CLUSTER ALGEBRAS AND CATEGORIFICATION TALKS: QUIVERS AND AUSLANDER-REITEN THEORY

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1 CLUSTER ALGEBRAS AND CATEGORIFICATION TALKS: QUIVERS AND AUSLANDER-REITEN THEORY ANDREW T. CARROLL Notes for this talk come primarily from two sources: M. Barot, ICTP Notes Representations of Quivers, research.html L.A. Hügel, ICTP Notes An Introduction to Auslander-Reiten Theory, http: //profs.sci.univr.it/~angeleri/publ.html For today s talk, we will fix k = k an algebraically closed field of characteristic 0.. Some Classical Questions in Linear Algebra Classify square matrices up to similarity (I.E. orbits in End k (k n ) under the action of GL(k n )) Classify n m linear maps up to conjugation (I.e. orbits in Hom k (k m, k n ) under the action of GL(k m ) GL(k n )) Classify pairs of linear maps up to simultaneous conjugation (I.e., orbits in Hom k (k m, k n ) Hom k (k m, k n ) under the action of GL(k m ) GL(k n )) Classify n-tuples of subspaces V, V 2,..., V n W under simultaneous change of basis (for example, triples of lines in two-space). There is no predetermined name for the equivalence relation under the appropriate action of the product of general linear groups, so we start to simply ask for a normal form. These problems all have a common thread: determine a normal form for collections of linear maps in some configuration. Quite simply, matrix problems. For many years, similarity of collections of matrices in certain prescribed block forms were studied intrinsically; but starting in the seventies, the point of view pivoted. The idea: square matrices, linear maps pairs of linear maps, objects in a category C n-tuples of subspaces } similarity, equivalence isomorphism class in C conjugation, change of basis Date: September 5, 20.

2 2 ANDREW T. CARROLL 2. Quivers and their Representations Definition 2.. A quiver Q = (Q 0, Q ) is a directed graph consisting of a set of vertices Q 0, and arrows Q. We further define two maps t, h : Q Q 0 with t(a) the tail and h(a) the head of the arrow a. Example 2.2. a 2 b 3 Definition 2.3. A representation V of a quiver Q is an assignment of a finite-dimensional vector space V x to each vertex x Q 0, and an assignment of a linear map V a : V t(a) V h(a) for each arrow a Q. Example 2.4. Below are a few quivers, and what a representation would look like: Q Representation V a 2 V V a V 2 a V a V What should a morphism in such a category be? Intuitively it should be a collection of linear maps at each vertex, but it should also respect the structure arising from the linear maps. Definition 2.5. A morphism between two representations V, W of Q is a Q 0 -tuple of linear maps (ϕ x : V x W x ) x Q0 such that for every a Q V ta ϕ t(a) W t(a) ϕ t(a) V h(a) W a ϕ h(a) W h(a) The category of representations of Q with morphisms so-defined is denoted Rep Q. Note. Notice that there is a natural direct sum operation. Namely, the representation V W defined by (V W ) x = V x W x [ ] Va 0 (V W ) a = 0 W a indeed satisfies the universal property of the coproduct. Proposition 2.6. We list, now, some properties of Rep Q, which will become clear after a remark to follow:

3 CLUSTER ALGEBRAS AND CATEGORIFICATION TALKS: QUIVERS AND AUSLANDER-REITEN THEORY3 i. Rep Q is an abelian category (i.e., Hom Q (V, W ) are abelian groups, the category admits finite coproducts, kernels and cokernels. In addition, monomorphisms are kernels and epimorphisms are cokernels); ii. In fact, Rep Q is a k-category (i.e. Hom Q (V, W ) are vector spaces); iii. Rep Q is a Krull-Schmidt category, i.e., every representation decomposes uniquely into a direct sum of indecomposable representations (i.e., which cannot be written as a non-trivial direct sum of two other representations). (i. and ii. are quite simple to prove on their own. iii. takes some work and uses Fittings lemma) Remark 2.7. Let us define by kq the following algebra: as a vector space kq is defined to be the free vector space generated by the set of directed paths in Q, including the lazy paths (i.e., paths of length zero commencing and terminating at a single vertex). The multiplicative structure is given as follows: { pq if h(q) = t(p) p q := 0 otherwise (Some notation: if e x is a lazy path at the vertex x, then h(e x ) = t(e x ) = x. Also, we have extended the functions t and h to the set of all directed paths. It should be clear what the head of a directed path is.) The algebra kq is called the path algebra of Q. Proposition 2.8. The category of (finite-dimensional) modules over the path algebra kq is equivalent to the category of (finite-dimensional) representations of Q. It is not difficult to construct such an equivalence: if V is a representation of Q, let V be the vector space x Q 0 V x. Then e x acts on V as the projection onto the subspace V x, and the arrow a acts by V a on the summand V t(a) and trivially on the other summands. Since kq is generated as an algebra by the lazy paths and the arrows (exercise), this data defines a module structure Exercise: If ϕ : V W is a morphism of representations, determine the corresponding homomorphism of modules. Now in order to understand such a category, we should look for its indecomposable objects, for from them comes everything else. Here s a first approximation. 3. Examples of Rep Q ad hoc techniques 3.. Vector Spaces. Let Q be the digraph consisting of a single vertex and no arrows. The objects of Rep Q are vector spaces, and Hom Q (V, W ) = Hom k (V, W ). Notice that if n >, k n = k n k, so the only indecomposable (non-trivial) representation is k.

4 4 ANDREW T. CARROLL a 3.2. Linear Maps. Let Q = 2. Then the objects of Rep Q are vector spaces V, V 2 together with a linear map V a : V V 2. Suppose that V a is of rank r. Then by row and column operations (i.e., change of basis in V and V 2, any such map can be written in the form: [ ] Ir 0 V a = 0 0 where I r is the r r identity matrix. Therefore, the representation splits into a direct sum: (k 0) n r (0 k) m r ( k k ) r Those in the parentheses are the non-trivial indecomposables, so indeed finitely many Endomorphism. Let Q = a. The objects of Rep Q are pairs (V, ϕ) where ϕ is a linear endomorphism of V. It is well known that by change of basis in V, such an endomorphism has the associated Jordan normal form: J r (λ ) J r2 (λ 2 ) J rk (λ k ) Such a representation splits into a direct sum of k indecomposable modules, and indeed (k r, J r (λ)), the r-dimensional vector space with a single r r Jordan block (associated to eigenvalue λ) is indecomposable. Notice that in this case, in each dimension of V, there is a one-parameter family of indecomposable modules (therefore, there are infinitely many non-isomorphic indcomposables) Triples of Linear Maps. Let Q = 2 be the so-called three-kronecker quiver. The objects are triples of linear maps, but look: 0 k 0 λ µ k 2 So there is a two-parameter family of indecomposable representations with k, k 2 as the assigned vector spaces. Module categories in which you get two-parameter families are BAD... impossible to determine/classify all of their indecomposable objects. Interesting note: if we could solve this problem (i.e., determine all indecomposables in this category), then we could do so for ANY category of representations of quivers. (see Barot)

5 CLUSTER ALGEBRAS AND CATEGORIFICATION TALKS: QUIVERS AND AUSLANDER-REITEN THEORY Classification problem. The classification problem amounts to determining a complete list of indecomposable modules/representations for such categories. It would be nice to also understand the morphisms between modules. For the former question, there are a number of results, not least of which is the result of Drozd, which essentially says that we just covered all of the types. Additionally, the methods we have used thus far have been quite ad hoc. It would be nice to have a systematic method for studying these categories. Thus, Auslander-Reiten theory is born. Definition 3.. A quiver Q (or the path algebra kq) is of finite representation type if there are only finitely many indecomposable representations of Q; tame representation type if in each dimension vector, there are at most finitely many one-parameter families of pairwise non-isomorphic indecomposable representations; wild representation type if there are dimension vectors for which there is a twoparameter family of pairwise non-isomorphic indecomposable representations. Theorem 3.2 (Drozd). A quiver Q can be of finite, tame, or wild representation type. If Q is of infinite representation type, then it is either tame or wild, but not both. (This is a very complicated proof. It involves the introdution of so-called BOCS, bimodules over categories with coalgebra structure.) 4. Gabriel s Theorem/ Kac s Theorem So which quivers are of finite and which are tame, and which are wild? Theorem 4.. Let Q be a quiver, and denote by (Q) its underlying undirected graph.

6 6 ANDREW T. CARROLL a. Q is of finite representation type if (Q) is a (simply laced) Dynkin diagram (A n, D n, E 6, E 7, E 8 ); A n =... D n =... E 6 = E 7 = E 8 = Furthermore, the dimension vectors of indecomposable representations are the positive roots of the associated root system.

7 CLUSTER ALGEBRAS AND CATEGORIFICATION TALKS: QUIVERS AND AUSLANDER-REITEN THEORY7 b. Q is of tame representation type if (Q) is an extended (simply laced) Dynkin diagram (Ãn, D n, Ẽ6, Ẽ7, Ẽ8); Ã n =... D n =... Ẽ 6 = Ẽ 7 = Ẽ 8 = Furthermore, the dimension vectors of indecomposable representations are the positive roots of the associated root system, and dimension vectors of one parameter families are the positive imaginary roots. c. Otherwise, Q is of wild representation type. 5. Auslander-Reiten Theory Definition 5.. A morphism ϕ : V W is called radical if, when written as a matrix of maps between indecomposable objects, no entry is an isomorphism. Note 2. Every map ϕ : V W with V, W, indecomposable non-isomorphic is radical. Proposition 5.2. rad Q (V, W ) is a vector space, and composition on either side with a radical morphism is again radical.

8 8 ANDREW T. CARROLL 5.. Example. Let Q = 2... n. The indecomposable modules can be written [i, j] with i < j n (i.e., [i, j] x = k if i x j, and 0 otherwise, and the maps are given by multiplication by ) Exercise: a. Show that Hom Q ([i, j], [i, j ]) 0 if and only if i i j j, and in this case, the dimension of the space of homomorphisms is. b. Show that End Q ([i, j]) = k. Let us take n = 4. Notice that Hom Q ([3, 4], [2, 4]) = k, and the morphisms here DO NOT factor through non-isomorphisms. Definition 5.3. Inductively define the space rad i Q(U, V ) := W Rep Q rad Q (W, V ) rad i (U, W ). A radical map f is called irreducible if f rad Q (U, V ) \ rad 2 Q(U, V ). Note 3. An irreducible morphism is either injective or surjective but not both (otherwise factor it through the image, which is not isomorphic to the domain since the morphism is assumed not injective). Definition 5.4. A short exact sequence 0 X f Y if g Z 0 is called almost split a. X and Z are indecomposable; b. f is not a section, and g is not a retraction; c. Any section X E factors through f, and any retraction E Y factors through g. Theorem 5.5. Auslander-Reiten Main Theorem For any quiver Q, there is a bijective map τ : ind(modq) ind(modq) such that for each non-projective indecomposable V, there is a representation E, and morphisms f, g such that 0 τv f E g V 0 is an almost split sequence. Corollary 5.6. Given V, τv, let E,..., E t be the set of indecomposable modules such that rad Q (τv, E i )/ rad 2 Q(τV, E i ) = n i 0. Write E = i E n i i. Let {f (i),..., f n (i) i } be a (i) basis for this space, and chose lifts f j. Then f (i) j i,j () coker( τv E ) = V ; (2) 0 τv E V 0 is an almost split sequence. It is convenient to organize this data in a directed graph.

9 CLUSTER ALGEBRAS AND CATEGORIFICATION TALKS: QUIVERS AND AUSLANDER-REITEN THEORY9 Definition 5.7. The Auslander-Reiten quiver Γ Q of Q is the directed graph with vertices corresponding to the set of isoclasses of indecomposables in Rep Q, and the number of arrows [M] [N] is dim rad Q (M, N)/ rad 2 Q(M, N). Example 5.8. Suppose Q = 2 3, denote by [i, j] the indecomposables (there are 6). Recall that the irreducible maps are [3, 3] [2, 3] [, 3] [2, 3] [2, 2] [, 2] [, 3] [, 2] [, ] We can draw the Auslander-Reiten Graph as follows: [, 3] [2, 3] τ [, 2] [3, 3] τ [2, 2] τ [, ] Notice that quadrilaterals are exact sequences Beautiful Combinatorics. I. Hom Q (C, τa) = D(Ext Q(A, C)), so dimensions of extension spaces can be read from homomorphism spaces (much easier to calculate); II. If M and N are two indecomposable modules, and [M], [N] are in the same connected component, then dim Hom Q (M, N) = #{ directed paths starting with M and ending with N} modulo the mesh relations BUT HOW DO YOU CALCULATE IT. So this is all quite nice, but how can these modules be calculated? Quite simply, in fact, once we know projective modules. It turns out that τ = D T r as functors (we should really say naturally isomorphic). But what is D and what is T r. First we need some properties: The category Rep Q is Hereditary, meaning subrepresentations of projective modules are projective (or equivalently that Ext i (V, W ) = 0 for all V, W Rep Q and i > ). Proof. Define d V W : x Q 0 Hom k (V (x), W (x)) a Q Hom k (V (ta), W (ha)) where d V W (ϕ(x) x Q 0 ) = (ϕ(ha)v (a) W (a)ϕ(ta)) a Q.

10 0 ANDREW T. CARROLL Exercise 5.9. The kernel of this map is isomorphic to Hom Q (V, W ) and the cokernel is Ext Q(V, W ). The projective cover of the simple module concentrated at the vertex x, which we denote P x has a basis consisting of paths starting with the vertex x, and left multiplication is given by left concatenation of paths; The injective cover of a simple module concentrated at the vertex x, which we denote I x has a basis consisting of paths ending at the vertex x, and left multiplication is given by right concatenation; Hom Q (P x, P y ) has a basis consisting of paths which start at y and end at x. Therefore Hom Q (P x, kq) has a basis consisting of paths which start at any vertex and end at x. This is actually a right kq module, or a left kq op module. Definition 5.0 (works in general for semiperfect rings R). The functor T r is defined on the full subcategory of non-projective modules as follows: let V be such a module, and let 0 V x Q 0 Px nx δ x Q 0 Px mx, where δ is a matrix whose entries are paths (recall that maps between projectives are paths). Apply Hom Q (, kq) to the resolution: x Q 0 Hom Q (P x, kq) nx Then T r(v ) := coker(hom Q (δ, kq)) Hom Q (δ,kq) x Q 0 Hom Q (P x, kq) mx For the sake of our discussion, the dual can be taken to be the functor ( ) op (although this is an oversimplification) Example. Let Q = a 2 b 3 4. Let us calculate τ(s 2 ). It has a projective resolution c which is really Apply Hom Q (, kq) to yield the sequence 0 2 P 2 P 3 P 4 P 2 b c P 3 P 4

11 CLUSTER ALGEBRAS AND CATEGORIFICATION TALKS: QUIVERS AND AUSLANDER-REITEN THEORY which are modules over Q op := a b 2 3 c The cokernel of this map (by dimension count) is of dimension vector. Apply the dual (op), to arrive at the representation id id (which, since τ(indecomposable) is indecomposable, we know the maps are as shown in above. Department of Mathematics, Northeastern University, Boston, MA 025 address: id

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