COUNTING INDECOMPOSABLE QUIVER REPRESENTATIONS AFTER CRAWLEY-BOEVEY AND VAN DEN BERGH

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1 COUNTING INDECOMPOSABLE QUIVER REPRESENTATIONS AFTER CRAWLEY-BOEVEY AND VAN DEN BERGH FORSCHUNGSSEMINAR AG HOSKINS/AG SCHMITT WINTERSEMESTER 2015/2016 Abstract. In this seminar, we would like to understand the work of Crawley- Boevey and Van den Bergh [2] on counting absolutely indecomposable representations of quivers over finite fields. It is the basis for the work of Schiffmann [14] on counting absolutely indecomposable vector bundles on curves over finite fields. Talk 1. Holomorphic triples over elliptic curves. Eva Martínez and Alejandra Rincón will report on their joint work on derived categories of holomorphic triples on elliptic curves and stability conditions. Part I. Basic representation theory of quivers and algebras Talk 2. In this talk, you will explain the role of indecomposable objects in classification problems and set up the basic formalism of quivers and their representations. The theorem of Krull Schmidt Remak Azumaya. This theorem reduces the problem of classifying objects in an abelian category (under certain technical assumptions) to the problem of classifying indecomposable objects. Please present this theorem, following [11], Section 4.8. You may give one or two easy examples. In any case, there will be plenty of examples in the part about quivers. Quivers and their representations. Define the notions of quivers and representations of quivers. Discuss (many) examples of indecomposable quiver representations. References: [1], Chapter II, in particular, Section II.1, [12], Chapter 1,2, [13], Chapter 1, in particular, Section 1.2. Path algebras. Define the path algebra of a quiver. Show that modules over the path algebra of a quiver are the same as representations of the quiver. (Be careful about your conventions. Since you will be dealing with non-commutative algebras, you have to distinguish between left and right modules.) References: [1], Chapter II, [12], Chapter 4, [13], Chapter 4, in particular, Definition 4.5 and the following. Talk 3 and 4. Talks 2 and 3 contain an important motivation for studying representations of quivers. It is result of Gabriel s [5] which asserts that the category of finite dimensional modules over a finite dimensional algebra over an algebraically closed field k is equivalent to the category of representations of a quiver Q, subject to certain relations. The basic idea is to decompose an algebra using idempotents. These idempotents should form the vertices of the quiver. This approach works only for certain algebras, called basic algebras. 1

2 2 FS WS 2015/206 Talk 3. Representations of finite dimensional algebras. Discuss the basic notions from the representation theory of algebras. Explain why it suffices, from the viewpoint of representation theory, to consider basic algebras ([1], Corollary 6.10). You may follow [1], Chapter I. Further references: [12], Section 4.5, [13], Chapter 4,5. Talk 4. Morita equivalence. In this part, you will prove that the category of finite dimensional representations of a basic finite dimensional k-algebra is equivalent to the category of representations of a quiver with relations ([1], Theorem 3.7). You will also explain that this quiver is unique. Give some examples. The basic reference is [1], Chapter II. Further references: [12], Section 4.5, [13], Chapter 5. Talk 5 and 6. Quivers of finite representation type. A basic observation in the representation theory of quivers is that there is a trichotomy, i.e., any quiver belongs to exactly one of the following classes: Quivers of finite representation type. These are quivers for which there exist only finitely many isomorphy classes of indecomposable objects. Tame quivers. Here, there exists a one dimensional family of isomorphy classes of indecomposable objects in any dimension. Wild quivers. These are quivers where there is no hope of classifying all indecomposable representations. The number of parameters necessary for describing indecomposable objects in a given dimension grows rapidly with the dimension. Quivers of finite representation type were classified by Gabriel [4] and tame quivers by Kac [6]. In both cases, the dimension vectors of indecomposable representations were also classified. First, you should explain the above trichotomy. In fact, one studies at least two quivers of finite representation type and one tame quiver in any beginner s course on linear algebra. The basic example of a wild quiver can also be easily described. The main part of the talks is devoted to the proof of Gabriel s theorem. You may follow the approach in [13], Chapter 8. At the end, you will briefly sketch the proof given in [1], Section VII.5. Further references: [12], Chapter 6. Talk 7 and 8. Geometric invariant theory of quiver representations. (Christoph Horst/Victoria Hoskins) As we have seen in the previous talk, quivers are usually wild, and one has little or no hope of writing down a complete list of indecomposable representations. Still, one may try to construct moduli spaces. Here, one has an obvious approach. Let Q be a quiver with vertex set V, arrow set A, and maps t,h: A V, assigning to an arrow its tail and head, respectively. Fix a dimension vector n = (n v,v V). Set U := Rep r (Q) := a AHom k (k n t(a),k n h(a) ). Then, any representation of Q with dimension vector n is isomorphic to one parameterized by U. Next, define G := GL n (k) := X v V GL nv (k).

3 Indecomposable quiver representations 3 The group G acts on U via σ: G U U ( (gv,v V),(f a,a A) ) (g h(a) f a g 1 t(a),a A). Two representations u,u U are isomorphic if and only if they lie in the same G-orbit. Geometric invariant theory grants the existence of a categorical quotient U//G. It parameterizes semisimple representations of dimension vector n. This quotient is usually not very satisfactory, because from the viewpoint of representation theory, semisimple representations are usually not what one would like to classify, simple representations are indecomposable, but quite rare. Geometric invariant theory allows to choose a linearization. This is given by a character χ: G G m (k). In the given set-up, this was worked out by King [8]. GIT gives a G-invariant open subset U χ U of χ-semistable representations, a categorical quotient U χ //G, a projective morphism U χ //G U//G. An important observation is that χ-stable representations are indecomposable. The moduli spaces of χ-stable representations of King thus parameterize indecomposable representations. In contrast to simple representations, stable representations are more frequent. More precisely, one may characterize the situations in which stable representations do exist ([8], Proposition 4.4). Explain why U//G parameterizes semisimple representations. Compare semisimple and indecomposable representations for the quiver Q =. Briefly summarize the results of Le Bruyn and Procesi [10] on generators and relations for the algebra k[u] G = k[u//g]. Give a detailed exposition of [8], 1-5. You may use [3] for examples and an outlook on further research. Part II. Kac Moody algebras Kac Moody algebras are (mostly infinite dimensional) generalizations of semisimple Lie algebras. They are attached to generalized Cartan matrices. In particular, quivers give rise to Kac Moody algebras. Although we probably won t work too much with Kac Moody algebras, they played an important role in Kac s work on quivers. So, it seems worthwhile to go at least through the basic definitions of Kac Moody algebras. We will follow Chapter I of [9]. Talk 9. Root space decomposition. The basic definitions of generalized Cartan matrices and associated Kac Moody algebras will be given and the decomposition of a Kac Moody algebra into root spaces will be explained. The basis of the talk are Sections I.1 and I.2 of [9]. Talk 10. Weyl groups. A summary of Sections I.3 and I.4 of [9] on Weyl groups of Kac Moody algebras should be presented. Probably, we have already seen many of those things. This gives the possibility to shorten.

4 4 FS WS 2015/206 Talk 11. Symmetrizable Kac Moody algebras. Here, we will look at Section I.5 of [9]. The class of symmetrizable Kac Moody algebras comprises semisimple Lie algebras and Kac Moody algebras associated with euclidean diagrams and, in general, those arising from quivers. Part III. The work of Crawley-Boevey and Van den Bergh The last five talks will be devoted to the actual work of Crawley-Boevey and Van den Bergh. However, the speakers should not feel compelled to finish in one session. If there is difficult material which needs to be carefully explained, the speakers should take their time to do this. Talk 12. The result of Kac and overview. Explain the work of Kac [7] which led to the formulation of his conjectures. After that give an overview over the work of CB/VdB, following the introduction to [2]. Please have a look at the references given there to fill in additional details. Special emphasis should be given to the one parameter family Ξ mentioned in the third paragraph on Page 539. This has a nice analog in Schiffmann s work [14] on Higgs bundles. Talk 13. Purity, invariant theory over the integers. We start with the contents of the two appendices to [2]. These contain some nice and interesting techniques of independent interest. Beware that Appendix A requires l-adic cohomology. Make sure to quote the results you need in a way that is understandable also for non-experts. In positive characteristic, the formation of GIT quotients does not always commute with base change. Appendix B shows how one may avoid this phenomenon in the setting of the paper. There are several references to different papers dealing with the representation theory of reductive groups. Again, pay special attention to including them in such a way that notation fits and everything is understandable. Talk 14. Preparations and the point count. Present Sections 2.1, 2.2. Maybe you can give more details on the deformed preprojective algebra. The point counting in Proposition should be presented in detail. It seems to be fairly elementary. Reference [24] is available from Giangiacomo. Talk 15 and 16. Proof of Theorem 1.1. A team of two people should present this. The first central topic are hyperkähler structures on certain moduli spaces of quiver representations. These are essential for proving the topological triviality of the one parameter family Ξ mentioned before. Section 2.4 shows that Conjecture B of Kac and Proposition 1.2 are equivalent. This is now almost self-contained. (An exception seems to be the reference to Slodowy s paper.) The last step is the proof of Proposition 1.2. A nice feature is that the Harder Narasimhan stratification plays a prominent role. Here, the scope goes again beyond the setting of quiver representations. At a first glance, there seem to be no further complications, here. References [1] I. Assem, D. Simson, A. Skowroński, Elements of the representation theory of associative algebras. 1: Techniques of representation theory, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006, x+458 pp. [2] W. Crawley-Boevey, M. Van den Bergh, Absolutely indecomposable representations and Kac Moody Lie algebras, with an appendix by Hiraku Nakajima, Invent. Math. 155(2004), [3] H. Derksen, J. Weyman, Quiver representations, Notices Amer. Math. Soc. 52 (2005),

5 Indecomposable quiver representations 5 [4] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), ; correction, ibid. 6 (1972), 309. [5] P. Gabriel, Indecomposable representations II, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), pp Academic Press, London, [6] V.G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), [7] V.G. Kac, Root systems, representations of quivers and invariant theory, in Invariant theory (Montecatini, 1982), , Lecture Notes in Math. 996, Springer, Berlin, [8] A.D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford (2) 45 (1994), [9] S. Kumar, Kac Moody groups, their flag varieties and representation theory, Progress in Mathematics 204, Birkhäuser Boston, Inc., Boston, MA, 2002, xvi+606 pp. [10] L. Le Bruyn, C. Procesi, Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), [11] B. Pareigis, Categories and functors, translated from the German, Pure and Applied Mathematics, Vol. 39, Academic Press, New York London, 1970, viii+268 pp. [12] C.M. Ringel, Introduction to the representation theory of quivers, sek/kau/. [13] R. Schiffler, Quiver representations, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, Cham, 2014, xii+230 pp. [14] O. Schiffmann, Indecomposable vector bundles and stable Higgs bundles over smooth projective curves, Ann. of Math. 183 (2016),