A note on noncommutative Poisson structures

We introduce a new type of noncommutative Poisson structure on associative algebras. It induces Poisson structures on the moduli spaces classifying semisimple modules. Path algebras of doubled quivers and preprojective algebras have noncommutative Poisson structures given by the necklace Lie algebra.Comment: 6 page

Normality of Marsden-Weinstein reductions for representations of quivers

We prove that the Marsden-Weinstein reductions for the moment map associated to representations of a quiver are normal varieties. We give an application to conjugacy classes of matrices.Comment: 20 pages; contains a new appendi

Kac's Theorem for weighted projective lines

We prove an analogue of Kac's Theorem, describing the dimension vectors of indecomposable coherent sheaves, or parabolic bundles, over weighted projective lines. We use a theorem of Peng and Xiao to associate a Lie algebra to the category of coherent sheaves for a weighted projective line over a finite field, and find elements of this Lie algebra which satisfy the relations defining the loop algebra of a Kac-Moody Lie algebra.Comment: 13 pages; minor changes onl

Quiver algebras, weighted projective lines, and the Deligne-Simpson problem

We describe recent work on preprojective algebras and moduli spaces of their representations. We give an analogue of Kac's Theorem, characterizing the dimension types of indecomposable coherent sheaves over weighted projective lines in terms of loop algebras of Kac-Moody Lie algebras, and explain how it is proved using Hall algebras. We discuss applications to the problem of describing the possible conjugacy classes of sums and products of matrices in known conjugacy classes.Comment: To appear in Proceedings of ICM 2006 Madrid (11 pages

Connections for weighted projective lines

We introduce a notion of a connection on a coherent sheaf on a weighted projective line (in the sense of Geigle and Lenzing). Using a theorem of Huebner and Lenzing we show, under a mild hypothesis, that if one considers coherent sheaves equipped with such a connection, and one passes to the perpendicular category to a nonzero vector bundle without self-extensions, then the resulting category is equivalent to the category of representations of a deformed preprojective algebra.Comment: 12 page

On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero

We determine those k-tuples of conjugacy classes of matrices, from which it is possible to choose matrices which have no common invariant subspace and have sum zero. This is an additive version of the Deligne-Simpson problem. We deduce the result from earlier work of ours on preprojective algebras and the moment map for representations of quivers. Our answer depends on the root system for a Kac-Moody Lie algebra.Comment: 11 pages. Only trivial changes since the last versio

Kac's Theorem for equipped graphs and for maximal rank representations

We give two generalizations of Kac's Theorem on representations of quivers. One is to representations of equipped graphs by relations, in the sense of Gelfand and Ponomarev. The other is to representations of quivers in which certain of the linear maps are required to have maximal rank.Comment: 4 pages; v2 corrects slightly garbled proof of Theorem 2.

General sheaves over weighted projective lines

We develop a theory of general sheaves over weighted projective lines. We define and study a canonical decomposition, analogous to Kac's canonical decomposition for representations of quivers, study subsheaves of a general sheaf, general ranks of morphisms, and prove analogues of Schofield's results on general representations of quivers. Using these, we give a recursive algorithm for computing properties of general sheaves. Many of our results are proved in a more abstract setting, involving a hereditary abelian category.Comment: 26 pages; one reference added and a change of notatio

Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity

We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n by n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a Kac-Moody Lie algebra. Our proofs use Ringel's theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the Riemann-Hilbert correspondence and an algebraic version, due to Dettweiler and Reiter, of Katz's middle convolution operation.Comment: 30 pages, various corrections and improvement

Representations of equipped graphs: Auslander-Reiten theory

Representations of equipped graphs were introduced by Gelfand and Ponomarev; they are similar to representation of quivers, but one does not need to choose an orientation of the graph. In a previous article we have shown that, as in Kac's Theorem for quivers, the dimension vectors of indecomposable representations are exactly the positive roots for the graph. In this article we begin by surveying that work, and then we go on to discuss Auslander-Reiten theory for equipped graphs, and give examples of Auslander-Reiten quivers.Comment: Submitted to Proceedings of the 50th Symposium on Ring Theory and Representation Theory (Yamanashi, 2017