Quiver Representations

Transcription

1 Quiver Representations Molly Logue August 28, 2012 Abstract After giving a general introduction and overview to the subject of Quivers and Quiver Representations, we will explore the counting and classification of indecomposable representations of general quivers and quivers with relations. 1 Introduction To begin, I will define what a quiver is. Essentially a quiver is a finite directed graph, where loops are allowed and there can be more than one arrow between two vertices. A more formal definition follows: Definition 1. A quiver is a set Q = (Q 0, Q 1, where Q 0 is a finite set of vertices and Q 1 is a finite set of arrows. Here are some examples of quivers: Example 2. Example 3. For notation s sake, if a is an arrow in a quiver, we will say that ta is the vertex at the tail of arrow a and ha is the vertex at the head of arrow a. The study of quivers themselves is left to graph theorists, since our main concern is representations of quivers. To represent a quiver means to fix a field, F, and to assign an F-vector space to each vertex of the quiver. Then each arrow in the quiver is assigned an F-linear map between the assigned vector spaces. A more formal definition follows: 1

2 Definition 4. Fix a F a field. A representation of a quiver Q = (Q 0, Q 1 is a set of vector spaces V x, where x is in Q 0, together with a set of F-linear maps V a (a Q 1 such that V a maps V ta to V ha. Example 5. Consider the quiver An example of a representation of this quiver would be C A C 2 C 2, C 4 ( 1 where A = 2 ( 1 0, = 2 C ( 0, and C = 1 Definition 6. If V is a representation of a quiver Q = (Q 0, Q 1 (where Q 0 = 1, 2, 3,..., n, then the dimension vector of Q is the n-tuple α = ({α 1, α 2,..., α n } such that α x is the dimension of V x. Example 7. In the above example, the dimension vector would be (1, 2, 2, 1 As it turns out, representations of quivers form a category, so we can define morphisms between quivers and study patterns and properties of quiver representations. The definition of a morphism is as follows: Definition 8. Let Q = (Q 0, Q 1 be a quiver, and V, W be representations of Q over a field F. Then a morphism between V and W is a set of F -linear maps {ψ x : V x W x } such that the following diagram commutes for all a Q 0 : V ta V a V ha I.e. ψ ta W a = V a ψ ha, for all a Q 0 ψ ta ψ ha W ta W a W ha If all of these ψ x maps are invertible, then we say the morphism ψ : V W is an isomorphism of representations of Q. With the definition of a morphism of representations of a quiver Q, we can better understand this category: Rep F (Q := {representations of a quiver Q over a field F}. Each different quiver or base field gives a new category of representations of quivers. Example 9. Consider the following quiver representation: V : C 2 A 1 C A2 ( C 2, 1 where A 1 = (1 2, and A 2 =. Consider another representation 0 of the same quiver: C 2 1 C 2 ( C 2 where 1 = (1 1, and 2 = i 2.Then φ = (φ 1 1, φ 2, φ 3, where is a morphism of these representations 2

3 ( 2 1 φ 1 = 0 3 commutes: ( i 0, φ 2 = (2, and φ 3 = 2 C 2 A 1 C A2 C, since the following diagram φ 1 φ 2 C 2 1 C 2 C Since φ 1, φ 2, andφ 3 are all invertible, this is an isomorphism of representations. Given an arbitrary quiver, Q, with no other restrictions, there are a huge number of representations, since each vertex could be represented by a vector space of any finite dimension, and there could be infinitely many possible linear maps between any two vector spaces. In order to narrow the field of study, we must first categorize quiver representations as either decomposable or indecomposable. To understand what this means, we must understand the direct sum of two quiver representations. Definition 10. If V and W are representations of a quiver Q over a field F, then their direct sum Z = V W is defined as follows: Z x = V x W x, for all x in Q 0 ( Va 0 Z a =, for all a in Q 0 W 0 a Definition 11. A representation of a quiver Q is decomposable if it can be expressed as the direct sum of two or more nontrivial representations of Q Example 12. Consider the quiver Q :. Let V be the following representation of Q. C 3 A C 3, where A is given by the matrix V decomposable, and is isomorphic to the following representation ( of Q: [ C A1 C ] [ C 2 A C 2 ], where A 1 = (2, and A 2 =. 5 3 This example shows how a representation can be decomposed into a direct sum of simpler representations. The simplest representations to study are those that cannot be decomposed at all: Definition 13. A representation of a quiver Q is indecomposable if it cannot be written as a direct sum of representations of Q. Example 14. The following quiver representations are indecomposable: C φ

4 R 2 F R 2 F 2 5 (Note: over C, this is not indecomposable These examples highlight an important definition: Definition 15. A representation of a quiver is absolutely indecomposable over a field F if it is indecomposable over the algebraic closure of F Example 16. So the quiver: R R 2 is not absolutely indecomposable, since over C, this can be decomposed into C i C C i C The definition of indecomposable leads us to an important theorem in representation theory of quivers: Proposition 17. Given a representation V of a quiver Q over a field F, V can be decomposed uniquely (up to isomorphism into a direct sum of indecomposable representations This theorem follows from the Krull-Remak-Schmidt Theorem [3], which states a more general result involving modules with local endomorphism rings. This theorem allows us to narrow the field of study to focus only on indecomposable representations of quivers, since all representations of quivers are direct sums of indecomposables. 2 Indecomposable Representations Since each representation of a quiver Q can be described as the direct sum of indecomposable representations, it makes sense to study indecomposable representations of specific quivers. In some cases, it is possible to describe all of the indecomposable representations of a quiver up to isomorphism. In other cases, a quiver may have infinitely many indecomposable representations. Consider the following example: Example 18. Consider representations the quiver over a fixed field, F. How does a general representation decompose into indecomposables? Fix two vector spaces, V and W and a map, φ, such that V injective, then we can decompose this representation into ker(φ 0 V φ W φ W. If φ is not 4

5 , where V is the compliment of ker(φ in V. If φ is not surjective, then we can decompose further: ker(φ 0 V φ im(φ 0 W, where W is the complement of im(φ. These components are multiples of the following indecomposable representations: 1 0, 1 1, 0 1 In this example, the quiver has exactly 3 indecomposable representations (over any field up to isomorphism, and we can explicitly describe what they are. The slightly more complicated quiver, also behaves nicely and has exactly 6 indecomposable representations. Not all indecomposable representations of quivers can be described so easily, however. For example, the quiver has infinitely many indecomposable representations. For some quivers, we can explicitly describe all of the indecomposable representations up to isomorphism. For other quivers, this is difficult or impossible. However, we may be able to count the indecomposables, even if we aren t able to describe them. 3 Counting Indecomposable Representations As we saw in the previous section, the quiver has exactly 3 indecomposable representations up to isomorphism, and they can be easily described. Other quivers, however, are not as simple to describe. For example, the quiver may look simple, but it has infinitely many indecomposable representations (since one arrow could be represented by the identity matrix, and the other any Jordan block. Certain types of quivers have finitely many indecomposable representations up to isomorphism, while others are not so wellbehaved. Pierre Gabriel proved the following theorem that explicitly classifies which quivers have finitely many indecomposables Theorem 19. A quiver has finitely many indecomposable representations up to isomorphism if and only if the underlying (undirected graph is a union of the following Dynkin diagrams: A n : D n : 5

6 E 6 : E 7 : E 8 : This theorem tells us explicitly which quivers will have finitely many indecomposable representations up to isomorphism. It gives an explicit way of counting the indecomposable representations of quivers that are one of these types. However, it does not tell us the number of indecomposables for more general quivers. Counting these indecomposables in general requires a more group theoretic approach. First, we restrict to the case where our base field is finite: F q. Fix a quiver Q and a dimension vector α (where α = (α 1, α 2,...α n. We can then consider the set, denoted Rep α (Q of all representations of Q with this dimension vector over the finite field, F p. Now we consider the action of GL α = GL α1 GL α2 GL αn on Rep α (Q by conjugation. This action can be better understood through an example: Example 20. Let Q :. Fix a dimension vector α = (2, 1, 3, 2. Let F 2 A F F 3 C F 2 be an indecomposable representation (call it V of this quiver. So A is a 1x2 matrix, is a 3x1 matrix, and C is a 2x3 matrix. Then GL α acts on V as follows: (φ 1, φ 2, φ 3, φ 4, (A,, C (φ 1 Aφ 2, φ 2 φ 3, φ 3 Cφ 4 This gives us a new representation of this quiver: F 2 φ 1Aφ 2 F φ 2φ 3 F 3 φ 3Cφ 4 F 2 This new representation is isomorphic to V. Consider the diagram: F 2 A F F 3 C F 2 F 2 φ 1 φ 1Aφ 2 F φ 2 φ 2φ 3 F 3 φ 3 φ 3Cφ 4 Sine this diagram commutes, this is a morphism of representations. And since each φ i is invertible, these two representations are isomorphic. F 2 φ 4 6

7 When an element of GL α acts on an element V of Rep α (Q, it yields a new representation that is isomorphic to V. This gives us the following fact: Proposition 21. The number orbits in Rep α (Q[F q ] (the set of indecomposable representations of Q over F q under the action of GL α (F q equals the number of indecomposable representations of Q over F q up to isomorphism. Now, instead of counting indecomposable representations, the problem is to count the number of orbits. To do this requires the use of generating functions. We can let One useful fact when counting indecomposable representations of quivers is that the orientation of the arrows of a quiver does not affect the number of indecomposables. Consider the following example: Example 22. The quiver has 6 indecomposable representations: 1 0 0, 0 0 1, 1 1 0, has the following indecomposable representations: 1 1 1, The quiver 1 0 0, 0 0 1, , 1 1 0, These quivers are the same except for the orientation of the arrows, and they have the same number of indecomposable representations. This example displays a more general fact: Theorem 23. The number of indecomposable representations of a quiver over a field F with a dimension vector α is independent of the orientation of the arrows in the quiver. [4] 4 Quiver Representations with Relations Over a finite field, for fixed dimension vector, there are only finitely many representations. ut quivers with relations are an interesting class of algebras for which we would like to understand the indecompsable representations. As stated previously, certain quivers, such as or have infinitely many indecomposable representations, in which case trying to count them all would be unfruitful. However, if we impose certain restrictions on the representations, we can simplify this question. 7

8 Example 24. Consider representations of the quiver over a fixed field, K, where both compositions are zero. I.e., consider representations Where C A = C = 0. V 1 A V 2 C V 3 Case 1: V 3 = 0. In this case, we can reduce to looking at indecomposable representations of this quiver: without any relations imposed. However, since this quiver is not a union of Dynkin graphs A n, D n, E 6, E 7, or E 8, we know that it has infinitely many indecomposable representations. Case 2: V 3 0, so C 0. We know that C must be bijective, because otherwise we could decompose this representation further (by splitting off the kernel and the image of C. If C is bijective, then with a change of basis C is the identity, which can be decomposed into F id F. So now we have that the indecomposable is V A 1 F id F. Since id A = 0 and id = 0, A = = 0. So this indecomposable is V = (0, F, F, (0, 0, id So there are infinitely many indecomposable representations of this quiver under this relation. The quiver in the previous example with the imposed relation was fairly simple to decompose and understand. The following quiver is similar, but even when a similar relation is imposed, counting and describing the indecomposables is much more difficult., and consider representations where the composition of the top arrows is zero and the composition of the bottom arrows is zero. The addition of another arrow makes these indecomposable representations much more difficult to describe. Instead of working with this quiver and this relation in general, we can simplify the problem by fixing a dimension vector, and trying to construct and indecomposable representation. Perhaps the easiest method of constructing an indecomposable relation is by describing where the basis elements go. Example 25. Describe an indecomposable representation of with dimension vector (2, 3, 2, where the composition of the top arrows and the composition of the bottom arrows are zero. Let s say the representation is A V : C 2 C 3 C C 2 How can we describe these maps? We start with a D 8

9 diagram like this: e 1 e 1 e 1 e 2 e 2 e 2 e 3 Now, to construct an indecomposable representation, we draw arrows representing maps A,, C, and D. We must make sure that there is no path such that A and C are connected or and D are connected, since the compositions of these maps must be zero. Also, all basis elements must be connected to the diagram, otherwise this representation would be decomposable. For example, consider the following diagram: e 1 e 2 A e 1 e 1 C e 2 D e 3 e 2 This diagram does not work to describe an indecomposable of this dimension vector, because this can be decomposed into a representation with dimension vector (2, 2, 2, since e 1 is not touched by any map. Also, this diagram fails because the composition of A and C yields e 1, not 0, and the composition of and D yields e 2, and not 0. Now, consider this diagram: e 1 A e 1 e 1 D e 2 A e 2 C e 2 D e 3 Note that this diagram fits the relation, since and D are not connected and A and C are not connected. Also, since all of the basis elements are touched by some map in the diagram, this representation is indecomposable. Once we know where the basis elements are mapped to, the representation can be described in matrices:. V : (C 2, C 3, C 2, , , ( ( 0 1 0,

10 This example gives us a method of how we could construct indecomposable representations with relations with fixed dimension vectors. However, it is not clear that this method would work to describe all indecomposable representations of a quiver with a fixed dimension vector and relation. References [1] Derksen, Harm, and Weyman, Jerzy, Quiver Representations, volume 52, number 2 of Notices of the AMS. American Mathematical Society, RI, [2] Etingof, Pavel, Golberg, Oleg, Hensel, Sebastian, Liu, Tiankai, Schwedner, Alex, Vaintrob, Dmitry, Yudovina, Elena, Introduction to Representation Theory, Volume 59, American Mathematical Society, RI, 2011 [3] Facchini, Alberto Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules, Chapter 2: The Krull-Schmidt- Remak-Azumaya Theorem, Modern irkhauser Classics, 1998 [4] Kac, V. G. Infinite Root Systems, Representations of graphs and invariant theory. Invent. Math 56 (1980, no. 1,