# The Lusztig automorphism of the $q$-Onsager algebra

## TL;DR

This paper provides a new explicit expression for the Lusztig automorphism of the $q$-Onsager algebra, proves its existence without computer assistance, and explores its implications for related algebraic structures and modules.

## Contribution

It introduces explicit formulas for the Lusztig automorphism, proves its existence independently, and extends the automorphism to related current algebras and modules.

## Key findings

- Explicit formulas for $L$ and $L^{-1}$ as quantum adjoint sums
- Proof of the existence of $L$ without computational methods
- Description of the automorphism's effect on finite-dimensional modules

## Abstract

Pascal Baseilhac and Stefan Kolb recently introduced the Lusztig automorphism $L$ of the $q$-Onsager algebra $\mathcal O_q$. In this paper, we express each of $L, L^{-1}$ as a formal sum involving some quantum adjoints. In addition, (i) we give a computer-free proof that $L$ exists; (ii) we establish the higher order $q$-Dolan/Grady relations previously conjectured by Baseilhac and Thao Vu; (iii) we obtain a Lusztig automorphism for the current algebra $\mathcal A_q$ associated with $\mathcal O_q$; (iv) we describe what happens when a finite-dimensional irreducible $\mathcal O_q$-module is twisted via $L$.