The $S_3$-symmetric $q$-Onsager algebra and its Lusztig automorphisms

TL;DR

This paper introduces Lusztig automorphisms for the $S_3$-symmetric $q$-Onsager algebra, explores their relations, and examines their effects on finite-dimensional modules, including a detailed 5-dimensional example.

Contribution

It constructs and analyzes Lusztig automorphisms for the $S_3$-symmetric $q$-Onsager algebra, extending the automorphism theory to this generalized setting.

Findings

Lusztig automorphisms are constructed for each standard generator.

The relations among the six Lusztig automorphisms are described.

Effects of twisting modules by Lusztig automorphisms are analyzed.

Abstract

The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. In 2019, Baseilhac and Kolb introduced two automorphisms of $O_q$, now called the Lusztig automorphisms. Recently, we introduced a generalization of $O_q$ called the $S_3$-symmetric $q$-Onsager algebra $\mathbb O_q$. The algebra $\mathbb O_q$ has six distinguished generators, said to be standard. The standard $\mathbb O_q$-generators can be identified with the vertices of a regular hexagon, such that nonadjacent generators commute and adjacent generators satisfy the $q$-Dolan/Grady relations. In the present paper we do the following: (i) for each standard $\mathbb O_q$-generator we construct an automorphism of $\mathbb O_q$ called a Lusztig automorphism; (ii) we describe how the six Lusztig automorphisms of $\mathbb O_q$ are related to each other; (iii) we describe what happens if a finite-dimensional irreducible $\mathbb O_q$-module is twisted by a Lusztig automorphism; (iv) we give a detailed example involving an irreducible $\mathbb O_q$-module with dimension 5.