Summer School Geometry of quiver-representations and preprojective algebras Isle of Thorns, September 2000 A few further recommendations for the lectures. Any queries to me at W.Crawley-Boevey@leeds.ac.uk This document may well change. Original version 5 April 2000 Changed 14 April 2000 (All lectures to be 50 minutes maximum) I suggest working with an arbitrary algebraically closed base field of characteristic zero (and the complex numbers when necessary). Lecture 1. Categorical quotients. 50 minutes maximum. Reductive groups. Reynolds operator. The notion of a categorical quotient. The fact that the affine quotient for a reductive group acting on an affine variety parametrizes the closed orbits. A good reference is sections 6.1-6.3 of J. Le Potier, Lectures on vector bundles. Maybe see also sections 2.2 and 2.3 of V. G. Kac, Root systems, representations of quivers and invariant theory, in: Invariant Theory, Proc. Montecatini 1982, ed. F. Gherardelli, SLN 996. Lecture 2. Representation spaces of quivers with relations. 50 minutes maximum. The space of representations of a quiver Rep(Q,alpha) and the action of the group GL(alpha). More generally, given an algebra A written as a quotient of a path algebra A = KQ/I, there is the space Rep(A,alpha) of representations of Q satisfying the relations in I. (In general we don't assume that A is finite dimensional, so there is nothing canonical about Q and I) (In my notes W. Crawley-Boevey, Geometry of representations of algebras, http://www.amsta.leeds.ac.uk/~pmtwc/geomreps.ps, I thought of A as being given as an algebra with a fixed complete set of orthogonal idempotents. This is equivalent.) Discussion of elementary properties. (There are various references. Everything for the variety Mod(A,n) generalizes. Maybe see C. Geiss, Geometric methods in representation theory of finite-dimensional algebras, in Representations theory of algebras and related topics, CMS Conf. Proc 19, Proc. Mexico City 1994, eds R. Bautista et al. Also P. Gabriel, Finite representation type is open, in: Representations of algebras, Proc Ottawa 1974, eds V. Dlab and P. Gabriel, SLN 488.) (Some of the elementary properties: orbits under GL(alpha) correspond to isomorphism classes of representations. The stabilizer of a point is the automorphism group of the representation. Modules with endomorphism ring of dimension at most d form an open subset. Indecomposables form a constructible subset. Modules with Ext^1(X,X)=0 correspond to orbits which are open subschemes. Modules with Ext^2(X,X)=0 give smooth points of Rep(A,alpha) (but not conversely).) An easy worked example. For example the Kronecker quiver with dimension vector (1,2)? The fact that polynomial invariants are generated by traces of oriented cycles. (Need characteristic zero assumption.) For quivers, this is a theorem of Le Bruyn and Procesi, Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), 585-598. For quivers with relations, note that the Reynolds operator (in Mumford's book on Geometric Invariant Theory) says that the fixed point functor from "rational" GL(alpha)-modules to vector spaces is exact. Apply this to the restriction map from the coordinate ring of Rep(Q,alpha) to the coordinate ring of Rep(A,alpha). The resulting map is still surjective. The fact that closed orbits correspond to semisimple modules, and the affine quotient Rep(A,alpha)//GL(alpha) classifies semisimple modules. (The proof in Gabriel is only for finite dimensional algebras. Is there a good reference for the general case? As a last resort see sections 3.3 and 3.4 of my notes for a DMV seminar, http://www.amsta.leeds.ac.uk/~pmtwc/dmvsing2.ps) Lecture 3. Degenerations. 50 minutes maximum. Discussion about connection between degenerations and extensions. Survey of some of the work of Zwara, Bongartz and Riedtmann. If possible it would be good to have a sketch of the theorem in G. Zwara, Degenerations of finite-dimensional modules are given by extensions, preprint. Lecture 4. Stable and semistable points. 50 minutes maximum. Section 2 of A. D. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford 45 (1994), 515-530. Notion of stable and semistable points; 1-parameter subgroups; the 'fundamental theorem' about 1-psgs (Theorem 1.4 of G. R. Kempf, Instability in invariant theory, Ann. of Math 108 (1978), 299-316); the fact that the quotient characterizes semistable points up to equivalence. Example. GL(V) acting on Hom(V,W) (with dim V < dim W). Taking the character to be the determinant character (or its inverse?), the semistable points are the injective linear maps. The quotient Hom(V,W) // (GL(V),chi) is therefore the Grassmannian of subspaces of W of dimension dim V. Other examples? Lecture 5. Moduli spaces of representations. 50 minutes maximum. Sections 3, 4.1 and 4.2 of A. D. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford 45 (1994), 515-530 (but done for arbitrary quivers with relations, not for finite dimensional algebras, so Prop 4.3 fails.) Example. Take the quiver with two vertices, n arrows from vertex 1 to 2, dimension vector alpha=(r,1) with r/I, suppose one wants to classify pairs (M,m) consisting of an n-dimensional cyclic B-module M and a generator m of that module, up to the natural equivalence (so that (M,m) is equivalent to (M',m') if there is an isomorphism from M to M' sending m to m'). Construct a quiver with two vertices 1,2, one arrow from 1 to 2, and loops x,y,... at the vertex 2. Take the relations for the quiver coming from the ideal I. Let alpha be the dimension vector (1,n). Choose theta suitably, so that every proper subrepresentation of a stable representation vanishes at vertex 1. Then the moduli space should classify the pairs (M,m). When B is commutative, this is the Hilbert scheme of n points in the scheme Spec B. The case where B is the polynomial ring in two variables is discussed in H. Nakajima, Lectures on Hilbert schemes of points on surfaces, Amer. Math. Soc. 1999. Lecture 6. Universal bundles. 50 minutes maximum. Definition of vector bundles. The universal bundle for a Grassmannian. Coarse and fine moduli spaces for representations of quivers with relations. Section 5 of A. D. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford 45 (1994), 515-530. Note: a family of representations of a quiver with relations, over a scheme X, consists of a vector bundle over X for each vertex, and a bundle homomorphism for each arrow in the quiver, satisfying the relations. For background on coarse and fine moduli spaces see the discussion at the start of Chapter 5 of Mumford's book on Geometric Invariant Theory. A good reference is P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Inst. of Fund. Research. Some of the arguments in King are very similar to those for moduli spaces of vector bundles on curves, so it may be useful to look at Section 8.4 of J. Le Potier, Lectures on vector bundles. Example. For the Grassmannian example of lecture 5, one has a fine moduli space. The universal bundle is given by the universal bundle of the Grassmannian, and its n maps to the trivial bundle (coming from the fact that it is a subbundle of the trivial bundle of rank n). Lecture 7. General representations of quivers. 50 minutes maximum. Some results from V. Kac, Infinite root systems, Representations of graphs and invariant theory II, J. Algebra 78 (1982), 141-162 and A. Schofield, General representations of quivers, Proc. London Math. Soc. 65 (1992), 46-64. The Ringel bilinear form and Ringel's formula for dim Hom - dim Ext. The numbers hom(alpha,beta) and ext(alpha,beta). (Section 1 of Schofield). Schofield's result that a general representation of dimension alpha+beta has a subrepresentation of dimension alpha if and only if ext(alpha,beta)=0. (Schofield, Theorem 3.3. We're only interested in the case when the base field has characteristic zero, so this is a tangent space computation. See W. Crawley-Boevey, Subrepresentations of general representations of quivers, Bull. London Math. Soc. 28 (1996), 363-366.) Kac's result that the general representation of dimension alpha+beta decomposes as a direct sum of representations of dimensions alpha and beta if and only if ext(alpha,beta)=0 and ext(beta,alpha)=0. The characterization of Schur roots. The canonical decomposition. The proofs in Kac use his main theorem, which we want to avoid at this stage. Instead see my slides dating from a Summer School in 1995. http://www.amsta.leeds.ac.uk/~pmtwc/quiver/krippen.ps The relevant part is page 20 "Tangent space computations" onwards. Lecture 8. Maps between general representations. 50 minutes maximum. Sections 1,2 of W. Crawley-Boevey, On homomorphisms from a fixed representation to a general representation of a quiver, Trans. Amer. Math. Soc. 348 (1996), 1909-1919. Deduction of Theorem 5.2 of A. Schofield, General representations of quivers, Proc. London Math. Soc., 65 (1992), 46-64. Lecture 9. Schubert varieties 50 minutes maximum. Let Gr(k,n) be the Grassmannian of k-dimensional subspaces of an n-dimensional vector space. The fact that Gr(k,n) is a projective variety should already have been done as an example of moduli spaces. The first aim of the lecture is to explain Griffiths and Harris, Principles of algebraic geometry, section 1.5, p193 - p197 (half way down). (Another good reference is S. L. Kleiman and D. Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061-1082.) In particular, the definition of the Schubert varieties (the closures of the W_{a_1,...,a_k}), and the fact that the homology of Gr(k,n) is freely generated as an abelian group by the Schubert cycles sigma_a. (If you want, instead of homology you could use cohomology, by Poincare duality, or the intersection ring.) The second aim is to point out (perhaps with minimal explanation) that if V, V', V'', ... are general flags in the n-dimensional space, then the intersection $\sigma_a(V) \cap \sigma_b(V') \cap \sigma_c(V'') \cap ...$ of Schubert varieties is proper and transversal. (For the notions of a proper and transversal intersection of two subvarieties of a nonsingular projective variety, see for example W. Fulton, Young Tableaux, appendix B.1.) (Is this fact obvious, or standard? I don't know. I would use the fact that GL(n) acts transitively on Gr(k,n) and on the set of flags, and the following result: If a connected algebraic group G acts transitively on an irreducible variety X, and Y and Z are irreducible closed subvarieties, then for g in some dense open subset of G the subvarieties gY and Z meet transversally. I don't know a reference for this. One can deduce it from S. L. Kleiman, the transversality of a general translate, Compositio Math. 28 (1974), 287-297. Letting Y_0 be the smooth locus of Y and Y_i the irreducible components of the singular locus, and similarly for Z, one wants to choose g such that gY \cap Z_i and gY_i \cap Z are all proper, and gY_0 \cap Z_0 is transversal. This needs characteristic zero, but this is ok since we're working over the complex numbers.) Finally, relate this to representations of quivers. Let sigma_a, sigma_b, ... be Schubert cycles in complementary dimension, so with $\sum_{i=1}^k a_i + \sum_{i=1}^k b_i + ... = k(n-k)$. If V, V', ... are general flags in the n-dimensional space, then the intersection $\sigma_a(V) \cap \sigma_b(V') \cap \sigma_c(V'') \cap ...$ is finite, and we claim it is the number of subrepresentations of dimension beta of the general representation of dimension alpha, for some quiver and some dimension vectors alpha and beta. The quiver one takes is a star with one arm for each Schubert cycle, and each arm of length k (excluding the central vertex), so mk+1 vertices in all if there are m Schubert cycles. Let alpha be the dimension vector which is n at the centre, on the arm for a is n-k+1-a_1, n-k+2-a_2, ..., n-a_k working towards the centre, and similarly for the other arms. Let beta be the dimension vector which is k at the centre, and on each arm is 1,2,...,k. Given general flags V, V', ... in the n-dimensional space, one can construct a representation (V,V',...) of the quiver of dimension alpha, in which all maps are inclusions. Conversely, in the general representation of the quiver of dimension alpha all maps are injective, so the representation arises in this way. We show that there is a 1-1 correspondence between subrepresentations of dimension beta in (V,V',...), and elements of $\sigma_a(V) \cap \sigma_b(V') \cap ...$. In a subrepresentation of dimension beta the vector space at the central vertex is a k-dimensional subspace of the n-dimensional space, so is a point in Gr(k,n). Because it is a subrepresentation, it is easy to see that this point is in $\sigma_a(V) \cap \sigma_b(V') \cap ...$. Conversely, since the dimensions are complementary and the flags are general, any point in the intersection $\sigma_a(V) \cap \sigma_b(V') \cap ...$ is actually in $W_a(V) \cap W_b(V') ...$ (with the obvious adaptation of the notation of Griffiths and Harris p195), and so extends in a unique way to a subrepresentation of (V,V',...) of dimension beta. Lecture 10. Schubert calculus. 50 minutes maximum. The object is to explain Griffiths and Harris, Principles of algebraic geometry, section 1.5, p197 (half way down) to p206. Another reference is W. Fulton, Young Tableaux, section 9.4. It would also be useful to explain that the coefficients delta(a,b;c) are the same as the Littlewood-Richardson coefficients for multiplying Schur polynomials. This connection is explained at the end of the section in Fulton. Lecture 11. Chern class calculations. 50 minutes maximum. Given a rank r vector bundle E on a smooth projective variety X there are Chern classes c_i(E) for i between 0 and r. Over an arbitrary algebraically closed field they can be considered as elements of the intersection ring (Hartshorne, Algebraic geometry, appendix A.3) Over the complex numbers they can be considered as elements of the integer cohomology H^{2i}(X) (see W. Fulton, Young Tableaux, appendix B.1.) The first aim of the lecture is to discuss properties of Chern classes. Some key properties are (a) Class of the zero scheme of a section. (Property (C6) in Hartshorne A.3, or equivalently Example 14.1.1 in W. Fulton, Intersection theory.) (b) Exact sequences of bundles (Fulton, Intersection Theory, Theorem 3.2(e)). (c) Using Chern roots to compute the Chern classes of dual bundles and tensor products (Fulton, Intersection Theory, Remark 3.2.3) or (Hartshorne, A.3 property (C5)). The second aim is to discuss the universal subbundle and universal quotient bundle for a Grassmannian. Just the definition and the Gauss-Bonnet Theorem (Griffiths and Harris, p410-411) that the Chern classes of the universal quotient bundle are special Schubert cycles. See also Fulton, Intersection Theory, section 14.7. The third aim is to show the connection with to quivers. The argument in the theorem in W. Crawley-Boevey, Subrepresentations of general representations of quivers, Bull. London Math. Soc., 28 (1996), 363-366, shows that the general representation of a quiver of dimension alpha has a subrepresentation of dimension beta if and only if the top Chern class of a certain vector bundle $E = \oplus_{a:i\to j} Hom(S_i,Q_j)$ is nonzero. This top Chern class can be calculated either as in the theorem, or (less explicitly) by Chern characters. (We're not interested in the application of this to characteristic p>0.) It would be good, if possible, to discuss the connection with degeneracy loci (if any?). See for example Fulton and Pragacz, Schubert varieties and degeneracy loci, Springer Lec. Notes 1689 (1998). Lecture 12. Kac-Moody algebras. 50 minutes maximum. Section 1.1 of Kac's book on Infinite Dimensional Lie Algebras. Notion of a Generalized Cartan Matrix A, and a realization. The corresponding Lie algebra \mathfrak{g}(A), which can be defined by generators and relations as in section 9.11 in the book. Symmetric GCMs correspond exactly to graphs without loops, with A_ij = - number of edges between i and j (for i,j distinct). (Because of this we only need to deal with symmetric GCMs.) It would be good to explain to 1-1 correspondence between roots for the Lie algebra and roots for the corresponding graph, as described for example in section 1.1 of V. G. Kac, Root systems, representations of quivers and invariant theory, Proc. Montecatini 1982, ed. F. Gherardelli, SLN 996. (Chapter 5 of the book.) Maybe discuss the classification of indecomposable GCMs (Chapter 4 of the book and Section 1.2 of the article.) Briefly, the integrable highest weight modules L(\Lambda) and the character formula (Chapter 10). Lecture 13. Kac's Theorem. 50 minutes maximum. As well as Kac, and the article of Kraft and Riedtmann, a suitable reference might be my lecture notes on Geometry of Representations of Algebras, http://www.amsta.leeds.ac.uk/~pmtwc/geomreps.ps Lecture 14. Symplectic forms and the preprojective algebra. 50 minutes maximum. The preprojective algebra \Pi of a quiver Q, given by generators and relations as the quotient of the path algebra of the "double" of Q by the relation that the sum over all arrows in Q of the commutator [a,a*] is zero. Sketch of the connection between this definition and the tensor algebra description. See Theorem A of C. M. Ringel, The preprojective algebra of a quiver. Algebras and modules, II (Geiranger, 1996), 467-480, CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, 1998, http://www.mathematik.uni-bielefeld.de/sfb343/preprints/pr97046.ps.gz or Theorems 0.1,0.2 of W. Crawley-Boevey, Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities, Comment. Math. Helv. 74 (1999), 548-574, http://www.amsta.leeds.ac.uk/~pmtwc/preproj2.dvi Symplectic forms on complex vector spaces. Symplectic varieties. Moment maps for group actions. Symplectic reduction. This is all standard over the reals. Adapting to the complex numbers (or varieties over an algebraically closed field of characteristic zero) is straightforward. See, perhaps, sections 1.1 and 1.4 of N. Chriss and V. Ginzburg, Representation theory and complex geometry. I'm only really looking for definitions and motivation, and the formula for the moment map when an algebraic group acts on a vector space preserving a symplectic form. The fact that Rep(\overline Q,\alpha) can be identified with the cotangent bundle of Rep(Q,\alpha), that it has a moment map for the action of GL(\alpha), and the fibre over zero is the space of representations of the preprojective algebra. The fibre over X of the map from Rep(\Pi,alpha) to Rep(Q,alpha) is isomorphic to the dual of Ext(X,X) (mentioned in Lemma 10.4 of W. C-B, loc.cit). Using Ringel's formula dim End(X) - dim Ext(X,X) = , it follows that the inverse image in Rep(\Pi,alpha) of any orbit in Rep(Q,alpha) has constant dimension. In case there are only finitely many orbits in Rep(Q,alpha), the closures of the inverse images of the orbits are the irreducible components of Rep(\Pi,alpha). Lecture 15. The nilpotent variety. 50 minutes maximum. Lagrangian subspaces of complex symplectic vector spaces and Lagrangian subvarieties of complex symplectic varieties (see for example Definition 1.3.24 of N. Chriss and V. Ginzburg, Representation theory and complex geometry). The nilpotent variety Lambda_v and its pure dimension, Theorem 12.3 of G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc., 4 (1991), 365-421. Statement of its Lagrangian property, Theorem 12.9 of loc.cit. Proof if possible (unpublished, but probably similar to Theorem 7.2 of H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J., 91 (1998), 515-560). Lecture 16. Crystal bases in quantized enveloping algebras. 50 minutes maximum. Sections 2, 3 of M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J., 89 (1997), 9-36. Lecture 17. Geometric construction of crystal bases. 50 minutes maximum. Section 5 of M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J., 89 (1997), 9-36. Lecture 18. Construction of the positive part for Dynkin Lie algebras. 50 minutes maximum. Construction of the positive part of the Lie algebra in the Dynkin case, Ch. Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994), 526-546. Lecture 19. Constructible functions and enveloping algebras. 50 minutes maximum. The formalism of constructible functions (section 10.18 of G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc., 4 (1991), 365-421, and section 1 of R. MacPherson, Chern classes for singular varieties, Ann. of Math. 100 (1974), 423-432). Convolution product for functions on representation spaces (Lusztig, 10.19). Statement, but not proof, of Schofield's Theorem (Lusztig, 10.20). Constructible functions on nilpotent varieties (Lusztig, 12.10 - 12.13). Lecture 20. Nakajima's quiver varieties. 50 minutes maximum. Definition and basic properties of Nakajima's quiver varieties mathfrak{M}(v,w) in section 3 of H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), 515-560. I think it is useful to show that they can be realized as moduli spaces of representations of a preprojective algebra. To construct the quiver variety \mathfrak{M}(v,w) associated to an oriented graph (I,\Omega), start by constructing a new oriented graph, adjoining a new vertex and, for each vertex i in I, adding w_i arrows from the new vertex to i. Let \Pi be the preprojective algebra for this new oriented graph, and consider representations of \Pi whose dimension vector is v_i at vertices i in I, and 1 at the new vertex. Choosing bases of the vector spaces W_i, and using these to rewrite linear maps to or from W_i as collections of w_i linear maps to or from a one-dimensional vector space, elements of M(v,w) correspond to representations of the "double" of the new oriented graph. Moreover the relation \mu(B,i,j) = 0 in loc. cit. section 3.1 corresponds exactly to the relations for the preprojective algebra. It appears that for representations of \Pi there should be one more relation, corresponding to the new vertex. But this holds automatically. (Consider the sum of the traces of all the other relations.) Finally, one can observe using loc. cit. Lemma 3.8 that the stable triples (B,i,j) correspond to representations which are "cogenerated" at the new vertex, i.e. have no proper subrepresentation which vanishes at the new vertex. Lecture 21. Construction of integrable representations. 50 minutes maximum. Sketch of H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), 515-560, trying to get to Theorem 10.2.