Xueqing Chen
chenx@uww.edu
800 West Main Street
Dept of Mathematical & Computer Sciences
University of Wisconsin–Whitewater
Whitewater, WI 53190.
Talk 1
Title: Hall algebras over triangulated categories
Abstract:
Hall algebras provided a framework involving the categorification and the
geometrization of Lie algebras and quantum groups. In this talk, we will start by
recalling Ringel’s work on the Hall algebra associated to abelian category which
provided the realization of the positive part of a quantum group, Peng-Xiao's
work on the construction of Kac-Moody algebra from the derived category of
hereditary algebra, and Toën's work on the construction of derived Hall algebra
from differential graded category under some finiteness conditions. Note that
Toën's formula has been simplified by Xiao-Xu on a triangulated category with
(left) homological-finite condition.
Then we will discuss some generalizations of the above results and prove an
analogue of Toën's formula which is used to define derived Hall algebras and
then to construct Lie algebras for odd-periodic triangulated categories. As an
example, the Hall algebra over the 3-periodic orbit triangulated category of a
hereditary abelian category will be described.
Finally, we shall go on to discuss some of the many further developments and
applications of the theory such as the recent work of Bridgeland on the
realization of the full quantum group from the Hall algebra of the category of Z_2graded complexes in an abelian category.
Part of this talk is based on joint works with F. Xu.
Talk 2
Title: Integral bases of quantum cluster algebras for affine valued quivers
Abstract:
Let Q be an acyclic valued quiver. Recently, Rupel proved the quantum CalderoChapoton formula which gives a bijection from the set of indecomposable rigid
valued representations of Q to the set of non-initial quantum cluster variables for
the corresponding quantum cluster algebra. This correspondence enables us to
obtain integral bases of the quantum cluster algebras for affine valued quivers by
using the standard monomials constructed by Berenstein and Zelevinsky in their
paper “Quantum cluster algebras”. The integral bases of quantum cluster
algebras of type A_2^{(2)} will be addressed in detail. Some recent work on the
different types of bases of quantum cluster algebras will be discussed.
This talk is based on a joint work with M. Ding and J. Sheng.