324 research outputs found
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
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
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
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.
Classification of modules for infinite-dimensional string algebras
We relax the definition of a string algebra to also include
infinite-dimensional algebras such as k[x,y]/(xy). Using the functorial
filtration method, which goes back to Gelfand and Ponomarev, we show that
finitely generated and artinian modules (and more generally finitely controlled
and pointwise artinian modules) are classified in terms of string and band
modules. This subsumes the known classifications of finite-dimensional modules
for string algebras and of finitely generated modules for k[x,y]/(xy). Unlike
in the finite-dimensional case, the words parameterizing string modules may be
infinite.Comment: Corrected and substantially revised, it now also includes the
classification of artinian module
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
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
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