n = dimEQα . If we fix a basis of EQ

α , then we can identify EQα with n-

dimensional affine space An using coordinates with respect to this basis. We will

2. Quivers 25

usually consider elements of EQα to be matrices and choose as our basis the set

eρij | 1 ≤ i ≤ α(t(ρ)), 1 ≤ j ≤ α(h(ρ))ρ∈A, where eρij denotes the α(t(ρ)) × α(h(ρ))

matrix whose (i, j)-entry is equal to 1, and all other entries are 0. Under this iden-

tification, the condition that a representation T ∈ EQα satisfies a given relation in

Q becomes a system of polynomial equations in the affine coordinates of T . Thus

Eϕ−1(R′)α is a Zariski-closed subset of An. In fact, N (E

ϕ−1(R′)α ) is also a Zariski-closed

subset of An as it is an intersection of closed sets.

Note that if α is of finite type, so is ϕ!(α), and moreover n = dimEQα ≤

dimEQ′

ϕ!(α) =: m. Thus we have:

EQ′

ϕ!(α) =⊕ρ′∈A′

Hom(Cϕ!(α)(t(ρ′)),Cϕ!(α)(h(ρ′)))

=⊕ρ′∈A′

Hom

⊕x:t(ρ′)=ϕ(x)

Cα(x),⊕

y:h(ρ′)=ϕ(y)

Cα(y)

=⊕ρ′∈A′

⊕x:t(ρ′)=ϕ(x)y:h(ρ′)=ϕ(y)

Hom(Cα(x),Cα(y))

=⊕ρ′∈A′

⊕ρ:ρ′=ϕ(ρ)

Hom(Cα(t(ρ)),Cα(t(ρ)))⊕

x:t(ρ′)=ϕ(x)y:h(ρ′)=ϕ(y)@ρ:x→y

Hom(Cα(x),Cα(y))

= EQα ⊕

⊕ρ′∈A′

⊕x:t(ρ′)=ϕ(x)y:h(ρ′)=ϕ(y)@ρ:x→y

Hom(Cα(x),Cα(y)).

In particular, EQα is naturally isomorphic to a subspace of EQ′

ϕ!(α). Hence we may

complete the set eρij mentioned above to a basis of EQ′

ϕ!(α), and thereby identify

EQ′

ϕ!(α) with Am. Under these identifications, the map ϕR′

! : N (Eϕ−1(R′)α )→ ΛQ∗

ϕ!(α) is the

restriction of the canonical injection An → Am to the variety N (EQ′

ϕ!(α)). This shows

that ϕR′

! is a morphism of affine varieties. In fact, we claim that ϕR′

! (N (Eϕ−1(R′)α ))

is isomorphic to a subvariety of ΛQ∗

ϕ!(α). Indeed, consider the pullback of ϕR′

! between

2. Quivers 26

the coordinate rings of N (Eϕ−1(R′)α ) and ΛQ∗

ϕ!(α):

C[ΛQ∗

ϕ!(α)]→ C[N (Eϕ−1(R′)α )]

f 7→ f ϕR′! .

If f ∈ C[N (Eϕ−1(R′)α )], then fix a representative of the coset of f (under the identifica-

tion (B.1)) and write f = [g], where g ∈ C[x1, . . . , xn]. Let g 7→ g′ under the canonical

injection C[x1, . . . , xn] → C[x1, . . . , xm] and define f ′ ∈ C[ΛQ∗

ϕ!(α)] by f ′ = [g′]. Then

f ′ ϕR′! = [g′ ϕR′! (x1, . . . , xn)] = [g′(x1, . . . , xn, 0, . . . , 0] = [g(x1, . . . , xn)] = f,

and therefore the pullback between the coordinate rings is surjective. It follows that

the image of the morphism ϕR′

! is isomorphic to a subvariety of ΛQ∗

ϕ!(α) by Lemma B.

We will now give a concise summary of how the functors ϕ∗, ϕ∗, and ϕ! act with

respect to certain subcategories of Rep(Q) and Rep(Q′).

First we consider the subcategories Rep(Q,ϕ−1(R′)) and Rep(Q′, R′) where R′ is

a set of relations in Q′. In Proposition 2.4.4, we showed that ϕ∗ maps Rep(Q′, R′) to

Rep(Q,ϕ−1(R′)) and that the functors ϕ∗ and ϕ! map Rep(Q,ϕ−1(R′)) to Rep(Q′, R′).

For any quiver Q, we will denote by NRep(Q) the subcategory of Rep(Q) consisting

of nilpotent representations of Q. We claim that the functors induced by quiver mor-

phisms on the categories of representations respect the property of being nilpotent.

Proposition 2.5.3. (i) If V ∈ NRep(Q′), then ϕ∗(V ) ∈ NRep(Q).

(ii) If V ∈ NRep(Q), then ϕ!(V ), ϕ∗(V ) ∈ NRep(Q′).

Proof:

(i) Let V ∈ NRep(Q′) and suppose ϕ∗(V ) were not nilpotent. Then for any m ∈ N,

there is some path ρ1 . . . ρm in Q with ϕ∗(V )(ρ1 . . . ρm) 6= 0. But then we have

0 6= V (ρ′1 . . . ρ′m), where ρ′i = ϕ(ρi) for each i = 1, . . . ,m, which contradicts the

fact that V is nilpotent.

2. Quivers 27

(ii) Let V ∈ NRep(Q). Let N be such that for any path τ of length N in Q,

V (τ) = 0. Then for any path τ ′ of length N in Q′, for any vertex x ∈ X, the

path ϕ−1(τ)x is also of length N . It follows from Lemma 2.4.3 that we have

ϕ!(V )(τ ′) =⊕

x:h(τ ′)=ϕ(x) V (ϕ−1(τ ′)x) = 0. The proof for ϕ∗ is similar.

We conclude that ϕ∗ maps NRep(Q′, R′) to NRep(Q,ϕ−1(R′)) and that ϕ! and

ϕ∗ map NRep(Q,ϕ−1(R′)) to NRep(Q′, R′), where NRep(Q,R) denotes the category

of nilpotent representations of Q satisfying the relations R.

Next we consider the subcategories Repfd(Q) and Repfd(Q′) consisting of finite

dimensional representations of Q and Q′ respectively.

Lemma 2.5.4. If V ∈ Repfd(Q) then ϕ!(V ), ϕ∗ ∈ Repfd(Q′).

Proof: If V ∈ Repfd(Q) and dimV = α, then the graded dimension of both ϕ!(V )

and ϕ∗(V ) is given by ϕ!(α)(x′) =∑

x∈ϕ−1(x′) α(x) for all x′ ∈ X ′. The result follows.

On the other hand, it is not difficult to see that ϕ∗ does not map Repfd(Q′)

to Repfd(Q) in general. Indeed, suppose ϕ is a morphism having the property that

there is some vertex x′ ∈ Q′ such that there are infinitely many vertices x ∈ X with

ϕ(x) = x′. If V is a finite dimensional representation of Q′ such that dimV (x′) > 0,

then ϕ∗(V ) will not be finite dimensional. This motivates the definition of locally

finite representations.

Definition 2.5.5 (Locally Finite Representation). For a quiver Q = (X,A), we will

say a representation V ∈ Rep(Q) is locally finite if V (x) is a finite dimensional vector

space for every x ∈ X. We denote by Replf(Q) the subcategory of Rep(Q) consisting

of locally finite representations.

2. Quivers 28

Lemma 2.5.6. If V ∈ Replf(Q′) then ϕ∗(V ) ∈ Replf(Q).

Proof: If dimV (x′) <∞ for every x′ ∈ X ′, then dimϕ∗(V )(x) = dimV (ϕ(x)) <

∞ for every x ∈ X, and so ϕ∗ preserves the property of being locally finite.

The extension functors ϕ! and ϕ∗, however, do not necessarily preserve the prop-

erty of being locally finite. Again, if ϕ maps infinitely many vertices of Q to a single

vertex in Q′, it is easy to construct a locally finite representation V of Q such that

ϕ!(V ) is not locally finite.

The discussion in the two preceeding paragraphs is summarized in the tables

below. The entries in the tables represent the subcategory of Rep(Q) in which the

image of representations having the properties specified by the row and column live.

ϕ∗ Satisfies Relations R′ Nilpotent Both

Finite dimensional Replf(Q,ϕ−1(R′)) NReplf(Q) NReplf(Q,ϕ

−1(R′))

Locally finite Replf(Q,ϕ−1(R′)) NReplf(Q) NReplf(Q,ϕ

−1(R′))

Neither Rep(Q,ϕ−1(R′)) NRep(Q) NRep(Q,ϕ−1(R′))

ϕ∗, ϕ! Satisfies Relations ϕ−1(R′) Nilpotent Both

Finite dimensional Repfd(Q,R′) NRepfd(Q) NRepfd(Q,R′)

Locally finite Rep(Q,R′) NRep(Q) NRep(Q,R′)

Neither Rep(Q,R′) NRep(Q) NRep(Q,R′)

We remark that all of the properties discussed in Section 2.3 (adjointness, faith-

fulness, exactness) are unaffected by the restriction of a functor to a subcategory.

Thus we may use the results of that section even when we are considering the action

of ϕ∗, ϕ! or ϕ∗ on one of the subcategories mentioned above.

Chapter 3

Lie Algebras

We begin this chapter with a brief review of some elementary notions of Lie algebras,

and recall the classification of three-dimensional Lie algebras over C. We then focus

on a specific infinite family of non-isomorphic three-dimensional Lie algebras, and

introduce modified versions of their universal enveloping algebras.

3.1 Elementary Notions

A Lie algebra over a field F is an F -vector space, L, together with a bilinear map,

the Lie bracket,

L× L→ L

(x, y) 7→ [x, y]

satisfying the following properties:

[x, x] = 0 ∀x ∈ L,

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 ∀x, y, z ∈ L (the Jacobi identity).

A Lie algebra is said to be abelian if L = Z(L), where

Z(L) = x ∈ L | [x, y] = 0 ∀y ∈ L

29

3. Lie Algebras 30

is the centre of L. We define the derived algebra of a Lie algebra L as

L′ = Span[x, y] | x, y ∈ L.

Given two Lie algebras, L1 and L2, a map ϕ : L1 → L2 is a Lie algebra ho-

momorphism if ϕ([x, y]) = [ϕ(x), ϕ(y)] for every x, y ∈ L1. An important Lie al-

gebra homomorphism is the adjoint homomorphism, ad : L → End(L) defined by

(adx)(y) := [x, y].

By a representation of a Lie algebra L, we shall mean a homomorphism ϕ : L→

gl(V ) for some vector space V . Here gl(V ) denotes the Lie algebra of endomorphisms

of V with Lie bracket given by [x, y] = xy − yx for all x, y ∈ gl(V ). If ϕ : L→ gl(V )

is a representation of a Lie algebra L, then V is an L-module with action given by

x · v = ϕ(x)v for all v ∈ V . By the usual abuse of terminology, we will often use the

terms representation and module interchangeably.

3.2 Lie Algebras of Low Dimension

In this section we study complex Lie algebras of dimension at most 3. We review

the classification of such Lie algebras, and then study one particular family of 3-

dimensional Lie algebras. This family depends on a continuous parameter µ ∈ C,

and we denote its members by Lµ. The Lie algebras Lµ are not as well understood

as the other 3-dimensional complex Lie algebras, and their representation theory is

the subject of the current paper.

First, we note that every 1-dimensional Lie algebra is abelian. It is not difficult

to see that two abelian Lie algebras are isomorphic if and only if they have the

same dimension, and hence all 1-dimensional Lie algebras are isomorphic. In the

2-dimensional case, there is a unique (up to isomorphism) non-abelian Lie algebra,

which has a basis x, y such that its Lie bracket is described by [x, y] = x. See for

example [3, Theorem 3.1] for a proof of this statement.

3. Lie Algebras 31

In dimension 3, all cases of non-abelian Lie algebras can be classified by relating

L′ to Z(L). First we will consider the cases dimL′ = 1 and dimL′ = 3.

Lemma 3.2.1 ([3, Section 3.2]). There are unique 3-dimensional Lie algebras having

the following properties:

(i) dimL′ = 1 and L′ ⊆ Z(L). This Lie algebra is known as the Heisenberg algebra.

(ii) dimL′ = 1 but L′ * Z(L). This Lie algebra is the direct sum of the 2-

dimensional non-abelian Lie algebra with the 1-dimensional Lie algebra.

(iii) L′ = L. In this case, L = sl(2,C), the Lie subalgebra of gl(2,C) := gl(C2)

consisting of trace zero operators.

We will now consider the case where dimL = 3 and dimL′ = 2. Let y, z be

a basis of L′ and let x ∈ L \ L′. We now have a basis of L, x, y, z. We need the

following lemma.

Lemma 3.2.2 ([3, Lemma 3.3]). Let L be a Lie algebra such that dimL = 3 and

dimL′ = 2, and let x ∈ L \ L′. Then:

(i) L′ is abelian.

(ii) The map adx : L′ → L′ is an isomorphism.

We can separate Lie algebras L having the properties dimL = 3 and dimL′ = 2

into two cases. The first happens when there is some β ∈ L\L′ such that ad β : L′ →

L′ is diagonalisable. The second case occurs if adx : L′ → L′ is not diagonalisable for

any x ∈ L \ L′.

In the latter case, we get that the Jordan canonical form of the matrix of adx

must be a single 2 × 2 Jordan block. Since adx is an isomorphism it must have an

3. Lie Algebras 32

eigenvector with nonzero eigenvalue, and by proper scaling of x we may assume that

this eigenvalue is 1. Thus the Jordan form of the matrix of adx acting on L′ is1 1

0 1

,

which completely determines the Lie algebra L.

The case of interest for us will be the one where there is some β /∈ L′ such

that the map ad β : L′ → L′ is diagonalisable. We will choose basis α1, α2 of L′

consisting of eigenvectors of ad β. Then the associated eigenvalues of α1 and α2 must

be nonzero by part (ii) of Lemma 3.2.2.

Since α1 and α2 are eigenvectors of ad β, [β, α1] = ηα1 and [β, α2] = µα2 for some

η, µ ∈ C∗. We have [η−1β, α1] = α1, therefore, with proper scaling, we may assume

that η = 1. With respect to the basis α1, α2, ad β : L′ → L′ has matrix1 0

0 µ

. (3.2.1)

We will call this Lie algebra Lµ. Furthermore, by [3, Exercise 3.2] we can see

that Lµ ∼= Lν ⇔ µ = ν or µ = ν−1. This is true for all µ ∈ C∗. First we will focus on

the case where µ ∈ Q∗. The matrix of adx is1 0

0 nm

,

where µ = nm

. With a simple change of basis we getm 0

0 n

. (3.2.2)

Therefore, Lµ has basis α1, β, α2 and commutation relations

[β, α1] = mα1, [β, α2] = nα2, [α1, α2] = 0. (3.2.3)

3. Lie Algebras 33

Remark 3.2.3. A special case of this algebra occurs when µ = −1 and we get L−1

which is the Euclidean algebra. See [12, Section 2] for a discussion of this Lie algebra

in the same context as the current paper.

Remark 3.2.4. Since µ−1 = ( nm

)−1 = mn

, we have L nm

∼= Lmn

.

On the other hand, if µ ∈ C \Q then by (3.2.1) the commutation relations are

[β, α1] = α1, [β, α2] = µα2, [α1, α2] = 0. (3.2.4)

3.3 The Universal Enveloping Algebra of Lµ and

its Representations

3.3.1 Universal Enveloping Algebras

If L is a Lie algebra, then the universal enveloping algebra of L is the pair (U, i),

where U is a unital associative algebra and i : L→ U is a map satisfying

i([x, y]) = i(x)i(y)− i(y)i(x) ∀x, y ∈ L. (3.3.1)

Moreover, the pair (U, i) is universal with respect to this property. That is, for

any pair (U ′, i′) satisfying (3.3.1) there exists a unique homomorphism ϕ : U → U ′

such that ϕi = i′. Since we have defined the universal enveloping algebra by a

universal property, it is unique up to isomorphism. Furthermore, this definition makes

it clear that the category of representations of a Lie algebra L is equivalent to the

category of representations of its universal enveloping algebra U . If we denote by T

the tensor algebra of the Lie algebra L and by IT the two sided ideal of T generated

by all elements of the form x⊗ y− y⊗ x− [x, y], x, y ∈ L, then it can be shown that

U ∼= T/IT . If xi | i ∈ I is a basis of L ordered by some indexing set I, then the

set xi1 · · ·xin | i1 ≤ · · · ≤ in forms a basis for the space T/IT . This is known as the

Poincare-Birkhoff-Witt (PBW) Theorem. See for example [7, Section 17] for a proof

of both the isomorphism U ∼= T/IT and the PBW Theorem.

3. Lie Algebras 34

3.3.2 Rational Case

Let µ ∈ Q∗, µ = nm

with gcd(m,n) = 1. Then for any indecomposable representation

V of Lµ, the eigenvalues of the action of β on V will come from a set of the form γ+Z

for some γ ∈ C. We will write Vλ to represent the eigenspace of β with eigenvalue λ.

This gives the following eigenspace decomposition:

V =⊕

k∈Z Vγ+k, Vλ = v ∈ V | β · v = λv.

Let Um,n be the universal enveloping algebra of Lµ and let U0, U1, U2 be the

subalgebras generated by β, α1, α2 respectively. Then, by the PBW Theorem, we

have

Um,n ∼= U1 ⊗ U0 ⊗ U2 (as vector spaces). (3.3.2)

From (3.2.3) we obtain the following relations in the universal enveloping algebra:

βα1 − α1β = mα1, βα2 − α2β = nα2, α1α2 = α2α1. (3.3.3)

Let x ∈ Vλ. Then βx = λx so we have:

β(α1x) = (βα1)x

= (mα1 + α1β)x by (3.3.3)

= (λ+m)(α1x)

⇒ α1Vλ ⊆ Vm+λ.

Similarly, we find that α2Vλ ⊆ Vn+λ.

Following [12, Section 2] we will consider the modified enveloping algebra Um,n

of Lµ by replacing U0 with a sum of 1-dimensional algebras:

Um,n = U1 ⊗(⊕

k∈ZCak)⊗ U2. (3.3.4)

Multiplication in the modified enveloping algebra is given by

aka` = δk`ak,

3. Lie Algebras 35

α1ak = ak+mα1, α2ak = ak+nα2,

α1α2ak = α2α1ak, (3.3.5)

where k, ` ∈ Z. We can think of ak as the projection onto the k-th weight space

Vγ+k. For any associative algebra A, we denote the category of representations of

A by Rep(A). If we denote by wt-rep(Um,n) the category of representations of Um,n

on which β acts semisimply with eigenvalues from the set γ + Z, then we have the

equivalence of categories wt-rep(Um,n) ∼= Rep(Um,n).

3.3.3 Non-Rational Case

Using the same notation as in Section 3.3.2, we have V a representation of Lµ and

Vλ the eigenspace of β with eigenvalue λ. However, when µ ∈ C \ Q, β will act on

indecomposable representations of Lµ with eigenvalues of the form γ + k for some

γ ∈ C, where k ∈ Z+ Zµ. Therefore, we have V =⊕

k∈Z+Zµ Vγ+k.

When µ ∈ C \ Q, we get the same decomposition of the universal enveloping

algebra as found in (3.3.2). However, the relations found in (3.3.3) become

βα1 − α1β = α1, βα2 − α2β = µα2, α1α2 = α2α1. (3.3.6)

As in the rational case, we find that α1Vλ ⊆ Vλ+1 and α2Vλ ⊆ Vλ+µ.

Again we consider the modified enveloping algebra, Uµ, in this case given by

Uµ = U1 ⊗(⊕

k∈Z+µZCak)⊗ U2. (3.3.7)

Multiplication in this modified enveloping algebra is given by

aka` = δk`ak,

α1ak = ak+1α1, α2ak = ak+µα2,

α1α2ak = α2α1ak, (3.3.8)

3. Lie Algebras 36

where k, ` ∈ Z + µZ. Since µ ∈ C \ Q, we have Z + µZ ∼= Z × Z, so we can reindex

the projections ak by defining aij = ai+jµ. In this notation the modified enveloping

algebra has the form

Uµ = U1 ⊗(⊕

i,j∈ZCaij)⊗ U2. (3.3.9)

The multiplication is given by:

aijast =

aij, if i = s, j = t,

0, otherwise,

α1aij = a(i+1)jα1, α2aij = ai(j+1)α2,

α1α2aij = α2α1aij. (3.3.10)

Remark 3.3.1. The importance of this last point is that we have eliminated any

dependence on µ. This shows that when µ ∈ C \Q the modified enveloping algebras

of all the Lie algebras Lµ are isomorphic. Thus from now on we will denote Uµ simply

by U when µ is not rational.

Again we have an equivalence of categories wt-rep(Uµ) ∼= Rep(U), where Uµ is the

universal enveloping algebra of Lµ. Note that the categories of weight representations

of Lµ are all equivalent whenever µ ∈ C \Q.

Chapter 4

The Quivers Qm,n and Q∞×∞

In this chapter we relate the quivers Qm,n and Q∞×∞ to the modified enveloping

algebras of the Lie algebras Lµ and use this relationship to study the representation

theory of these Lie algebras.

4.1 Relation to the Lie Algebras Lµ

In this section we prove how the modified enveloping algebra Um,n is related to the

path algebra CQm,n. We have seen in Section 3.3.2 that a basis of Um,n is given by the

elements αr1akαs2, and so in order to establish a connection between the two algebras,

we need a way of associating a path in Qm,n to such an element. Recall that in Um,n,

the operator ak corresponds to projection onto the kth weight space, α1 moves the

weight i to the weight i + m, and α2 moves the weight i to the weight i + n. Since

the vertices of Qm,n correspond to the weights in Um,n, the idea is to map αr1akαs2 to

the path in Qm,n pictured below.

... ...

k-sn k-(s-1)n k-n k k+m k+(r-1)m k+rm

37

4. The Quivers Qm,n and Q∞×∞ 38

Of course, such paths do not form a basis of CQm,n, and so this map will not be

surjective. In order to obtain an isomorphism, we need to quotient by a certain ideal

in CQm,n. Loosely speaking, this quotient identifies the path above with any path

with the same endpoints that travels along upper arrows r times and lower arrows s

times. That is, the equivalence class is uniquely determined by r, s, and k, and hence

will establish an isomorphism with Um,n. A similar strategy can be employed in the

non-rational case, but since the proof runs analogously to the rational case, we do

not include the details here.

4.1.1 Rational Case

Consider the linear map ϕ determined by:

ϕ : CQm,n → U ,

ρk+σ(j1)+···+σ(js−1)js

· · · ρkj1 7→ αjs · · ·αj1ak. (4.1.1)

Lemma 4.1.1. The map ϕ is a homomorphism of algebras.

Proof: Since ϕ is linear, it suffices to show that it commutes with the multiplica-

tion. Let


· · · ρkj1 , ρl+σ(i1)+···+σ(ir−1)ir

· · · ρli1 ∈ CQm,n.

Then

(ρk+σ(j1)+···+σ(js−1)js

· · · ρkj1) · (ρl+σ(i1)+···+σ(ir−1)ir

· · · ρli1)

=


· · · ρkj1ρl+σ(i1)+···+σ(ir−1)ir

· · · ρli1 if l + σ(i1) + · · ·+ σ(ir) = k

0 otherwise

=⇒ ϕ(

(ρk+σ(j1)+···+σ(js−1)js

· · · ρkj1) · (ρl+σ(i1)+···+σ(ir−1)ir

· · · ρli1))

=

αjs · · ·αj1αir · · ·αi1al if l + σ(i1) + · · ·+ σ(ir) = k,

0 otherwise.


On the other hand,

ϕ(ρk+σ(j1)+···+σ(js−1)js

· · · ρkj1)ϕ(ρl+σ(i1)+···+σ(ir−1)ir

· · · ρli1)

= (αjs · · ·αj1ak)(αir · · ·αi1al)

= αjs · · ·αj1αirak−σ(ir)αir−1 · · ·αi1al

= αjs · · ·αj1αirαir−1ak−σ(ir)−σ(ir−1)αir−2 · · ·αi1al...

= αjs · · ·αj1αir · · ·αi1ak−σ(ir)−···−σ(i1)al

=

αjs · · ·αj1αir · · ·αi1al if l = k − σ(i1)− · · · − σ(ir),

0 otherwise.

And hence

ϕ(

(ρk+σ(j1)+···+σ(js−1)js

· · · ρkj1) · (ρl+σ(i1)+···+σ(ir−1)ir

· · · ρli1))

= ϕ(ρk+σ(j1)+···+σ(js−1)js

· · · ρkj1)ϕ(ρl+σ(i1)+···+σ(ir−1)ir

· · · ρli1).

We now establish a relationship between the path algebra CQm,n and the modified

enveloping algebra Um,n.

Proposition 4.1.2. For any m,n ∈ Z∗ there is an isomorphism of algebras Um,n ∼=

CQm,n/Im,n, where Im,n is the two-sided ideal generated by elements of the form

ρk+n1 ρk2 − ρk+m

2 ρk1.

Proof: We claim that Im,n ⊆ kerϕ. Indeed:

ϕ(ρk+n1 ρk2 − ρk+m

2 ρk1) = ϕ(ρk+n1 ρk2)− ϕ(ρk+m

2 ρk1)

= α1α2ak − α2α1ak

= α1α2ak − α1α2ak (by (3.3.5))

= 0


=⇒ ρk+n1 ρk2 − ρk+m

2 ρk1 ∈ kerϕ ∀k ∈ Z.

Thus ϕ induces a morphism

ϕ : CQm,n/Im,n → U ,

x+ I 7→ ϕ(x) ∀x ∈ CQm,n.

We will also consider the linear map determined by:

ψ : U → CQm,n/Im,n

α1rakα2

s 7→ ρk+(r−1)m1 · · · ρk1ρk−n2 · · · ρk−sn2 + I.

We will show that ψϕ and ϕψ are identity maps. Seeing as both maps are linear, it will

suffice to show that this is the case for basis elements. Let ρk+σ(j1)+···+σ(js−1)js

· · · ρkj1 +

I ∈ CQ/Im,n. Then

(ψϕ)(ρk+σ(j1)+···+σ(js−1)js

· · · ρkj1 + I)

= ψ(ϕ(ρ

k+σ(j1)+···+σ(js−1)js

· · · ρkj1 + I))

= ψ(ϕ(ρ

k+σ(j1)+···+σ(js−1)js

· · · ρkj1))

= ψ(αjs · · ·αj1ak)

= ψ(αr1αt2ak), for some r + t = s

= ψ(αr1ak+tnαt2)

= ρk+tn+(r−1)m1 · · · ρk+tn

1 ρk+(t−1)n2 · · · ρk2 + I

= ρk+σ(j1)+···+σ(js−1)js

· · · ρkj1 + I

=⇒ ψϕ = id : CQm,n/Im,n → CQm,n/I

m,n.

In the last line of the computation we used the relation ρk+n1 ρk2 − ρk+m

2 ρk1 ∈ Im,n to

reorder the terms into a new member of the same equivalence class.

Now let α1rakα2

s ∈ U . Then

(ϕψ)(α1rakα2

s) = ϕ (ψ(α1rakα2

s))


= ϕ(ρk+(r−1)m1 · · · ρk1ρk−n2 · · · ρk−sn2 + I

)= ϕ

(ρk+(r−1)m1 · · · ρk1ρk−n2 · · · ρk−sn2

)= α1

rα2sak−sn

= α1rakα2

s

=⇒ ϕψ = id : U → U .

Combining this result with Lemma 4.1.1, we see that ϕ is a bijective homomor-

phism of algebras, which completes the proof.

Thus there is an equivalence of categories Rep(Um,n) ∼= Rep(CQm,n/Im,n).


Consider the linear map Ω defined by:

Ω : CQ∞×∞ → U ,

ρk+θ(d1)+···+θ(ds−1)ds

· · · ρkd1 7→ αds · · ·αd1aij, (4.1.2)

where k = (i, j).

Lemma 4.1.3. The map Ω is a homomorphism of algebras.

Proof: The proof is similar to that of Lemma 4.1.1.

Proposition 4.1.4. There is an isomorphism of algebras U ∼= CQ∞×∞/I∞×∞ where

I∞×∞ is the two-sided ideal generated by elements of the form ρk+(0,1)1 ρk2 − ρ

k+(1,0)2 ρk1,

where k ∈ Z× Z.

Proof: The proof is similar to that of Proposition 4.1.2.

So we have an equivalence of categories Rep(U) ∼= Rep(CQ∞×∞/I∞×∞).


4.2 Representation Theory

4.2.1 Rational Case

Let µ = nm

, gcd(m,n) = 1. We have the equivalences

Rep(Um,n) ∼= Rep(CQm,n/Im,n) ∼= Rep(Qm,n, R

m,n),

where Rm,n = ρk+n1 ρk2 − ρk+m

2 ρk1 | k ∈ Z. If we define R = ρi+1ρi − ρi−1ρi | i ∈

Z/sZ, then we can study the representations of Um,n by relating the category

Rep(Qm,n, Rm,n) to the category Rep(Qm+n, R), where Qs is the quiver defined in

Example 2.3.3. This is an interesting connection, and moreover much is known about

the category Rep(Qm+n, R), see [4] for example.

Let V = (Vk, V (ρki )) ∈ Rep(Qm,n) and let j ∈ Z/(m+n)Z. If k ∈ Z, we will write

k ≡ j mod (m+n) simply as k ≡ j. Then if g ∈ Hom(Qm,n, Qm+n) is the morphism

described in Example 2.3.3, the representation g!(V ) has vector spaces given by

g!(V )(j) =⊕

k≡jm Vk.

The linear maps of the representation are given by g!(V )(ρj) =⊕

k≡jm V (ρk1)

and g!(V )(ρj) =⊕

k≡jm V (ρk2). Note that g!(V )(ρj) maps g!(V )(j) to g!(V )(j + 1),

and g!(V )(ρj) maps g!(V )(j) to g!(V )(j − 1). Moreover, if V, U ∈ Rep(Qm,n) and

ϕ ∈ HomQm,n(V, U) then g!(ϕ) = g!(ϕ)(j) : g!(V )(j) → g!(U)(j), where g!(ϕ)(j) =⊕k≡jm ϕk.

Let V, U ∈ Rep(Qm,n). Then g!(V ) g!(U) ⇒ V U since g! is a functor.

Further, it follows from Corollary 2.3.11 that if g!(V ) is indecomposable then V

is indecomposable, since additive functors preserve finite coproducts, which in the

categories Rep(Qm,n) and Rep(Qm+n) are finite direct sums.

Note that the preimage of the relations Rm+n in Qm+n under g! are exactly

the relations Rm,n, that is, g−1! (Rm+n) = Rm,n. Thus, by Proposition 2.4.4, we can

restrict the functor g! to the subcategory Rep(Qm,n, Rm,n) of Rep(Qm,n) to get a


functor gRm+n

! : Rep(Qm,n, Rm,n)→ Rep(Qm+n, Rm+n). Further, all of the properties

that were proven for g! still hold for the restricted functor gRm+n

! , and we can therefore

relate the categories Rep(Um,n) and Rep(Qm+n, Rm+n). It is natural to ask whether

or not gRm+n

! gives an equivalence of categories, and it turns out that this is true only

when µ = −1, in which case g! is the identity functor. When µ 6= −1, the functor

gRm+n

! is neither full nor essentially surjective, as the following examples illustrate:

Example 4.2.1. Let V ∈ Rep(Qm,n, Rm,n) be the representation given by V (0) =

V (n + m) = C, and V (i) = 0 for all other i ∈ Z. This representation is pictured in

Figure 4.1, where any vector space or linear map not pictured is assumed to be zero.

The endomorphism space of V is HomQm,n(V, V ) ∼= Hom(C,C) ⊕ Hom(C,C) ∼= C2.

C 0 0 C

Figure 4.1: The functor gRm+n

! is not full

The representation gRm+n

! (V ) ∈ Rep(Qm+n, Rm+n) is the representation such that

gRm+n

! (V )0 = C2, and all other vector spaces are zero. Thus

HomQm+n(gRm+n

! (V ), gRm+n

! (V )) ∼= Hom(C2,C2) ∼= Mat2×2(C).

Since the dimension of the endomorphism space of gRm+n

! (V ) is greater than the

dimension of the endomorphism space of V , the map gRm+n

!V V is not surjective. Hence

gRm+n

! is not full.

Example 4.2.2. Let U ∈ Rep(Qm,n, Rm,n). Then if U is finite dimensional, there

are only finitely many nonzero U(k), and so there exists some t ∈ Z such that

U(ρk+tm1 ρ

k+(t−1)m1 · · · ρk1) = 0

for any k ∈ Z. Then⊕

k≡jm U(ρk+tm1 ρ

k+(t−1)m1 · · · ρk1) = 0, and so there is a path in

the representation gRm+n

! (U) ∈ Rep(Qm+n, Rm+n) that acts by zero. Consider the


representation V ∈ Rep(Qm+n, Rm+n) defined by (V (i), V (ρi), V (ρi)) = (C, λ, 1) for

all i ∈ Z/(m + n)Z, where λ ∈ C. For m = 2 and n = 1 this representation is

pictured in Figure 4.2. Suppose there were some representation U ∈ Rep(Qm,n, Rm,n)

C

CC

λλ

λ

11

1

Figure 4.2: The functor gRm+n

! is not essentially surjective

such that gRm+n

! (U) ∼= V . Then since V is finite dimensional, U must also be finite

dimensional. However, since there are no paths in the representation V that act by

zero, this is a contradiction. Hence gRm+n

! is not essentially surjective.

If I denotes the two sided ideal of CQm+n generated by the relations R, then we

have CQm+n/I = P(Q∗m+n), where Q∗m+n is a subquiver of Qm+n obtained by omitting

the arrows ρi | i ∈ Z/(m + n)Z. The functor gRm+n

! provides an embedding of the

category Rep(Cm,n/Im,n) in Rep(P(Q∗m+n)).

Let α ∈ NZ and consider the space EQm,nα . Since the relations Rm+n are ex-

actly the Gelfand-Ponomarev relations (2.1.1) in Qm+n, the functor gRm+n

! induces

a morphism of affine varieties N (EQm,nα ) → Λ

Q∗m+n

g!(α) that identifies N (EQm,nα ) with a

subvariety of ΛQ∗m+n

g!(α) (see Section 2.5).

While the category Rep(Qm,n, Rm,n) can be quite difficult to study in general, if

we restrict our attention to representations which are supported on certain numbers

of vertices, we can obtain certain classification results.

Theorem 4.2.3. Let µ = m ∈ Z and let a, b ∈ Z be such that 0 ≤ b − a ≤ m. Let

Ca,b denote the subcategory of Rep(Qm,1, Rm,1) consisting of representations V such

that V (x) = 0 whenever x < a or x > b. Then Ca,b is of tame representation type.


Proof: Note that any representation V ∈ Rep(Qm,1) which is supported on at

most m+ 1 consecutive vertices automatically satisfies the relations Rm,1 in a trivial

way. Thus we may identify representations V ∈ Ca,b with representations of a quiver

whose underlying graph is an extended Dynkin diagram of type Am. The result then

follows from Theorem 2.2.4.

Theorem 4.2.4. Let µ = mn

, gcd(m,n) = 1, n 6= 1. Let a, b ∈ Z be such that 0 ≤

b− a ≤ m. Then there are only finitely many isomorphism classes of indecomposable

representations V ∈ Rep(Qm,n, Rm,n) such that V (x) = 0 whenever x < a or x > b.

Proof: As previously noted, any such representation V trivially satisfies the

relations Rm,n. Thus we may identify V with a representation of a quiver whose un-

derlying graph is a union of Dynkin diagrams of type A, and the result follows from

Theorem 2.2.4.


When µ ∈ C \Q we are interested in representations of the quiver Q∞×∞ introduced

in Example 2.1.2. Then we have the equivalences Rep(U) ∼= Rep(CQ∞×∞/I∞×∞) ∼=

Rep(Q∞×∞, R∞×∞), where R∞×∞ = ρk+(0,1)1 ρk2 − ρ

k+(1,0)2 ρk1 | k ∈ Z × Z. In order

to study the representations of Q∞×∞ we will relate the category Rep(Q∞×∞) to

the category Rep(Q∞). Here Q∞ denotes the quiver (Z, ρi, ρi), where t(ρi) = i =

h(ρi+1), and h(ρi) = i + 1 = t(ρi+1). We will then be able to relate the category

Rep(Q∞×∞, R∞×∞) to the category Rep(Q∞, R∞), where R∞ = ρi+1ρi−ρi−1ρi | i ∈

Z. We will obtain a relationship between these two categories which is similar to

the relationship we studied in Section 4.2.1. The representation theory of the quiver

Q∞ subject to relations R∞ is well known, see [12] for a summary of the results.


Remark 4.2.5. Given any representation V ∈ Rep(Q∞×∞, R∞×∞) and any i ∈ Z,

we can consider the representation of the quiver of type A∞ given by the ith row of

V , that is, the representation Vi ∈ Rep(A∞) with Vi(j) = V (i, j). Then the fact that

V satisfies the relations R∞×∞ implies that the collection V (ρij2 ) | j ∈ Z defines

a morphism of A∞ representations Vi → Vi+1. Thus we may think of elements of

Rep(Q∞×∞, R∞×∞) as chains of representations of the quiver of type A∞.

Let V be a representation of the quiver Q∞×∞ and let f ∈ Hom(Q∞×∞, Q∞)

be the morphism described in Example 2.3.4. Then the functor f! has vector spaces

given by

f!(V )(k) :=⊕

i−j=k V (i, j).

The linear maps between these spaces are given by f!(V )(ρk) =⊕

i−j=k V (ρij1 )

and f!(V )(ρk) =⊕

i−j=k V (ρij2 ). Note that f!(V )(ρk) maps f!(V )(k) to f!(V )(k + 1)

and f!(ρk) maps f!(V )(k) to f!(V )(k − 1). The functor f! acts on morphisms of

Rep(Q∞×∞) as follows: if ϕ = ϕ(i, j) is a morphism between two representations

V and U of Q∞×∞, where ϕ(i, j) : V (i, j) → U(i, j), then f!(ϕ) = f!(ϕ)(k), where

f!(ϕ)(k) =⊕

i−j=k ϕ(i, j).

Once again Corollary 2.3.11 implies that f! is an additive functor. If two objects

f!(V ), f!(U) ∈ Rep(Q∞) are non-isomorphic, the objects V, U ∈ Rep(Q∞×∞) must

be non-isomorphic. Also, if an object f!(V ) ∈ Rep(Q∞) is indecomposable, then the

object V ∈ Rep(Q∞×∞) is also indecomposable, as in the rational case.

The relations R∞ are the Gelfand-Ponomarev relations in Q∞. Thus f! can be

restricted to the subcategory Rep(Q∞×∞, R∞×∞) of Rep(Q∞×∞) to yield a functor

fR∞! : Rep(Q∞×∞, R∞×∞) → Rep(Q∞, R∞), and this restricted functor shares the

properties proven for f!. However, the following examples illustrate that fR∞! is

neither full nor essentially surjective:

Example 4.2.6. Consider the representation V ∈ Rep(Q∞×∞, R∞×∞) pictured in

Figure 4.3, where all vector spaces not shown in the diagram are zero. The endo-


0 −−−→ Cx xC −−−→ 0

Figure 4.3: The functor fR∞! is not full

morphism space of V is given by HomQ∞×∞(V, V ) ∼= Hom(C,C)⊕ Hom(C,C) ∼= C2.

The object fR∞! (V ) is the representation of Q∞ such that fR∞! (V )0 = C2, and all

other vertices are 0. The endomorphism space of this representation, however, is

HomQ∞(fR∞! (V ), fR∞! (V )) ∼= Hom(C2,C2) ∼= Mat2×2(C). Since the dimension of

HomQ∞(fR∞! (V ), fR∞! (V )) is greater than the dimension of HomQ∞×∞(V, V ), the in-

duced functor fR∞!V V is not surjective, and hence fR∞! is not full.

Example 4.2.7. Next, consider the representation V ∈ Rep(Q∞, R∞) given by

(V (i), V (ρi), V (ρi)) = (C, λ, 1) for all i ∈ Z, where λ ∈ C is nonzero. This representa-

tion is pictured in Figure 4.4. Suppose V ∼= fR∞! (U) for some U ∈ Rep(Q∞×∞, R∞×∞).

C C C

C C C

C C C

λ λ

λ λ

λ λ

1

1

1

1

1

1

. . .

. . .

. . .

. . .

. . .

. . .

.

.. ...

.

..

..

....

..

.

Figure 4.4: The functor fR∞! is not essentially surjective

Recall that the vertical maps U(ρij2 ) of the representation U correspond to the left-

ward maps V (ρi) of V . Since each V (i) is one-dimensional, and each V (i) maps to

each V (i − 1) through the identity map, all nonzero U(i, j) must lie along the same

column. Relabelling if necessary, we may assume it’s the first column. Then we must


have U(1, j) ∼= V (j) and U(ρ1j2 ) ∼= 1. A similar argument shows that all nonzero

U(i, j) must lie along the first row, with U(i, 1) ∼= V (i) and U(ρi11 ) ∼= λ. Clearly, no

such U exists, and hence fR∞! is not essentially surjective.

We will now consider an example which shows that the representation theory of

the quiver Q∞×∞ is at least of tame type. To do this, we will show that there exists

a family of pairwise nonisomorphic indecomposable representations of Q∞×∞ which

depend upon a continuous parameter.

Example 4.2.8. For any λ ∈ C, let Vλ ∈ Rep(Q∞×∞, R∞×∞) denote the repre-

sentation pictured in Figure 4.5, where all vector spaces and maps not displayed

are assumed to be zero. We will assume that Vλ(0, 0) = C2, and label all other

C 1−−−→ C

1

x x(1 1)

C −−−→(1 0)T

C2 (1 λ)−−−→ C

(0 1)T

x x1

C −−−→1

C

Figure 4.5: The category Rep(Q∞×∞, R∞×∞) is at least tame

vertices accordingly (see Example 2.1.2). First we will show that Vλ is indecompos-

able for all λ ∈ C. Suppose Vλ = U ⊕ W . Then we may assume U(−1, 1) = C.

Since Vλ(ρ−1,11 ) = Vλ(ρ

−1,02 ) = 1, we must have U(0, 1) = U(−1, 0) = C. We

then have (1 λ)(1 0)T (C) =⊆ U(0, 1), and it follows that U(0, 1) = C. But then

U(0,−1) = U(1,−1) = C since Vλ(ρ0,−11 ) = Vλ(ρ

1,−12 ) = 1. Finally, we have

(1 0)T (C) ⊆ U(0, 0) and (0 1)T (C) ⊆ U(0, 0), and we conclude that U = Vλ, so

Vλ is indecomposable.

Now suppose Vλ ∼= Vµ. Then there exists an invertible 2×2 matrix A and nonzero


complex numbers z1, z2, z3, z4 ∈ C such that the following equations hold:

(1 0)T z1 = A(1 0)T ,

(0 1)T z2 = A(0 1)T ,

z3(1 1) = (1 1)A,

(1 µ)A = z4(1 λ).

The first two equations insist that A is a diagonal matrix. The third equation then

implies that it is a scalar matrix, and then the fourth equation forces λ = µ. Hence

when λ 6= µ, Vλ and Vµ are nonisomorphic. One can show in a similar manner

that the images of these representations under the functor fR∞! gives a family of

indecomposable pairwise nonisomorphic representations in Rep(Q∞, R∞).

While we have seen that it is neither full nor essentially surjective, the functor

fR∞! can still be used to study the category Rep(Q∞×∞, R∞×∞). First, we note that

the group Z acts on the vertices and arrows of Q∞×∞ via

z · (i, j) = (i+ z, j + z), z · ρij1 = ρ(i+z)(j+z)1 , z · ρij2 = ρ

(i+z)(j+z)2 .

Given any representation V ∈ Rep(Q∞×∞, R∞×∞), we denote by V (z) the repre-

sentation obtained from V by twisting with the action of Z. More precisely, the

representation V (z) is defined by

V (z)(i, j) = V (i+ z, j + z), V (z)(ρij1 ) = V (ρ(i+z)(j+z)1 ), V (z)(ρij2 ) = V (ρ

(i+z)(j+z)2 ).

Lemma 4.2.9. Let a, b ∈ Z be integers such that 0 < b − a ≤ 4. Let V ∈

Repfd(Q∞, R∞) be a finite dimensional representation such that V (k) = 0 whenever

k < a or k > b. Then V is isomorphic to fR∞! (U) for some U ∈ Rep(Q∞×∞, R∞×∞),

which is unique up to translation U 7→ U (z) by the group Z.

Proof: Since the functor fR∞! is additive, we may assume that V is indecompos-

able. Any indecomposable representation V ∈ Repfd(Q∞, R∞) such that V (k) = 0


whenever k < a or k > b is supported on at most 5 vertices, and hence may be

thought of as a representation of the preprojective algebra of the quiver of type A5.

The lemma then follows from [6, Lemma 9.1], which states a similar result in the case

of preprojective algebras of type An for 2 ≤ n ≤ 5.

The translation V 7→ V (z) by Z on representations of Q∞×∞ induces an action of Z

on the collection of isomorphism classes of representations of Uµ admitting a weight

space decomposition via the equivalence Rep(Q∞×∞, R∞×∞) ∼= wt-rep(Uµ). We then

have the following result:

Proposition 4.2.10. For a, b ∈ Z with 0 ≤ b − a ≤ 3, there are a finite number of

Z-orbits of isomorphism classes of indecomposable Uµ-modules V such that Vij = 0

whenever i− j < a or i− j > b.

Proof: By [12, Theorem 4.3], there are a finite number of isomorphism classes of

indecomposable modules V ∈ Rep(Q∞, R∞) such that V (k) = 0 for k < a or k > b.

The proposition then follows from the equivalence Rep(Q∞×∞, R∞×∞) ∼= Uµ -Mod

and Lemma 4.2.9.

Corollary 4.2.11. Let A be a finite subset of Z with the property that A does not

contain any five consecutive integers. Then there are a finite number of Z-orbits of

isomorphism classes of indecomposable Uµ-modules V such that Vij = 0 whenever

i− j /∈ A.

If I∞ denotes the two sided ideal of CQ∞ generated by the relations R∞, then

we have CQ∞/I∞ = P(Q∗∞), where Q∗∞ is a subquiver of Q∞ obtained by omitting

the maps ρi | i ∈ Z. The functor fR∞! provides an embedding of the category

Rep(CQ∞×∞/J) in Rep(P(Q∗∞)). The category Rep(P(Q∗∞)) is well understood,

and it is known that every finite dimensional representation of P(Q∗∞) is nilpotent,


see ([12]). Thus the functor fR∞! embeds the finite dimensional representations of

CQ∞×∞/I∞×∞ inside the Lusztig quiver varieties defined in Section 2.5. More specif-

ically, if α ∈ NZ×Z, then fR∞! defines a morphism of quiver varieties EQ∞×∞α → Λ

Q∗∞f!(α)

which identifies EQ∞×∞α with a subvariety of Λ

Q∗∞f!(α).

Appendix A

Category Theory

This chapter is intended to provide a terse review of some of the notions from category

theory that we have used throughout this work. The reader interested in category

theory can consult [9] for more information.

A.1 Categories and Functors

In this section we will review some of the basic definitions from category theory.

Definition A.1.1 (Category). A category C consists of:

1. a class of objects Ob(C),

2. for every X, Y ∈ Ob(C) a class of morphisms from X to Y , HomC(X, Y ), and

3. a binary operation (called a composition law)

HomC(X, Y )× HomC(Y, Z)→ HomC(X,Z)

for all X, Y, Z ∈ Ob(C) denoted by (σ, τ) 7→ τ σ

such that the following axioms hold:

52

A. Category Theory 53

(i) ( associativity) if σ ∈ HomC(X, Y ), τ ∈ HomC(Y, Z), and ρ ∈ HomC(Z,W ) then

ρ (τ σ) = (ρ τ) σ, and

(ii) ( identity) for every X ∈ Ob(C), there exists idX ∈ HomC(X,X) such that for

every σ ∈ HomC(Y,X) and every τ ∈ HomC(X,Z) we have idX σ = σ and

τ idX = τ .

Let σ ∈ HomC(X, Y ). Then σ is called a monomorphism if for every Z ∈ Ob(C)

and every τ, ρ ∈ HomC(Z,X) we have σ τ = σ ρ implies τ = ρ. The morphism

σ is called an epimorphism if for every Z ∈ Ob(C) and every τ, ρ ∈ HomC(Y, Z) we

have τ σ = ρ σ implies τ = σ. The morphism σ is called an isomorphism if there

exists τ ∈ HomC(Y,X) such that σ τ = idY and τ σ = idX . In some categories

there can exist morphisms that are monomorphisms and epimorphisms but are not

isomorphisms.

If C is a category, we define its opposite category, Cop, to be the category with

Ob(Cop) = Ob(C) and HomCop(A,B) = HomC(B,A) for all A,B ∈ Ob(C). By a

covariant functor between two categories, F : C → D, we shall mean a mapping

which associates to every object A ∈ C an object F(A) ∈ D and associates to each

morphism σ ∈ HomC(A,B) a morphism F(σ) ∈ HomD(F(A),F(B)) such that these

associations preserve identity morphisms and composition of morphisms. By a con-

travariant functor, F : C → D, we shall mean a covariant functor F : Cop → D. A

category is said to be small if Ob(C) is a set and HomC(X, Y ) is also a set for every

X, Y ∈ Ob(C). The collection of all small categories along with functors between

them forms a category, which we denote Cat.

If F and G are covariant functors between the categories C and D, then a natural

transformation from F to G, η : F → G, is a collection of morphisms ηA : F(A) →

G(A) | A ∈ Ob(C) such that for every σ ∈ HomC(A,B) the following diagram


commutes:

F(A)F(σ)−−−→ F(B)

ηA

y yηBG(A) −−−→

G(σ)G(B)

If ηA is an isomorphism for every A ∈ Ob(C) then we say η is a natural isomorphism.

Let C and D be two categories, and denote by IC and ID the endofunctors of C

and D respectively that send every object and every morphism to themselves. An

equivalence of categories between C and D consists of a covariant functor F : C → D,

a covariant functor G : D → C, and two natural isomorphisms ε : FG → ID and

η : IC → GF . We say C and D are equivalent if there exists an equivalence of categories

between them.

A.2 Abelian Categories and Adjunctions

Throughout this work we will deal exclusively with abelian categories, which are well-

behaved categories. In this section we will briefly review the definition of an abelian

category as well as some basic results concerning functors in abelian categories. In

order to define abelian categories we will first need to review some other categorical

definitions.

An object O in a category C is called a zero object if for any X ∈ Ob(C) there is

exactly one morphism O → X and for any Y ∈ Ob(C) there is exactly one morphism

Y → O. If C has a zero object O then for any two objects X, Y ∈ Ob(C) we define

the zero map from X to Y , 0XY : X → Y , to be the (unique) map X → O → Y .

For any morphism σ : X → Y , the kernel of σ is a morphism κ : K → X such that

σκ = 0KY and for any other morphism τ : Z → X such that στ = 0ZY there exists

a unique morphism ς : Z → K such that κ ς = τ . This definition can be visualized

by the commutativity of the following diagram.


K Y

X

Z

0KY

κσ

ς

τ

0ZY

The kernel of σ is often written ker(σ) → X. The definition of a cokernel can be

obtained by reversing the arrows in the definition of the kernel.

Let I be an index set, and let Xi ∈ Ob(C) for all i ∈ I. Then the product of the

family of objects Xii∈I is an object∏

i∈I Xi along with a collection of morphisms

πi :∏

i∈I Xi → Xi such that for every object Y and every I-indexed family of mor-

phisms σi : Y → Xi there exists a unique morphism σ : Y →∏

i∈I Xi such that the

following diagram commutes for all i ∈ I.

Y Xi

∏i∈I Xi

σi

σπi

The definition of the coproduct of a family of objects can be obtained by reversing

the arrows in the definition of the product. The coproduct of a family of objects is

denoted∐

i∈I Xi.

We are now ready to define abelian categories.

Definition A.2.1 (Abelian Category). A category C is said to be abelian if it satisfies

the following properties

1. there exists a zero object O ∈ Ob(C),

2. for any two objects X, Y ∈ Ob(C), the product X∏Y and the coproduct X

∐Y

both exist,


3. for any morphism σ : X → Y , the kernel of σ and the cokernel of σ both exist,

4. any monomorphism is the kernel of some morphism and any epimorphism is

the cokernel of some morphism.

Let C and D be abelian categories, and let F : C → D be a covariant functor.

Then for any A,B ∈ Ob(C), the functor F induces a map FAB : HomC(A,B) →

HomD(F(A),F(B)). If FAB is a group homomorphism for all A,B, we say F is

additive. If FAB is injective (resp. surjective) for all A,B, we say F is faithful (resp.

full). If F is additive, then it preserves finite coproducts; F(A∐B) ∼= F(A)

∐F(B),

see for example [11, Corollary 5.88] for a proof of this statement. If for any short exact

sequence 0 → A → B → C → 0 in C the sequence 0 → F(A) → F(B) → F(C) is

exact, then we say the functor F is left exact. Similarly, if for any short exact sequence

0→ A→ B → C → 0 in C the sequence F(A)→ F(B)→ F(C)→ 0 is exact, then

we say F is right exact. One can show that a covariant functor is left exact if and

only if it preserves kernels, that is, F(kerσ) = ker(F(σ)) for every morphism σ ∈ C,

and similarly a covariant functor is right exact if and only if it preserves cokernels, i.e

F(cokerσ) = coker(F(σ)) for any morphism σ (see for example [9, Section VIII.3]).

Any functor which is both left exact and right exact is said to be exact.

If F : C → D and G : D → C are two (covariant) functors then we can consider

the functors HomD(F−,−) and HomC(−,G−) between the categories Cop × D and

Set. If there exists a natural isomorphism between these two functors, then we say

F is left adjoint to G, and G is right adjoint to F . More explicitly, F is right adjoint

to G if there exists a collection of isomorphisms

ΦA,B : HomD(F(A), B)→ HomC(A,G(B)) | A ∈ C, B ∈ D

such that for any morphisms σ ∈ HomC(A′, A) and τ ∈ HomD(B,B′) the following


diagram commutes:

HomD(F(A), B)HomD(F(σ),τ)−−−−−−−−→ HomD(F(A′), B′)

ΦA,B

y yΦA′,B′

HomC(A,G(B)) −−−−−−−−→HomC(σ,G(τ))

HomC(A′,G(B′)).

Here the horizontal arrows correspond to morphisms induced by composition with

σ and τ . If a functor F has two right (or left) adjoints, G and G ′, then G and G ′

are naturally isomorphic. Adjoint functors have many important properties. For

example, if C and D are abelian categories, then two adjoint functors F and G are

necessarily additive and a left (resp. right) adjoint functor is right (resp. left) exact

(see for example [9, Section V.5, Theorem 1]).

A.3 Kan Extensions

In this section we describe briefly the notion of left and right Kan extensions. Kan

extensions are very powerful tools in category theory (indeed, [9, Section X.7] is titled

“All Concepts Are Kan Extensions”), and their use in this work, while not imperative,

would simplify the proof of Theorem 2.3.9. The interested reader is encouraged to

consult [9, Chapter X].

Given two categoriesA and C, we can consider the categoryAC, whose objects are

all functors F : C → A, and whose morphisms are natural transformations between

such functors. Then for any category B and any covariant functor K : B → C, we can

construct a functor AK : AC → AB given by σ 7→ σ K. The study of Kan extensions

amounts to attempting to construct right and left adjoints to this functor.

Definition A.3.1 (Kan Extension). Let A,B, C be categories, and let K : B → C

and F : B → A be covariant functors. Then the right Kan extension of F along K

consists of a functor R : C → A and a natural transformation ε : RK → F such that


for any functor H : C → A and any natural transformation µ : HK → F , there exists

a unique natural transformation δ : H → R making the following diagram commute:

RK

HK,F

δKε

µ

where δK is the natural transformation such that

δK(X) = δ(K(X)) : HK(X)→ RK(X)

for every X ∈ Ob(A). We often write RanKF for the right Kan extension of F along

K, and ε is called the counit of the extension. The definition for a left Kan extension

of F along K can be obtained by reversing all arrows in the above diagram, and we

write LanKF for the left Kan extension of F along K. In the left case we use the

symbol η instead of ε, and η is called the unit of the extension.

From the universality of this definition, if RanKF exists for every F ∈ AB,

then the functor RanK : AB → AC given by F 7→ RanKF is a right adjoint to AK.

Similarly, if LanKF exists for every F ∈ AB, the functor F 7→ LanKF provides a left

adjoint for AK.

While this definition of Kan extensions in terms of their universal property is

the most general, it is not always convenient for computing them explicitly, nor does

it guarantee their existence. In order to introduce a more computationally useful

characterization of Kan extensions, we first need to introduce the notions of limits

and colimits.

Let F : C → D be a covariant functor. A cone over F is an object N ∈ Ob(D)

along with a family of morphisms σX : N → F(X), X ∈ Ob(C), such that for every

morphism τ : X → Y in C, we have D(τ) σX = σY . The limit of F , lim←−F , is a cone

(N, σXX∈C) over F such that for any other cone (L, ρXX∈C) over F there exists

a unique morphism ς : L → N such that σX ς = ρX for all X ∈ C. This can be

visualized by the commutativity of the following diagram.


N F(Y )

F(X)

L

ρY

ρXF(τ)

ς

σX

σY

The definition of the colimit of F , lim−→F can be obtained by dualizing the definition

of the limit. If D is a category such that for any small category C and any functor

F : C → D the limit (resp. colimit) of F exists, then we say D is complete (resp.

cocomplete).

In the case that A is a small category and C is a complete category, the following

theorem gives a more explicit description of Kan extensions (see for example [9,

Section X.3, Theorem 1]).

Theorem A.3.2. Suppose A,B, and C are categories with B small and A complete.

Let K : B → C and F : B → A be covariant functors. Then the right Kan extension

of F along K exists and is given by

RanKF(X) = lim←−((X ↓ K)QX−−→ B F−→ A).

for all X ∈ C.

In Theorem A.3.2 above, X ↓ K denotes the comma category , which has as

objects pairs (Y, σ) where Y ∈ Ob(B) and σ ∈ HomC(X,K(Y )). A morphism in

X ↓ K(Y ) between (Y, σ) and (Y ′, σ′) is given by a morphism τ ∈ HomC(Y, Y′) such

that the following diagram commutes

XidX−−−→ X

σ

y yσ′K(Y ) −−−→

K(τ)K(Y ′).

(A.3.1)


Along with every comma category there is a projection functor QX : X ↓ K → B that

takes (σ, Y ) to Y . The limit in Theorem A.3.2 is the limit of the composition of F

with this projection functor.

A similar result to Theorem A.3.2 holds for left Kan extensions, though we

suppose that C is cocomplete, and then LanKF(X) is the colimit over the comma

category K ↓ X (a description of which can be obtained from the description of

X ↓ K by reversing all of the arrows).

The relationship between Kan extensions and limits of functors goes even deeper

than Theorem A.3.2, as shown by the following result.

Theorem A.3.3. [9, Theorem X.7.1] A functor F : B → A has a limit if and only if

it has a right Kan extension along the functor K1 : B → ∗ (where ∗ is the category with

a single object and a single morphism), in which case lim←−F is the value of RanK1 F

on the unique object of ∗.

A similar theorem holds for left Kan extensions and colimits of functors.

One of the strengths of Theorem A.3.2 is that it guarantees the existence of Kan

extensions under certain conditions. Thus, if one is interested in a particular functor

that can be interpreted as being of the form AK for some suitable category A and

some suitable functor K, then we can guarantee the existence of adjoints to that

functor. In fact, we can use Kan extensions to prove a deeper result concerning the

existence of adjoints.

Theorem A.3.4. [9, Theorem X.7.2] A functor G : A → C has a left adjoint if and

only if RanG(IA) exists with counit ε and G RanG(IA) = RanG G with Gε as the

counit. When this is the case, RanG(IA) is the left adjoint of G.

This theorem can be used to simplify the proof of Theorem 2.3.9.

Appendix B

Affine Varieties

Here we quickly review some of the theory of affine varieties. The material presented

here will be very concise, for a more detailed exposition the reader can consult [13].

The main objects we will study in this section are the solution sets to systems

of polynomials over C. First we define affine n-space to be An := (x1, . . . , xn) | xi ∈

C for all i. We use the notation An rather than Cn to emphasize that we are not

considering this set as being equipped with its natural vector space structure. If

f ∈ C[x1, . . . , xn], then a point p ∈ An will be called a zero of f is f(p) = 0. We then

define

Z(f) = p ∈ An | p is a zero of f.

We can naturally extend this idea to any set T ⊆ C[x1, . . . , xn]:

Z(T ) = p ∈ An | p is a zero of every f ∈ T.

It follows immediately from these definitions that for any T ⊆ C[x1, . . . , xn], we

have Z(T ) = Z(〈T 〉), where 〈T 〉 denotes the ideal of C[x1, . . . , xn] generated by T .

Since C[x1, . . . , xn] is a Noetherian ring (this is known as Hilbert’s Basis Theorem), it

follows that for any ideal I ⊆ C[x1, . . . , xn], we have Z(I) = Z(f1, . . . , fm), where

I = 〈f1, . . . , fm〉.

61

B. Affine Varieties 62

It is not difficult to see that the collection of all sets Y ⊆ An such that Y = Z(T )

for some T ⊆ C[x1, . . . , xn] form the closed sets of a topology on An. This topology

is known as the Zariski Topology. This motivates the following definition:

Definition B.1 (Affine Variety). Let Y ⊆ An be such that there exists T ⊆ C[x1, . . . , xn]

with Y = Z(T ). Then Y is called an affine variety.

Remark B.2. Many authors use the term algebraic set for the objects defined above

and reserve the term affine variety for algebraic sets which are irreducible with respect

to the Zariski Topology.

On the other hand, given a subset Y ⊆ An we can consider the set

I(Y ) := f ∈ C[x1, . . . , xn] | f(Y ) = 0.

Clearly, I(Y ) is a radical ideal of C[x1, . . . , xn] for any Y . This gives two functions

between ideals I of C[x1, . . . , xn] and subsets Y of affine space, namely I 7→ Z(I)

and Y 7→ I(Y ). It turns out that there exists a very important fact about these two

functions known as Hilbert’s Nullstellensatz :

Theorem B.3. Let U be an ideal of C[x1, . . . , xn]. Then I(Z(U)) =√U, where

√U

denotes the radical of U.

Proof: See, for example, [13, Section 2.3].

Thus, if we restrict ourselves to radical ideals of C[x1, . . . , xn], the maps I 7→ Z(I)

and Y 7→ I(Y ) are inverse to each other. This correspondence allows us to translate

back and forth between affine varieties and ideals of C[x1, . . . , xn].

Now that we have defined the objects we wish to study, it is natural to consider

certain classes of functions on them. For any affine variety V , the restriction of

any f ∈ C[x1, . . . , xn] to V gives a function f |V : V → C. The restrictions of all

polynomials in C[x1, . . . , xn] to V form a C-algebra under pointwise addition and


multiplication, which we denote by C[V ]. We call C[V ] the coordinate ring of V .

Clearly we have a natural surjection C[x1, . . . , xn] C[V ] that has kernel I(V ), and

hence for any affine variety V we have

C[V ] ∼= C[x1, . . . , xn]/I(V ), (B.1)

which we will often view as an identification. We now turn our attention to morphisms

between affine varieties. The simplest example of a morphism between two affine

varieties is a map of the form

An → Am

x 7→ (F1(x), F2(x), . . . , Fm(x)),

where Fi is a polynomial in the coordinates x1, . . . , xn for every i. Such a map is

called a polynomial map. More generally, we make the following definition:

Definition B.4 (Morphism of Affine Varieties). Let V ⊆ An and W ⊆ Am be affine

varieties. A map F : V → W is a morphism of affine varieties if it is the restriction

to V of a polynomial map on the ambient affine spaces An → Am.

A morphism of affine varieties is said to be an isomorphism if it has an inverse

morphism. That is, a morphism is an isomorphism if it is bijective and its inverse map

is a morphism of affine varieties. Given a morphism of affine varieties, F : V → W ,

we get a naturally induced map between the coordinate rings C[W ] and C[V ] given

by

C[W ]→ C[V ],

g 7→ g F.

This induced map is called the pullback of F . It is easy to check that the pullback

of F defines a C-algebra homomorphism from C[W ] to C[V ]. Thus for any affine

variety V we get a finitely generated, reduced C-algebra C[V ], and for any morphism


of affine varieties, we get a morphism of C-algebras. In fact, this association defines

an equivalence of categories.

Theorem B.5. There is an equivalence of categories between the category of affine

varieties and the category of finitely generated, reduced C-algebras.


Thus we may freely pass back and forth between the geometric concept of affine

varieties and the algebraic concept of C-algebras. The following lemma will prove

useful.

Lemma B.6. A morphism of affine varieties F : V → W defines an isomorphism

between V and some algebraic subvariety of W if and only if the pullback of F is

surjective.


Bibliography

[1] Ibrahim Assem, Daniel Simson, and Andrzej Skowronski. Elements of the rep-

resentation theory of associative algebras. Vol. 1, volume 65 of London Mathe-

matical Society Student Texts. Cambridge University Press, Cambridge, 2006.

Techniques of representation theory.

[2] Edgar E. Enochs and Ivo Herzog. A homotopy of quiver morphisms with appli-

cations to representations. Canad. J. Math., 51(2):294–308, 1999.

[3] Karin Erdmann and Mark J. Wildon. Introduction to Lie algebras. Springer

Undergraduate Mathematics Series. Springer-Verlag London Ltd., London, 2006.

[4] Igor B. Frenkel and Alistair Savage. Bases of representations of type A affine

Lie algebras via quiver varieties and statistical mechanics. Int. Math. Res. Not.,

(28):1521–1547, 2003.

[5] Peter Gabriel. Unzerlegbare Darstellungen. I. Manuscripta Math., 6:71–103;

correction, ibid. 6 (1972), 309, 1972.

[6] Christof Geiss, Bernard Leclerc, and Jan Schroer. Semicanonical bases and pre-

projective algebras. Ann. Sci. Ecole Norm. Sup. (4), 38(2):193–253, 2005.

[7] James E. Humphreys. Introduction to Lie algebras and representation theory,

volume 9 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1978.

Second printing, revised.

65

BIBLIOGRAPHY 66

[8] G. Lusztig. Quivers, perverse sheaves, and quantized enveloping algebras. J.

Amer. Math. Soc., 4(2):365–421, 1991.

[9] Saunders Mac Lane. Categories for the working mathematician, volume 5 of

Graduate Texts in Mathematics. Springer-Verlag, New York, second edition,

1998.

[10] E. A. Makedonskiı. On wild and tame finite-dimensional Lie algebras. Funkt-

sional. Anal. i Prilozhen., 47(4):30–44, 2013.

[11] Joseph J. Rotman. An introduction to homological algebra. Universitext.

Springer, New York, second edition, 2009.

[12] Alistair Savage. Quivers and the Euclidean group. In Representation theory,

volume 478 of Contemp. Math., pages 177–188. Amer. Math. Soc., Providence,

RI, 2009.

[13] Karen E. Smith, Lauri Kahanpaa, Pekka Kekalainen, and William Traves. An

invitation to algebraic geometry. Universitext. Springer-Verlag, New York, 2000.

Index

affine variety, 59

category, 49, 50

abelian, 52, 53

cocomplete, 56

complete, 56

cokernel, 52

comma category, 56

cone, 55

coordinate ring, 60

coproduct, 52

covering morphisms, 11

double quiver, 7

epimorphism, 50

equivalence

of categories, 51

functor

additive, 53

contravariant, 50

covariant, 50

exact, 53

faithful, 53

full, 53

Gelfand-Ponomarev relations, 7

graded dimension, 21

finite type, 21

isomorphism, 50

kernel, 51

left

Kan extension, 55

adjoint, 53

exact, 53

extension functor, 12

Lie algebra, 27

abelian, 27

homomorphism, 28

representation, 28

limit, 55

Lusztig quiver varieties, 22

modified enveloping algebra, 32, 33

monomorphism, 50

morphism, 49

of affine varieties, 60

67

INDEX 68

of quiver representations, 8

of quivers, 9

natural

isomorphism, 51

transformation, 50

opposite category, 50

path, 6

algebra, 6

Poincare-Birkhoff-Witt Theorem, 31

preprojective algebra, 7

product, 52

quiver, 5

morphism, 9

relation, 17

representation, 8

quiver representation

finite dimensional, 21

locally finite, 25

nilpotent, 22

representation space, 21

restriction functor, 12

right

Kan extension, 54

adjoint, 53

exact, 53

extension functor, 12

universal enveloping algebra, 31

Zariski Topology, 59

zero object, 51

quivers and three-dimensional lie algebras · 2017-01-31 · chapter 1 introduction in the late...

Documents