Weyl group action and semicanonical bases

TL;DR

This paper explores the compatibility between the Weyl group action and the semicanonical basis within the positive subalgebra of the enveloping algebra of a symmetric Kac-Moody algebra, revealing their deep interrelation.

Contribution

It demonstrates the high degree of compatibility between the Weyl group action and the semicanonical basis in the algebra U^+ of a symmetric Kac-Moody algebra.

Findings

Weyl group acts on U up to a sign

Semicanonical basis has remarkable properties

Compatibility between Weyl group action and semicanonical basis

Abstract

Let U be the enveloping algebra of a symmetric Kac-Moody algebra. The Weyl group acts on U, up to a sign. In addition, the positive subalgebra U^+ contains a so-called semicanonical basis, with remarkable properties. The aim of this paper is to show that these two structures are as compatible as possible.