Geometric approach to Hall algebras and character sheaves

Date

2012-08-07

Journal Title

Journal ISSN

Volume Title

Publisher

Kansas State University

Abstract

A representation of a quiver [Gamma] over a commutative ring R assigns an R-module to each vertex and an R-linear map to each arrow. In this dissertation, we consider R = k[t]/(t[superscript]n) and all R-free representations of [Gamma] which assign a free R-module to each vertex. The category, denoted by Rep[superscript]f[subscript] R([Gamma]), containing all such representations is not an abelian category, but rather an exact category. In this dissertation, we firstly study the Hall algebra of the category Rep[superscript]f[subscript] R([Gamma]), denote by Eta, for a loop-free quiver [Gamma]. A geometric realization of the composition subalgebra of Eta is given under the framework of Lusztig's geometric setting. Moreover, the canonical basis and a monomial basis of this subalgebra are constructed by using perverse sheaves. This generalizes Lusztig's result about the geometric realization of quantum enveloping algebra. As a byproduct, the relation between this subalgebra and quantum generalized Kac-Moody algebras is obtained. If [Gamma] is a Jordan quiver, which is a quiver with one vertex and one loop, each representation in Rep[superscript]f[subscript] R([Gamma]), gives a matrix over R when we fix a basis of the free R-module. An interesting case arises when considering invertible matrices. It then turns out that one is dealing with representations of the group GL[subscript]m(k[t]/(t[superscript]n)). Character sheaf theory is a geometric character theory of algebraic groups. In this dissertation, we secondly construct character sheaves on GL[subscript]m(k[t]/(t[superscript]2)). Then we define an induction functor and restriction functor on these perverse sheaves.

Description

Keywords

Geometric realization, Hall algebras, Character sheaves, Quiver representations, Quantum groups

Graduation Month

August

Degree

Doctor of Philosophy

Department

Department of Mathematics

Major Professor

Zongzhu Lin

Date

2012

Type

Dissertation

Citation