Ringel-Hall algebras and applications to moduli - I

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Description:	Reineke, M (Bergische Universität Wuppertal) Wednesday 12 January 2011, 11:30-12:30

Created:

2011-01-14 09:20

Collection:

Moduli Spaces

Publisher:

Isaac Newton Institute

Reineke, M

Language:

eng (English)

Credits:

Author:	Reineke, M
Producer:	Tim Cooper


Abstract:	Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry. The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representation-theoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields. We will develop several such identities and discuss more recent applications to wall-crossing formulae. Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wall-crossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147.

Abstract:

Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry.

The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representation-theoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields.

We will develop several such identities and discuss more recent applications to wall-crossing formulae.

Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wall-crossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147.

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