Generic Subalgebras of Hall Algebras and Canonical Basis

 Zongzhu Lin
(Kansas State University)

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.