Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / 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.
| Identifer | oai:union.ndltd.org:KSU/oai:krex.k-state.edu:2097/14136 |
| Date | January 1900 |
| Creators | Fan, Zhaobing |
| Publisher | Kansas State University |
| Source Sets | K-State Research Exchange |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
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