Algebras, Graphs and Algorithms

Peter Draxler

February 2012

Contents

1 Introduction 1

2 k-Categories and their Modules 2

3 Multilocal and Local k-Categories 6

4 k-Spectroids and Their Modules 8

5 Coherent Modules, Dualising k-Spectroidsand Auslander-Reiten Sequences 10

6 The Quiver of a k-Spectroid 12

7 Representation Types 15

8 Algebras with Preprojective Components 15

9 Ray Categories and their Coverings 17

10 Criteria and Classification Lists 19

11 Computer Algebra in Representation Theory of Algebras 23

1 Introduction

Motivation and PrerequisitesThese notes are based on a series of lectures which I gave in February 2012

at the University of Bielefeld. I am grateful to Henning Krause for providingthe opportunity to give this course.

The aim of the lectures was to present a survey of a mathematical frameworkwhich was used for the development and implementation of algorithms for do-ing computer algebra concerning the representation theory of finite-dimensionalalgebras over a field k. The framework is built around the notion of finite-dimensional k-categories on the algebraic side and around the notion of quiverson the combinatorial side. As (historical) central theme serves the study ofalgebras of finite and tame representation type. As it turns out, for this topicfinite partially orderd sets (posets) are frequently even more suited then quiversbecause they provide sort of ”computationally canonical” bases.

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The framework was used for building the computer algebra system CREP(Combinatorial REPresentation theory) in the years 1991 to 2000. In general,I refrained from including more recent material with one exception, namelythe paper [BPS] which allows now to formulate various algorithms on tamerepresentation type in an appropriate generality.

I will not deal with concrete implementations but only will provide somehints and comments about this topic at the end. In addition, in the text Iwill sometimes insert some comments and ask some questions about computeralgebra and/or software engineering aspects. These remarks will be presentedunder the headline CA.

Needed PrerequisitesWe assume that the basic theory of modules over rings is known including

elementary category theory and homological algebra.

CategoriesWe work inside a Grothendieck universe which contains the set N of natural

numbers as element. Consequently, sets are elements of this universe, classesare subsets of the universe.

A category C is a pair C = (C0, C1) where C0 is a class, the class of objects, andC1 = (C1(x, y))x,y∈C0 is a family of sets. The set C1(x, y) is the set of morphismsfrom x to y.

Frequently, we write C instead of C0 and C(x, y) instead of C1(x, y).A category C is called small resp. skeletally small if the class C0 resp. a

skeleton of C0 is a set.If C and D are two categories a covariant functor F : C → D is a pair

F = (F0, F1) where F0 is a map F0 : C0 → D0 and F1 is a family of maps F1 :C(x, y)→ D(F0(x), F0(y)) with index set C0×C0 such that for each compositionba in C the equation F1(ba) = F1(b)F1(a) holds.

In contrast for a contravariant functor the family F1 is a family of mapsF1 : C(x, y) → D(F0(y), F0(x)) where the equation F1(ba) = F1(a)F1(b) holds.Clearly, contravariant functors C → D can be considered as covariant functorsCop → D where Cop is the opposite category of C.

Usually we write F for F0 and F1.

Natural Transformations and Functor CategoriesFor two functors F : C → D a natural transformation φ : F → G is a family

φ = (φx)x ∈ C of morphisms φx : F (x)→ G(x) in D such that for any morphisma : x→ y in C the equation φyF (a) = G(a)φx holds.

Note, that for two categories C, D where C is skeletally small we obtain thecategory of functors C → D by choosing the functors F : C → D as object classand for two functors F,G the set of natural transformations F → G as set ofmorphisms.

Quivers and Path CategoriesA directed graph or briefly quiver Q is a pair Q = (Q0, Q1) where Q0 is a

set, the set of points, and Q1 = (Q1(x, y))x,y∈Q0is a familiy of sets. The set

Q1(x, y) is the set of arrows from x to y.

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Frequently, we write Q instead of Q0 and Q(x, y) instead of Q1(x, y).

For an arrow α ∈ Q(x, y) we define the start point as α• := x and the endpoint as •α := y.

If Q and R are two quivers, then a quiver morphism F : Q → R is a pairF = (F0, F1) where F0 : Q0 → R0 and F1 is a family of maps F1 : Q1(x, y) →R1(F0(x), F0(y)) with index set Q0 ×Q0.

Given a quiver Q we construct the path category pth(Q) by choosing Q0 asset of objects pth(Q)0 and pth(Q)(x, y) as the set of all paths from x to y whichare the sequences α = αn . . . α1 where α•1 = x, •αn = y and α•i+1 =• αi for alli = 1, . . . , n− 1. Note, that n ≥ 0, thus for each point x of Q there is a trivialpath 1x from x to x which serves as identity.

We consider a category C as a quiver by forgetting the composition. If Q isa quiver and F : Q → C is a quiver morphism, then F extends uniquely to afunctor pth(Q)→ C which we denote by F as well.

2 k-Categories and their Modules

k-CategoriesA category A is said to be a k-category if for all x, y in A the morphism sets

A(x, y) are k-spaces and the composition in A is k-bilinear.In addition A is called finite-dimensional if all these k-spaces A(x, y) are

finite-dimensional.

Well-known examples of k-categories are the category Mod−k of all vectorspaces over k and its full subcategory mod−k of all finite-dimensional vectorspaces over k which is obviously finite-dimensional.

k-algebras considered as categories with a single object are k-categories aswell as their categories of modules.

An ideal I of a k-category A consists of a family (I(x, y))x,y∈A of subspacesI(x, y) of A(x, y) such that for all a in I(x, y), all x′, y′ in A and all b in A(x′, x)and alle c in A(y, y′) the composition cab lies in I(x′, y′).

For a family (ρi)i∈I of morphisms in A the symbol < (ρi)i∈I > denotes theideal of A generated by this family.

k-LinearisationIf C is a small category, then the objects of its k-linearisation kC are the

objects of C, the morphism space kC(x, y) has the set C(x, y) as basis and thecomposition is the obvious linear extension.

Popular examples are the group algebra kG for a group G or the the polyno-mial algebra k[X] = k{1, X,X2, . . .} considered as categories with one object.

An example occurring in the representation theory of algebras of finite typewhich we will encounter later is kP for a ray catgory P .

Of major importance for us will be the category kQ := kpth(Q) for a quiverQ.

Crucial will also be the incidence k-category kS for a poset S = (S,≤). Weconsider a poset as category with objects set S, morphism set {(y, x)} if x ≥ y,

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∅ otherwise, and (z, y)(y, x) := (z, x). Let us call such a morphism (y, x) a directneighbour pair provided there is no element z of S satisfying x > z > y.

We say that the poset S is discrete if for all elements x,y with x > y thereis a sequence (x0, . . . , xn) with n ∈ N, x = x0, y = xn and (xi, xi−1) a directneighbour pair for all i = 1, . . . , n.

We obtain the Hasse quiver QS of a discrete poset S by putting an arrow x→y for each direcht neighbour pair (y, x). Using this Hasse quiver we can identifykS with the factor category of kQS by the ideal generated by all differencesu− v where u and v are paths of length at least 2 with u• = v• and •u =• v.

Any Hasse quiver is narrow meaning that it is directed and for every arrowα : x→ y there does not exist a path u with u• = x and •u = y which is differentfrom α. It is obvious that conversely any narrow quiver defines a discrete poset.

Modules over k-CatgoriesFor given k-catgories A, B a functor F : A → B is said to be k-linear if the

maps F : A(x, y)→ B(F (x), F (y)) are k-linear for all objects x, y of A.Any functor F : C → A where C is a category and A is a k-category can be

extended uniquely to a k-linear functor kC → A which we denote by F as well.Conversely, any k-linear functor kC → A yields a restricted functor C → A.Thus we can identify the functors C → A with the k-linear functors kC → A.

For a skeletally small k-category A the full subcategory of the category ofcontravariant functors A → Mod−k formed by the k-linear functors is denotedby Mod−A.

In view of our remark above for a category C we can identify Mod−kC withthe category of all functors C → Mod−k which we denote by Mod−C. Con-sequently, for an ideal I of kC generated by a family (%i)i∈I we can identifyMod−kC/I with the category Mod−C(%i)i∈I of functors annihilating the mor-phism %i.

Representations of QuiversAs seen above for a quiver Q we may identify Mod−kQ with Mod−pthQ

which as observed in an earlier section can be identified with the category ofquiver morphisms Q → Mod−k denoted by Mod−Q. The category Mod−Q iscalled the category of representations of Q.

Again for an ideal I of kC generated by a family (%i)i∈I we consequentlycan identify Mod−kQ/I with the category Mod−Q(%i)i∈I of reprsentationsannihilating the morphisms %i.

For an arbitrary small k-category A there is obviously a quiver Q and afull and dense k-linear functor F : kQ → A. If the kernel I of F is gener-ated by (%i)i∈I , then we can identify A with kQ/I and therefore Mod−A withMod−Q(%i)i∈I .

CA. The above calculations show that the study of modules over k-categoriesby using quivers Q and relations (%i)i∈I can be reduced to k-linear algebra.

Modules over PosetsIn a similar fashion for a poset S we my identify Mod−kS with Mod−S.

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For a convex subset T of S we define the indicator module kδ(T ) by definingkδ(T )(x) := k for x ∈ T , kδ(T )(x) := 0 for x 6∈ T , kδ(T )(y, x) := idk for y ≤ xin T and kδ(T )(y, x) := 0 else.

Of particular interest will be the convex subsets x↑ := {y ∈ S : y ≥ x},x↓ := {y ∈ S : y ≤ x}, x := {x}.

Free Modules over k-CategoriesFor a small k-category A the category Mod−A is a complete and cocomplete

abelian category. The necessary constructions are performed ”pointwise” on thevectorspaces representing the objects. Hence, concerning the abelian structurewe can treat the category Mod−A like the cateogory of modules over a ring. Inaddition, we can do homological algebra using free and ”cofree” resolutions aswe will point out in the next lines.

For convenience of notation we put x := A(−, x) in Mod−A. Modules whichare direct sums of modules of the shape x are called free.

Theorem 1 (Yoneda lemma). The map HomA(x,M) → M(x) sending ϕ toϕx(idx) for all x in A and M in Mod−A is a natural isomorphism of bifunctorsA×Mod−A → Mod−k.

Note, that the inverse image under the Yoneda isomorphisms of an elementm in M(x) is a homomorhims whose image is the smallest submodule of Mcontaining m.

Corollary 2. • The functor A → Mod−A mapping an object x to x :=A(−, x) is a full embedding A → Mod−A.

• For each object x of A the functor HomA(x,−) is exact i.e. x is projective.

• Every module is a factor of a free module. (This is what is needed to formprojective resolutions.)

The Dual ModuleLet A be a small k-category, then we obtain a contravariant faithful and

exact functor D : Mod−A → A−Mod by sending M to Homk(M,k).Define for M in Mod−A the morphism ιM : M → D2M by defining

(ιM )x(m) := φ(m) for m ∈ M(x) and φ ∈ Homk(M(x),k), then (ιM ) :IdMod−A → D2 is a natural monomorphism.

Again for convenience of notation we put x := DA(x,−) in Mod−A. Mod-ules which are direct products of modules of the shape x are called cofree.

Theorem 3 (Dual Yoneda Lemma). The map HomA(M, x)→ DM(x) sendingϕ to the map m 7→ (ϕx(m))(idx) for all x in A and M in Mod−A is a naturalisomorphism of bifunctors Mod−A×A → Mod−k.

Corollary 4. • The functor A → Mod−A mapping an object x to x :=DA(x,−) is a full embedding A → Mod−A.

• For each object x of A the functor HomA(−, x) is exact i.e. x is injective.

• Every module is a submodule of cofree module. (This is what is needed toform injective resolutions.)

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Modules of Finite LengthLet A be a small k-category. A non-zero module S in Mod−A is said to be

simple if the only submodules of S are the 0-module and S.For every simple module S there exists an object x of A and a maximal

submodule U of x such that S ∼= x/U .By mapping U to U(x) we obtain a bijection from the set of maximal sub-

modules of x to the right ideals of the algebra A(x, x).

A module M in Mod−A is called of finite length if there is a compositionseries 0 = U0 ⊆ U1 . . . Ul−1 ⊆ Ul = M of M meaning a tower of submodules ofM such that Ui/Ui−1 is simple for all i = 1, . . . , l.

As usual, any two composition series are equivalent. Hence the lengthl(M) := l is well-defined.

Let us denote by modfl−A the full subcategory of Mod−A supported by themodules of finite length.

Of great technical significance is:

Theorem 5 (Fitting). If M is a module of finite length and ϕ ∈ EndA(M) anendomorphism, then there are submodules M0 and M1 such that M = M0⊕M1,

ϕ =

(ϕ0 00 ϕ1

), ϕ0 : M0 → M0 is nilpotent and ϕ1 : M1 → M1 is an

automorphism.

The Radical of a k-Category

Theorem 6. If x, y are two objects of a k-category A, then for a morphism fin A(x, y) the following statements are equivalent:

• f is in the intersection of all maximal submodules of A(x,−).

• For all g ∈ A(y, x) the morphism idx +gf is an automorphism.

• f is in the intersection of all maximal submodules of A(−, y).

• For all g ∈ A(y, x) the morphism idy +fg is an automorphism.

The morphisms f for alle x, y defined by the above theorem form an idealwhich is denoted by radA(−,−) and called the (Jacobson) radical of A

Observe that a full k-functor F : A → B reflects isomorhisms iff KerF ⊆radA.

Note also, that for a full subcategory B of A and two objects x, y of B theequation radB(x, y) = radA(x, y) holds. However, this is not true for the higherpowers of the radical.

For each n ∈ N the power radnA := (radA)n, hence also the intersection ofalle these powers are ideals of A. This intersection is denoted by rad∞A.

3 Multilocal and Local k-Categories

Additive k-categories with splitting idempotentsA category k-category A is called

• additive if A as finite biproducts.

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• with splitting idempotents if for each object x in A and each idempotentelement e (i.e. 0 6= e = e2) of A(x, x) there is an object y of A andhomomorphisms ι : y → x and π : x→ y such that ιπ = idx and πι = e.

If A is a k-category, then

• the category addA having all tuples (x1, . . . , xn) with n ∈ N0 as objectsand the obvious matrices of morphisms of A as morphisms is a model forthe smallest additive k-category containing A.

• the category having the pairs (x, e) where x is an object of A and e is anidempotent morphism in A(x, x) as objects and the products e′ae wherea : x → x′ is a morphism in A as morphisms is a model for the smallestk-category with splitting idempotents containing A.

• the category splt addA is an additive k-category with splitting idempo-tents.

Local AlgebrasA non-trivial k-algebra A is said to be local if the non-units in A are closed

under addition. Equivalent characterisations are:

• the set of non-units coincides with radA

• A/radA is a skew field

Clearly, for an object x with local endomorphism algebra idx is the onlyidempotent of A(x, x), hence x is indecomposable. Using the Fitting Lemmaone shows that for finite-dimensional algebras also the converse is true:

Lemma 7. A finite-dimensional k-algebra A is local iff 1A is the only idempo-tent in A.

Multilocal k-catgoriesA k-category A is called multilocal if A is additive and each object of A is

a finite biproduct of objects with local endomorphism algebras.

Note, that these decompositions are unique and satisfy the exchange prop-erty. In a multilocal k-category the indecomposable objects coincide with theobjects with local endomorphism algebra. Moreover, A has splitting idempo-tents.

A k-category S is called local if each object x has a local endomorphismalgebra S(x, x) and any two different objects are non-isomorphic.

If A is a multilocal k-category and S is a skeleton of splt addA, then S is alocal k-category which we denote by S(A). The category S(A) is well-defindedup to equivalence.

Theorem 8. If A is a finite-dimensional k-category, then splt addA is a finite-dimensional multilocal k-catgory.

The category S(A) := S(splt addA) is called the basic category of A.Obviously, the categories Mod−A, Mod− splt addA and Mod−S(A) are

equivalent. This is known as Morita equivalence.

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CA. S(A) can be constructed from A in the following way:

• For each object x of A find a decomposition idx =∑n(x)i=1 ei(x) such that

eiA(x, x)ei is local.

• Inside the full subcategory of spltA supported by all (x, ei(x)) find a skele-ton.

Modules of Finite Length IIIf A is a small k-category, then modfl−A is a multlilocal k-category. We

define for all d ∈ N the category S(modfl−A, d) as the full subcategory ofS(modfl−A) given by all objects M satisfying l(M) ≤ d.

Theorem 9 (Harada-Sai). The radical of S(modfl−A, d) is nilpotent with index2d − 1.

A module M in Mod−A is called finite-dimensional if its dimensiondimM :=

∑x∈A dimAM(x) is finite. Of course every finite-dimensional module

is of finite length whereas the converse generally is not true.

Minimal and Almost Split MorphismsSuppose A is a finite-dimensional mulitlocal k-category with S := S(A) and

g : y → z is a morphism in A. g is said to be right minimal if every endomorhismh of y satisfying gh = g has to be an automorphism. Equivalently this meansthat any section s : y′ → y satisfying gs = 0 has to be 0. Clearly, ”left minimal”is defined dually.

Two morphisms g : y → z and g′ : y′ → z are said to be right lift equivalentprovided there are morphisms h, h′ such that g = g′h′ and g′ = gh. In everyright equivalence class there is right minimal representative g : y → z. For anyother member g′ : y′ → z there is a retraction t : y′ → y such that g′ = gt, inparticular, for two right minimal representatives g : y → z and g′ : y′ → z thereis an isomorphism h such that g = g′h.

A morphism g : y → z in A with z in S is said to be right almost split ifg is in radA(y, z) and for every h : u → z in radA(u, z) there is a morphismh′ : u→ y satisfying h = gh′. ”Left almost split” is defined dually.

If there is a right almost split morphism g′ : y′ → z, then there is even a rightminimal almost split morphism g : y → z which is unique up to isomorphism asseen above.

A is said to have almost split morphisms if for each y in S there is a rightminimal almost split morphism x→ y and a left minimal almost split morphismy → z.

Obviously, A has almost split morphisms provided that S is finite.

4 k-Spectroids and Their Modules

Locally Finite-Dimensional ModulesFrom now on we will mainly consider small finite-dimensional local k-

categories although many of the concepts and results may still be valid in a

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more general setup. Following [GR] we will call these categories k-spectroids.For abbreviation for a k-spectroid S we put JS := radS.

For a k-spectroid S the category Mod−S contains the contravariant k-linearfunctors S → mod−k as full subcategory which is called the category of locallyfinite-dimensional modules and denoted by modlfd−S. The category modlfd−Sis skeletally small. As observed above the restrictionD : modlfd−S → S−modlfd

is a duality, i.e. a full, faithful and dense contravariant functor.With each object x the associated projective module x, the injective mod-

ule x and the simple module x− := x/JS(−, x) ∼= soc x belong to modlfd−S.Consequently, modlfd−S is an abelian full subcategory of Mod−S with enoughprojectives and injectives to do homological algebra.

CA. If S is presented by a quiver Q with relations (ρi), then the objects inmodlfd−S can be described by attaching a matrix over k to each arrow. Hencewe are in the realm of linear algebra and also of affine varieties because therelations define closed subsets of the matrix varieties.

On the other hand, calculations inside kQ are basically also a matter oflinear algebra where the paths are the basis. That these base vectors have acertain monoidal structure is the starting point to apply Groebner basis methods.

Projective, Injectives and Simple Modules over SpectroidsFor a k-spectroid S in general modlfd−S is neither a mulitlocal nor a finite-

dimensional k-category but at least proj−S := add{x : x ∈ S}, resp. inj−S :=add{x : x ∈ S} resp. add{x− : x ∈ S} are finite-dimensional multilocal k-categories with spectroids {x : x ∈ S} resp. {x : x ∈ S} resp. {x− : x ∈ S}.These subcategories are called the categories of projective, resp. injective, resp.semisimple modules.

For the simple module x− holds x−(y) = 0 for all y 6= x and x−(x) =S(x, x)/ radS(x, x). Hence the simple modules are finite-dimensional and weobtain that each module of finite length is finite-dimensional. It follows thatfor a spectroid S the equation modfd−S = modfl−S holds. This category is afinite-dimensional multi-local k-category.

Note also, that the duality D restricts to a duality D : modfd−S →S−modfd.

Modules over PosetsObviously, for a poset S the k-category kS is k-spectroid. More general, if

I is an ideal of kS then also kS/I is a spectroid and we may assume that I isgenerated by a set R of morphims (y, x) with x ≥ y.

Using the identification of Mod−kS with Mod−S it will be easy to defineinteresting modules using convex subsets of S. Extending the notion of convexitywe say that a convex subset T of S is I-convex if for all x, y in T with y ≤ x.the morphism (y, x) does not belong to R.

For any I-convex subset T of S the module kδ(T ) lies in modlfd−S. Par-

ticular examples of I-convex subsets are x↑I := {y ∈ x↑ : (x, y) 6∈ I} and

x↓I := {y ∈ x↓ : (y, x) 6∈ I}. Note, that kδ(x↑I)∼= x, kδ(x↓I)

∼= x, kδ(x) ∼= x− forall elements x of S with (x, x) 6∈ I.

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Finitely Presented ModulesLet S be a k-spectroid. A module M in Mod−S is called finitely presented

if there is an exact sequence P1 −→ P0 −→M → 0 with P1, P0 in proj−S. Thefull subcategory of modlfd−S given by the finitely presented modules is a finite-dimensional multilocal k-category which is denoted by modfp−S. Dually thecategory D(S−modfp) is called the catgegory of finitely copresented modules.

Since the multilocal k-category modfp−S will be of major interest for us, weintroduce the notation ind−S := S(modfp−S).

There is no general relation between modfp−S, D(S−modfp) and modfd−S.This situation changes if S is locally bounded which means

∑y∈S dimk S(x, y)+

dimk S(y, x) <∞ for each object x of S . Because this means that all projectivesx as well as all injectives x are finite-dimensional, it follows that modfp−S =D(S−modfp) = modfd−S.

From the Harada-Sai lemma follows:

Lemma 10. If S is locally bounded, then for all x in S there is nx ∈ N0 suchthat J nx

S (x,−) = 0 = J nx

S (−, x).

Finite spectroids are a special case of locally bounded spectroids. Clearly,the finite k-spectroids can be idenitfied with the finite-dimensional basic k-algebras.

5 Coherent Modules, Dualising k-Spectroidsand Auslander-Reiten Sequences

Coherent ModulesRemember that a module M over a k-spectroid S is finitely generated if

there is an epimorphism p : P → M where P is in proj−S. Of course, everyfinitely generated module is locally finite-dimensional. In addition, M is calledcoherent if M is finitely generated and each finitely generated submodule U ofM is finitely presented. The coherent modules form a full abelian subcategoryof mod−S.S itself is called right coherent if x is coherent for all objects x of S. Note,

that the following holds:

• S is right coherent iff modfp−S coincides with the full subcategory ofcoherent modules.

• If S is right coherent, then all x belong to modfp−S iff D(S−modfp) ⊆modfp−S.

• If S is right coherent and x belongs to modfp−S then x− belongs tomodfp−S.

Dualising SpectroidsA k-spectroid is called dualising if D(S−modfp) = modfp−S.

Theorem 11. For a k-spectroid S the following assertions hold:

• S is dualising if and only if S is right and left coherent and moreoverx = D(S(x,−)) and D(S(−, x)) are coherent for all x in S.

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• If S is dualising then modfl−S ⊆ modfp−S.

• If S is locally bounded, then S is dualising.

• If S is dualising, then ind−S is dualising.

Almost Split Morphisms and Auslander-Reiten Sequences inmodfp−S

Let S be a dualising k-spectroid and recall that JS := radS. It es easy tosee that for the objects x of S the canonical injections ιx : xJS ↪→ x and duallythe canonical projections πx : x� x/ soc(x) are right minimal almost split resp.left minimal almost split morphisms in modfp−S.

Theorem 12. For a short exact sequence ε = 0→ Xf−→ Y

g−→ Z → 0 inmodfp−S the following assertions are equivalent:

• X, Z are indecomposable and f is left almost split.

• X, Z are indecomposable and g is right almost split.

• X is indecomposable and f is left minimal almost split.

• Z is indecomposable and g is right minimal almost split.

A short exact sequence ε = 0→ Xf−→ Y

g−→ Z → 0 in modfp−S satisfyingthe conditions of the theorem is called Auslander-Reiten sequence.

In an Auslander-Reiten sequence X is non-injective and Z is non-projective.

Corollary 13. Let Auslander-Reiten sequences ε = 0→ Xf−→ Y

g−→ Z → 0

and ε′ = 0→ X ′f ′−→ Y ′

g′−→ Z ′ → 0 in modfp−S be given. Then the followingassertions are equivalent:

• ε ∼= ε′

• X ∼= X ′

• Z ∼= Z ′.

The Existence of Auslander-Reiten Sequences

Theorem 14. Let S be a dualising k-spectroid and ε =

0→ Xf−→ Y

g−→ Z → 0 a short exact sequence in modfp−S such that Zis indecomposable. The following assertions are equivalent:

• Y g−→ Zπ−→ Z− → 0 is a minimal projective presentation in the category

modfp−(ind−S).

• ε is an Auslander-Reiten sequence.

• 0 6= ε ∈ soc Ext1S(−, X) and X is indecomposable.

We denote by P(−,−) the ideal of modfp−S formed by all morphisms fac-toring through an object in proj−S. Dually I(−,−) is the ideal of modfp−Sformed by all morphisms factoring through an object in inj−S.

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Corollary 15. Let ε = 0→ Xf−→ Y

g−→ Z → 0 be an Auslander-Reiten se-quence in modfp−S. Then ε can be used to define the following natural isomor-phisms of functors modfp−S → mod−k:

• Ext1S(−, X) ∼= D(HomS(Z,−)/P)

• Ext1S(Z,−) ∼= D(HomS(−, X)/I)

We denote by indS \ P the non-projective objects and by indS \ I thenon-injective objects in indS.

Theorem 16. If S is a dualising k-spectroid, then modfp−S has almost splitmorphisms. In particular, for each Z in indS \ P there is an Auslander-Reitensequence ending in Z and for each X in indS \ I there is an Auslander-Reitensequence starting in X.

Using the uniqueness of Auslander-Reiten sequences

0→ Xf−→ Y

g−→ Z → 0 we obtain a bijective map τ : indS \ P → indS \ Iby sending Z to τZ := X. The inverse of this map is usually denoted by τ−.

The Transposition FunctorGiven a k-spectroid S we obtain a contravariant k-functor −∗ : Mod−S →

S−Mod by sending M to M∗ as follows: M∗(x) := HomS(M, x) and M∗(a) :=HomS(M, a).

The restriction of−∗ to proj−S yields a duality−∗ : proj−S → S−proj. Thecompositions ν+ := D(−∗) : Mod−S → Mod−S and ν− := (D−)∗ : Mod−S →Mod−S are called Nakayama functors. Their restrictions to proj−S resp. inj−Syield mutually inverse equivalences proj−S → inj−S and inj−S → proj−S.

We construct a multilocal finite-dimensional k-category mat(proj−S) bydefining the morphisms p : P1 → P0 with P0, P1 in proj−S as objects andfor two such objects p : P1 → P0, p′ : P ′1 → P ′0 the set of pairs (f1, f0) with f1

in HomS(P1, P′1), f0 in HomS(P0, P

′0), p′f1 = f0p as space of morphisms.

By passing to cokernels we obtain a full and dense functor Φ :mat(proj−S)→ modfp−S. Denoting by P the ideal of mat(proj−S) consistingof the maps factoring through finite products of objects of the shape 0 → x,

x → 0, xidx−−→ x the functor Φ induces an equivalence Φ : mat(proj−S)/P →

modfp−S/P.The functor −∗ induces a duality mat(proj−S) → mat(S−proj) and

also a duality mat(proj−S)/P → mat(S−proj)/P. The composition Tr :=Φ(−)∗Φ−1 : modfp−S/P → S−modfp/P is a duality which is called the trans-position functor.

The Calculation of the Auslander-Reiten TranslationThere is some ambiguity in the construction of Tr which we want to use in a

convenient way: If we start with a non-projective indecomposable module M in

mod−S and choose a minimal projective presentation P1p−→ P0 −→M → 0,

then we can identify TrM with Coker p∗ and this cokernel turns out to beindecomposable. Therefore we always choose TrM := Coker p∗.

Theorem 17. Let S be a dualising k-spectroid.

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• If Z in mod−S \ proj−S, then τZ = DTrZ.

• If X in mod−S \ inj−S, then τ−X = TrDZ.

CA. The proof is constructive. It is worked out in [Ga3].

6 The Quiver of a k-Spectroid

Splitting FieldLet S be a k-spectroid. For each object x we have the canonical algebra

injection k → S(x, x). We identify k with its image in S(x, x) and say that kis splitting field for S if S(x, x) = k⊕ radS(x, x) for all objects x of S.

From now on we will assume that k is algebraically closed because this forcesall finite-dimensional skew field extension Dx := S(x, x)/ radS(x, x) ⊇ k to betrivial and automatically makes k to a splitting field for any k-spectroid S.

In general, even if we start with a spectroid S where k is splitting field, thenk will only rarely be a splitting field for ind−S.

Note however, that there is the intermediate case that S(x, x) = Dx ⊕radS(x, x) leading to the notion of species.

Admissible Ideals of Path CategoriesLet Q be a quiver. For all n ∈ N the ideal knQ of kQ is generated by all

paths u of length at least n. We observe that knQ = (k1Q)n.An ideal I of kQ is called admissible if I ⊆ k2Q and for all pairs x, y of

elements of Q there is ν(x, y) ∈ N such that kν(x,y)Q(x, y) ⊆ I(x, y)

Lemma 18. If I is an admissible ideal of kQ, then S := kQ/I is a localk-category with radS = k1Q/I.

Note, that for every pair x, y the set Q(x, y) yields a k-basis of radS/ rad2 S.Hence alle Q(x, y) are finite provided S is a k-spectroid. The converse in generalis not true. However, if Q is locally finite, meaning that in every point onlyfinitely many arrows start and finally many arrows stop, then S = kQ/I isactually a spectroid.

In analogy to the quiver case we define admissibility of ideals I of kS where Sis a poset and k+S is the ideal generated by all morphisms (y, x) where x > y.

The Quiver of a SpectroidFor a k-spectroid S and all pairs x, y of objects we define irrS(x, y) :=

radS(x, y)/ rad2 S(x, y). A morphism f in radS(x, y) is said to be irreducibleif the image of f is non-zero in irrS(x, y).

We define the quiver QS of S by choosing the objects of S as points of QS . Inaddition we insert dimk irrS(x, y) arrows from x to y. Note, that for a discreteposet S the Hasse quiver coincides with QkS .

A quiver morphism Φ : QS → S is called presentation of S if it is the identityon the objects and maps the arrows in QS(x, y) to a k-basis of irrS(x, y).

As seen above a presentation Φ : QS → S can be extended uniquely toa dense functor kQS → S which we denote be Φ as well. We observe thatΦ(knQS) ⊆ radn S.

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The k-spectroid S is said to be radical reduced if rad∞ S = 0. It is easy tosee that S/ rad∞ S is always radical reduced, S/ rad∞ S has the same quiveras S and any presentation Φ : QS → S yields a presentation Φ : QS → S →S/ rad∞ S.

Theorem 19. Let Φ : QS → S be a presentation of a spectroid S. Then S isradical reduced iff Φ is full and Ker Φ is an admissible ideal of kQS .

Remember that for a dualising spectroid S the simple modules are finitelypresented which shows that QS is locally finite. In addition, if S is locallybounded, then S is radical reduced.

Translation QuiversA translation quiver is a triple (Γ,PΓ, τΓ) satisfying the following properties:

• Γ is a locally finite quiver

• PΓ ⊆ Γ0

• τΓ : Γ0 \ PΓ → Γ0 is an injective map

• For each z in Γ0 \ PΓ and each y in Γ0 the equation card Γ(τz, y) =card Γ(y, z) holds.

The elements of PΓ are called projective, the elements of IΓ := Γ0 \ τΓ(Γ0 \PΓ)are called injective.

By abuse of notation we write Γ for the whole triple. Γ is called proper if forall z in Γ0 \ PΓ there exits y in Γ0 such that Γ(y, z) 6= ∅. Moreover Γ is calledstable if PΓ = ∅ = IΓ.

Theorem 20. If S is a dualising spectroid, then ΓS := (QS , {x : x ∈ S}, τS) isa proper translation quiver such that {x : x ∈ S} is the set of injective vertices.

Preprojective ComponentsFrequently the Auslander-Reiten quiver is not connected. Then it makes

sense to consider its components as translation quivers on their own right.Popular examples are the preprojective components where a translation

quiver Γ is said to be preprojective if Γ is directed, proper, PΓ is finite andor each x in Γ there exists n ∈ N0 satisfying τn(x) ∈ PΓ.

Preinjective translation quivers and components are defined dually.

The orbit graph O(Γ) of a preprojective translation quiver Γ has the pro-jective vertices of Γ as points and there are n edges from y to x if there is apredecessor z of x and t ∈ N0 such that τ tz = y and there are n arrows from zto x in Γ.

In particular, in this way the orbit graph O(C) of a preprojective componentC of the Auslander-Reiten quiver of a spectroid S is defined. Note, that sucha preprojective component C is called complete if all projectives x belong to C.In this case the vertex set of O(C) can be identified with S0.

Using right almost split sequences one sees, that if X belongs to a pre-projective component C of ΓS and Y is another module in indS satisfyingHomS(Y,X) 6= 0, then Y belongs to C and is a predecessor of X.

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The Grothendieck GroupFor a finite k-spectroid S the Grothendieck group of K0(S) with respect to

short exact sequences is the factor of the free abelian group on modlfd−S bythe subgroup generated by all differences Y − X − Z where there is a shortexact sequence 0→ X −→ Y −→ Z → 0. If we define the dimension vector ofa module M in modlfd−S as dim := (dimkM(x))x∈S , then dim induces anisomorphism K0(S)→ ZS .

In particular, for any Auslander-Reiten sequence 0→ X −→ Y −→ Z → 0we obtain dimX + dimZ = dimY . In a preprojective component this meansthe we can ”knit” all modules from the dimension vectors of the projectives andthe indecomposable summands of their radicals.

The x-component of the the dimension vector of M is given as dimkM(x) =dimk HomS(x,M). Using the defining property of Auslander-Reiten sequenceswe can generalise the above additivity formula in the following way: If Uis in indS and 0→ X −→ Y −→ Z → 0 is an Auslander-Reiten sequence inmodfp−S, then the following equation holds:

dimk HomS(U,X) + dimk HomS(U,Z) = dimk HomS(U, Y ) + δU,Z

7 Representation Types

Finite Representation TypeA finite k-spectroid S is said to be representation-finite if ind−S is a finite

category. More general, a locally bounded k-spectroid S is said to be locallyrepresentation-finite if for each object x of S there are only finitely many objectsM in indS satisfying M(x) 6= 0.

Theorem 21. If C is a component of the Auslander-Reiten quiver of a finitek-spectroid S with connected quiver QS . Then the following assertions are equiv-alent:

• S is representation-finite.

• C is finite.

• There is a bound n ∈ N such that dimk C ≤ n for all modules C in C.

For the case of a locally bounded spectroid the middle statement does not makesense but the first and third statement are still equivalent.

Tame and Wild Representation TypeFor the definition of tame representation type we write the given finite k-

spectroid S as kQS/I where I is an admissible ideal generated by a family (ρi).As noted above we can identify the set modd

lfd−S of modules M in modlfd−Ssatisfying dimM = d with a closed subvariety of the matrix representations ofQS defined by the polynomial equations imposed by the relations ρi.

S is said to be (representation)-tame provided for each possible dimen-sion vector d ∈ NS0 there exist finitely many 1-parameter families γ1, . . . , γµ(d)

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that up to isomorphism and finitely many exceptions each indecomposable inmodd

lfd−S lies in the image of one of the γi : k→ moddlfd−S.

S is said to be (representation)-wild provided there is an embedding functormodfd−k{X,Y } → modfd−S which preserves indecomposabiliy and isomor-phism classes

Theorem 22 (Drozd). Any finite k-spectroid is tame or wild but not both.

8 Algebras with Preprojective Components

The Separation ConditionIf we look for finite spectroids S whose Auslander-Reiten quivers have pre-

projective components then it is obvious that the points x such that x lies in sucha component must no lie on a cycle in QS . Therefore we study spectroids whosequivers QS are directed. Since we are mainly interest in finite type we mayassume that the considered spectroids are schurian meaning dimk S(x, y) ≤ 1for all x, y in S.

Since QS is directed we obtain a partial order ≤S on S0 by putting y ≤S x ifx is a predecessor of y. For each x in S on the set [x,S] := {x ∈ S : S(y, x) 6= 0}we impose another partial order ≤x by y ≤x z provided that S(y, x)S(z, y) 6= 0.Observe that ≤S is stronger than ≤x. Let us put ]x,S] := [x,S] \ x. Thepoint x is said to be separating if any two different components of ]x,S]≤x

liein different connected components of S0\]∞, x]S . The spectroid S itself is saidto be separating if all points x of S are separating.

Theorem 23 (Bongartz; Bautista, Larrion). Let S be a finite spectroid withdirected quiver. The following assertions hold:

• If S is separating then the Auslander-Reiten quiver ΓS has a preprojectivecomponent C whose orbit graph O(C) is a tree.

• Conversely, if ΓS has a complete preprojective component C whose orbitgraph O(C) is a tree, then S is separating.

Also algebras where the separation condition fails may have preprojectivecomponents.

Completely Separating SpectroidsA finite spectroid S is said to be completely separating if all full convex

subspectroids T of S are separating. That this is not a one-sided conditionfollows from:

Lemma 24. S is completely separating iff H1(T ) = 0 for all full convex sub-spectroids T of S.

Examining the simplicial homology and cohomology in more detail leads toH2(S,k∗) = 0 and one obtains:

Theorem 25. If S is a completely separating spectroid and S := (S0,≤S) thenthere is an admissible ideal I of kS such that S ∼= kS/I.

For S = kS/I the orders ≤S and ≤x coincide. Moreover, there are criteriawhich avoid to run through all convex subcategories. In fact 2n+1 subcategorieshave to be examined if S has n objects.

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Thin ModulesLet S be a spectroid. A module M in modlfd−S is said to be thin if

dimkM(x) ≤ 1 for all objects x of SFor a module M in modlfd−S we put suppM := {x ∈ S0 : M(x) 6= 0}.

Theorem 26. If S = kS/I is completely separating and M in modlfd−S isan indecomposable thin module with T := suppM , then T is I-convex andM ∼= kδ(T ).

The existence proof of preprojective components for separating spectroids isactually constructive and would yield even a ”knitting” algorithm on the levelof dimension vectors provided that the indecomposable summands of JS(−, x)are uniquely determined by their dimension vectors. This is true in general formodules in preprojective components. Nevertheless, for completely separatingspectroids also the above theorem serves for this purpose.

9 Ray Categories and their Coverings

Distributive spectroidsA k-category S is called distributive if S is locally bounded and the ideal

lattice is distributive. Distributivity is equivalent to:

• For all objects x of S there is dx ∈ N0 such that S(x, x) ∼= k[T ]/(T dx).

• For all objects x, y the bimodule S(x, y) is uniserial over S(x, x) or S(y, y).

Note, that for a distributive spectroid the inequality cardQS(x, y) ≤ 1 holds.

Theorem 27 (Jans). If S is a locally representation-finite, locally boundedspectroid then S is distributive.

Ray CategoriesA small category P is called base category if the following properties are

satisfied:

• Different objects are non-isomorphic.

• P is a category with zeros i.e. there is a zero morphism 0 = 0y,x in eachmorphism set P(x, y).

• idx 6= 0x,x for all objects x of P.

• f is nilpotent for all f ∈ P(x, x) where idx 6= f and x in P.

• In each object x of P only finitely many non-zero morphism start and onlyfinitely many non-zero morphisms stop.

A base category P is even a ray category if additionally the following propertiesare satisfied:

• P(x, x) is a cyclic semigroup for all x in P.

• Each P(x, y) is cyclic over P(x, x) or P(y, y).

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• If x, y, z are in P and morphims f ∈ P(x, x), g ∈ P(y, y) and h ∈ P(y, z)are given, then the equation 0 6= hgf = hf implies that g = idy.

A base category can be described by a path category of a quiver enrichedwith zeros and then subjected to a congruence relation.

CA. Base categories seem to offer themselves for some ”base field free” repre-sentation theoretic calculations. Up to my knowledge this was never pursued.

The Ray-Category of a Distributive Spectroid

Let S be a distributive spectroid. We obtain a ray category−→S in the fol-

lowing way:

• The objects of−→S are the objects of S.

• The morphism are the equivalence classes−→f of morphism f of S where

two morphisms f and g in S(x, y) are defined to be equivalent if there areautomorphisms t of y and s of x such that g = tfs.

• If f in S(x, y) and g in S(y, z) are given then there are two cases:

case 1 gJS(y, y)f ⊆ JS(z, z)gf + gfJS(x, x):

In this case we put −→g−→f :=

−→gf .

case 2 gf ∈ gJS(y, y)f 6= 0:

In this case we put −→g−→f := 0.

Conversely, if P is a ray-category, then k(P) = kP/(0y,x) is a distributivespectroid.

Theorem 28 (see [GR]). For a distributive spectroid S the following assertionshold:

• S is locally representation finite iff−→S is locally representation-finite.

• If S is locally representation finite and char k 6= 2 then S ∼= k(−→S ).

• If S is locally representation finite and char k = 2 then S is isomprphic to

a certain ”deformation” of k(−→S ).

CA. There does not seem to exist an implementation of the passage from S to−→S .

Zigzag-Chain Finite Ray-CategoriesA zigzag-chain in a base category P is by definition a finite sequence of non-

zero morphisms of the shape x0f1←− y1

g1−→ x1f2←− . . .

fn←− yn+1 or x0f1←− y1

g1−→x1

f2←− . . .gn−→ xn such that hfi 6= gi, fi 6= hgi, fi+1 6= gih, fi+1h 6= gi holds for

all possible morphism h.The category S is called zigzag-chain finite provided in every point x of S

only finitely many zigzag-chains start.

Theorem 29 (Jans Roiter Criterion). If S is a representation-finite distributive

spectroid, then−→S is zigzag-chain finite.

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The Universal Cover of a Ray-CategoryGiven a connected ray category P and an object x of P the universal cover

P and the fundamental group Π(P, x) are constructed as usual: We introduceformal inverse morphisms and a homotopy relation on the associated path cate-gory which respects non-zero compositions. The objects of P are the homotopyclasses of paths starting in x and the elements of Π(P, x) are the homotopy

classes of paths starting and ending in x. Note, that P is again a ray-category.

Theorem 30 (see [GR]). Let P be a ray-category. Then the following assertionshold:

• P is zigzag-chain finite iff P is zigzag-chain finite.

• If P is zigzag-chain finite then the canonical functor P → P is a Galoiscovering with free non-commutative automorphism group Π(P, x).

• If P is zigzag-chain finite then P is locally representation-finite iff P islocally representation finite.

CA. The calculation of the universal covering and the fundamental group canalso be based on the universal covering of QP which seems to be a possiblecomputational approach which was already used for zero-relation algebras.

Zigzag-Chain Finite, Simply Connected Ray-Categories

Corollary 31. If P is a zigzag-chain finite, simply connected ray-category, theneach of its full convex subcategories is simply connected and the quiver of QP isinterval finite. In particular, QP is directed.

In order to close the link to the previous section we define that a basecategory P is schurian if for x, y in P there is at most one non-zero morphimsin P(x, y).

Theorem 32 (Bautista Larrion). Let P be a base category. Then the followingtwo statements are equivalent:

• P is a simply connected, zigzag-chain finite ray category

• P is a schurian, zigzag-chain finite base category with directed, intervalfinite and locally finite quiver which in addition is separating.

Using the interval-finiteness it can be shown that being locallyrepresentation-finite can be decided inside finite convex subcategories. In fact,if P is the universal cover of a finite spectroid with n objects, then there aren finite ”fundamental domains” which only need to be examined. Returningto k(P) we are in the realm of completely separating algebras as consideredabove.

10 Criteria and Classification Lists

Critical Completely Separating SpectroidsA completely separating spectroid is called critical if it is not representation-

finite but each proper full convex subcategory is.

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Theorem 33 (Bongartz; Happel, Vossieck; see [Bo3] or [HV]). The criticalcompletely separating spectroids coincide with the preprojective tilting spectroidsT in ind−k∆ where ∆ is an extended Dynkin tree, i.e. the underlying graph isone of Dn, n ≥ 4 or En, 8 ≥ n ≥ 6.

There are 3 series arising from Dn, n ≥ 4, and 4302 exceptional spectroidsarising from En, 8 ≥ n ≥ 6. For visualisation they are usually condensed to 141frames.

Sincere Representation-Finite Completely Separating SpectroidsA module M in Mod−S is said to be sincere if M(x) 6= 0 for every x in S.

A representation-finite completely separating spectroid S is said to be sincereif there is an indecomposable sincere module in modlfd−S.

Theorem 34 (Bongartz; Ringel; see citeBo1 or citeRi2). The representation-finite, sincere, completely separating spectroids form a subset of the schuriantilting spectroids T in ind−k∆ containing preprojective and preinjective objectswhere ∆ is a tree with at most 4 end points.

There are 24 infinite series and 16344 isomorphism classes of exceptionalcases with at most 6 and at least 13 vertices which break down to 1664 frames.

The above theorem did not help for the classification of the exceptional cases,an inductive algorithm was used instead.

Since for an indecomposable module M in modlfd−S for a representation-finite, completely separating spectroid S the support suppM yields a convexsubcategory of S, by the above theorem all these modules are M classified.

Quadratic FormsLet I be a finite set, then χ ∈ Z[Xi : i ∈ I] is called unit form if χ =∑i∈I X

2i +

∑i,j∈I,i6=j χ{i,j}XiXj . A vector x ∈ ZI is called 1-root if χ(x) = 1.

It is called positive if x 6= 0 and xi ≥ 0 for all i ∈ I. The unit form χ iscalled weakly positive (resp. weakly non-negative) if χ(x) > 0 (resp. χ(x) ≥ 0)for all positive x in ZI . We denote by σk the reflection w.r.t. k defined byσk(x) := x− ∂kχ(x)e(k) where e(k) is the k-th canonical base vector.

Theorem 35 (Drozd). For a weakly positive unit-form χ the following asser-tions hold:

• χ has only finitely many positive 1-roots.

• For each positive 1-root x there exists a sequence j1, . . . , jm and k ∈ Isuch that x = σj1 . . . σjm(e(k)).

CA. The second statement can actually be used for testing weak positivity.

The Tits FormLet S be a finite k-spectroid with directed quiver QS . The Tits form qS of

S is the unit form qS :=∑x,y∈S(

∑2ν=0 dimk ExtνS(x−, y−))XxXy.

Lemma 36. Suppose that S = kQ/I where I is admissible and generated by aminimal family (ρx,y)x,y∈S . Then for all x, y in S the following assertions hold:

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• dimk Ext1S(x−, y−) = cardQ(y, x).

• dimk Ext2S(x−, y−) = dimk(I/k1QI + Ik1Q)(y, x) = card ρ(y, x).

Theorem 37 (Bongartz). Let S be a finite spectroid with directed quiver QSwhose Auslander-Reiten quiver has a preprojective component. Then the follow-ing assertions are equivalent:

• S is representation-finite.

• For each d ∈ NS0 there are only finitely many modules M in modlfd−Ssuch that dimM = d.

• The Tits form qS is weakly positive.

In addition, if the above assertions are satisfied, then the map dim induces abijection from ind−S onto the set of positive 1-roots of qS .

Note, that another interesting quadratic form is the Euler form χS :=∑x,y∈S(

∑∞ν=0 dimk ExtνS(x−, y−))XxXy which is closely related to the Cartan

matrix of S and coincides with qS if gl.dimS ≤ 2.

Excursion on Tame TypeThe analogous result on the Tits form concerning tame representation type

and weak non-negativity was recently proved in [BPS] for the class of stronglysimply connected spectroids.

From this it follows as well that the hypercritical completely separating spec-troids coincide with the schurian preprojective tilting spectroids T in ind−k∆where ∆ is a hyperextended Dynkin tree, i.e. the underlying graph is one of˜Dn, 8 ≥ n ≥ 4, T2,2,2,2,2 or

˜En, 8 ≥ n ≥ 6.

A completely separating spectroid is called hypercritical if it is not tame buteach proper full convex subcategory is.

These spectroids were calculated by Unger (see [Un]). There are 13747

isomorphism classes coming from the˜En cases and a small list coming from the

remaining cases.

Subspace CategoriesA vector space category is a pair (S,M) where S is a spectroid and M is in

addS−modfd. It is called faithful if M is a faithful module.The subspace category U(S,M) of (S,M) is a finite-dimensional multilocal

k-category whose objects are the triples V = (Vω, γV , V0) where Vω is a finite-dimensional k-space, V0 is an object of addS and γV : Vω →M(V0) is a k-linearmap whereas the morphisms V →W are the pairs (fω, f0) such that fω : Vω →Wω is k-linear, f0 : V0 →W0 is a morphism in addS and γW fω = M(f0)γV .

By passing from U(S,M) to U(S/KerM,M) we only loose the indecompos-able objects (0, 0, Z) where Z in S and M(Z) = 0. Thus for understanding theindecomposables in U(S,M) it is no harm to assume that M is faithful.

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Classifications for Linear Vectorspace CategoriesA faithful vectorspace category is said to be linear provided dimkM(x) =

1 for all x in S. It is immediate to see that a linear vectorspace category(S,M) is isomorphic to (kS, can) where S is the partially ordered set (S,≤S)as introduced earlier and can is the indicator module of S.

The subspace category of a linear vectorspace category (kS, can) is alsoknown as the category of representations of the poset S.

An indecomposable subspace V in the subspace category U(kS, can) is saidto be exact if each object x of S appears as summand of V0. A finite poset issaid to be exact if there exists an indecomposable exact subspace.

Theorem 38 (Kleiner). Let S be a finite poset, then the following assertionshold.

• (kS, can) is subspace-finite iff S does not contain one of the 5 posets ofKleiner’s first list (see [Kl]) as full subposet.

• If (kS, can) is exact and subspace-finite, then it is one of the 13 posets ofKleiner’s second list (see [Kl], where even the matrices for the correspond-ing maps γV are displayed).

Theorem 39 (Nazarova). If S is a finite poset, then (kS, can) is subspace-tameiff S does not contain one of the 6 posets of Nazarova’s list (see [Na]) as fullsubposet.

CA. The significance of the poset access to representation types is on one handthat the respective lists are short and on the other hand that posets give rise tomuch more efficient data structures and algorithms than quivers with relations.

One-Point Extensions and Fiber Sum FunctorsLet S be a finite spectroid with directed quiver and x an object of S. Define

Kx to be the subcategory of objects Z in ind−S satisfying τSZ(x) = 0. The fibersum functor Fx : U(Kx,HomS(x,−))→ mod−S which maps V to the cokernelof the adjoint homomorphism Vω ⊗k x → V0 of γV ∈ Homk(Vω, HomS(x, V0))is full and dense. Moreover, its kernel is an ideal generated by finitely manyindecomposable objects.

Theorem 40. If S is a finite spectroid with directed quiver and x an object ofS, then S is representation-finite iff (Kx,HomS(x,−)) is subspace finite.

There is an analogous result concerning tameness where one has to involvethe tameness of S/ < idx >.

The point is that frequently one can reduce calculations to spectroids withless objects. The most obvious case is that x is a sink. Then we arrive at theclassical one-point extension technique, i.e. (Kx,HomS(x,−)) can essentially beidentified with (ind−T , HomT (R,−)) where T is the convex subcategory of Sobtained by removing x from S and setting R := radS(−, x).

Thin Start ModulesUnfortunately there is no guarantee that the faithful version of the vec-

torspace category (Kx,HomS(x,−)) happens to be linear.

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Let us denote by Lx the full subcategory of Kx given by the thin modules.If we require that S = kS/I is completely separating, then we now that thethin modules are given by I-convex subsets of S which leads to the followingobservations:

• Lx can be computed from (S, I) combinatorially.

• The poset Lx underlying the linear vectorspace category (Lx,HomS(x,−))can be computed from (S, I) combinatorially.

Theorem 41. Let S be a finite, completely separating spectroid. Then S isrepresentation-finite (resp. representation-tame) if and only if (Lx,HomS(x,−))is subspace-finite (resp. subspace-tame) for all objects x of S.

In addition, if S is representation-finite, then each M in ind−S lies inFx(U(Lx,HomS(x,−))) for some x.

The last statement shows that the inner structure of any indecomposablemodule over a representation-finite, completely separating spectroid can actuallybe derived from Kleiner’s second list. Using the associated pushdown functorof the module categories along the universal covering this carries over to anyindecomposable over a representation-finite spectroid.

11 Computer Algebra in Representation The-ory of Algebras

CREPThe CREP system used mainly Maple as user interface, Pascal and C as

programming language and flat files as data bases. The following kernel packageswhere based on the framework introduced above:

• CREP base package, containing all basic routines for completely separat-ing algebras, poset algebras and posets.

• esrd package, dealing with the sincere, completely separating algebras offinite representation type.

• heralg package, dealing with heraditary algebras.

• xpreproj interface, providing graphical access to algorithms for calculationof preprojective components.

• tubular package, dealing with tubular algebras.

• repetit package, dealing with repetitive algebras.

• tpe package, dealing with critical, pg-critical and hypercritical completelyseparating algebras.

• uforms package, dealing with quadratic forms.

• hall polynomials package, dealing with Ringel-Hall algebras.

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Possible Topics To Be ApproachedThe following topics seem to offer themselves for computer calculations, but

to my knowledge they were not yet tackled:

• Computationally canonical multiplicative bases in relation to Groebnerbases

• Distributive and ray categories

• Strongly simply connected and (hyper-)canonical algebras

• Tubes, coils, directing components, A∞-components

• String algebras

The Point of View of Industrial Software EngineeringIt may be interesting to look at computer algebra in the representation

theory of algebras under the experience of 10 years work with industrial ITlifecycle processes. We do this in the form of questions which would have to beanswered if one would try to apply an industrial approach:

• Is there a market (industry, mathematics community) for computer alge-bra in representation theory of algebras and/or is it even a core compe-tency of representation theory of algebras community?

(If no, there may still be local computer algebra tools, but they cannot be”sold” as services.)

• Does the representation theory of algebras community have a ”businesscase” to support ”Plan, Build, Run” for computer algebra?

• Which conceptual architecture for the algorithms is to be used?

• Which system and process environment is to be used?

• Will current IT trends and best practices like mobility via tablets, 3-tierarchitecture, Web2.0, etc. be addressed?

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Peter DraxlerIT Service Center, Kassel UniversityMonchebergstr. 11, D-34109 Kassel, GermanyEMail: peter.draexler@uni-kassel.de

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