On Cacti and Crystals

Abstract

In the present work we study actions of various groups generated by involutions on the category Oqint(𝔤) of integrable highest weight Uq(𝔤 ) -modules and their crystal bases for any symmetrizable Kac–Moody algebra 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 Oqint(𝔤) closely related to the remarkable quantum twists discovered by Kimura and Oya (Int Math Res Notices, 2019).