On cacti and crystals

TL;DR

This paper explores the actions of various involution-generated groups, including the cactus and Weyl groups, on categories of integrable modules and their crystal bases for symmetrizable Kac-Moody algebras, revealing new symmetries and connections.

Contribution

It introduces and studies the actions of cactus and Weyl groups on integrable modules and crystal bases, uncovering new involutive symmetries and their relation to quantum twists.

Findings

Cactus group actions on crystal bases are established.

Connections between cactus group generators and quantum twists are identified.

Conjectural Weyl group actions are proposed for integrable modules.

Abstract

In the present work we study actions of various groups generated by involutions on the category $\mathscr O^{int}_q(\mathfrak g)$ of integrable highest weight $U_q(\mathfrak g)$-modules and their crystal bases for any symmetrizable Kac-Moody algebra $\mathfrak g$. The most notable of them are the cactus group and (yet conjectural) Weyl group action on any highest weight integrable module and its lower and upper crystal bases. Surprisingly, some generators of cactus groups are anti-involutions of the Gelfand-Kirillov model for $\mathscr O^{int}_q(\mathfrak g)$ closely related to the remarkable quantum twists discovered by Kimura and Oya.