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

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.