## Generic Subalgebras of Hall Algebras and Canonical Basis

**Zongzhu Lin**

(Kansas State University)

Abstract:

In this talk, we construct a class of subalgebras of the Hall

algebra that has the generic property following Lusztig's

geometric construction and then show that (with certain integral

form), these subalgebras can obtained from a positive part of

quantum enveloping algebra of generalised Kac-Moody Lie algebra.

This algebras has a Lusztig type canonical basis with certain

interesting irreducible representations. When $Q$ is a cycle, one

can obtain the entire Hall algebra ${\Cal H}(Q,q)$. Note that in

this case, the Hall polynomials exist.

This is a joint work with N. Jing and J. Xiao.