Geometric approach to Hall algebras and character sheaves
dc.contributor.author | Fan, Zhaobing | |
dc.date.accessioned | 2012-08-07T20:58:55Z | |
dc.date.available | 2012-08-07T20:58:55Z | |
dc.date.graduationmonth | August | en_US |
dc.date.issued | 2012-08-07 | |
dc.date.published | 2012 | en_US |
dc.description.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](R[Gamma]), for a loop-free quiver [Gamma]. A geometric realization of the composition subalgebra of [Eta](R[Gamma]) 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. | en_US |
dc.description.advisor | Zongzhu Lin | en_US |
dc.description.degree | Doctor of Philosophy | en_US |
dc.description.department | Department of Mathematics | en_US |
dc.description.level | Doctoral | en_US |
dc.identifier.uri | http://hdl.handle.net/2097/14136 | |
dc.language.iso | en_US | en_US |
dc.publisher | Kansas State University | en |
dc.subject | Geometric realization | en_US |
dc.subject | Hall algebras | en_US |
dc.subject | Character sheaves | en_US |
dc.subject | Quiver representations | en_US |
dc.subject | Quantum groups | en_US |
dc.subject.umi | Mathematics (0405) | en_US |
dc.title | Geometric approach to Hall algebras and character sheaves | en_US |
dc.type | Dissertation | en_US |