Canonical bases of loop algebras via quot schemes II

Canonical Bases for Hecke Algebra Quotients

We establish the existence of an IC basis for the generalized Temperley--Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra is finite dimensional.

Canonical bases of loop algebras via Quot schemes, I

We propose a construction of some canonical bases for quantum loop algebras of Kac-Moody algebras. We consider a smooth projective curve X, a group of automorphism G of X such that X/G=P^1, and we consider some Quot schemes of G-equivariant coherent sheaves on X. We view these spaces as loop analogues of spaces of representations of quivers and following Lusztig, we consider a convolution algebra of (semisimple, equivariant) perverse sheaves on the collection of these Quot schemes. We relate this algebra to a quantum loop algebra of some Kac-Moody algebra. In particular, when X=P^1 and G is a …

曲面Dehn广群的典范商的有限性

Finiteness of canonical quotients of Dehn quandles of surfaces

本文研究闭可定向曲面Dehn广群的典范商有限性，刻画其2-广群，确定n-广群有限的n值，计算最小非平凡商大小，还证明Artin广群对合商为Coxeter广群并确定辫子广群最小非平凡商。

Canonical bases and quantum coordinate ring

Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group $U_q(\mathfrak g)$ are constructed in \cite{LZ:2009} and one can use them to define some ideals of the modified quantized enveloping algebra. It is shown that the quotient algebras inherit canonical bases from the modified quantized enveloping algebra and are dual to the quantum coordinate ring defined by Kashiwara for symmetrizable Kac-Moody algebra $\mathfrak g$.