Double canonical bases

TL;DR

This paper introduces a new class of bases called double canonical bases for quantized universal enveloping algebras, which unify existing bases and exhibit symmetry properties, also revealing parts of the algebra's center.

Contribution

The authors develop double canonical bases for $U_q(rak g)$ that encompass dual canonical bases and are invariant under many symmetries, including Lusztig's, extending the understanding of algebraic structures.

Findings

Double canonical bases contain dual canonical bases of upper and lower halves.

These bases are invariant under many symmetries, including Lusztig's symmetries.

A part of the basis spans the center of $U_q(rak g)$.

Abstract

We introduce a new class of bases for quantized universal enveloping algebras $U_q(\mathfrak g)$ and other doubles attached to semisimple and Kac-Moody Lie algebras. These bases contain dual canonical bases of upper and lower halves of $U_q(\mathfrak g)$ and are invariant under many symmetries including all Lusztig's symmetries if $\mathfrak g$ is semisimple. It also turns out that a part of a double canonical basis of $U_q(\mathfrak g)$ spans its center.