Canonical bases and quantum coordinate ring

TL;DR

This paper explores the construction of filtrations in tensor products of modules over quantum groups, demonstrating how they lead to ideals with inherited canonical bases and duality with quantum coordinate rings.

Contribution

It introduces new filtrations in tensor products of modules over quantum groups and shows their role in defining ideals with canonical bases and duality properties.

Findings

Filtrations of tensor products lead to ideals with canonical bases.

Quotient algebras inherit canonical bases from the modified quantum algebra.

The quotient algebras are dual to Kashiwara's quantum coordinate ring.

Abstract

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 $\mathfrak g$.