The Lusztig automorphism of the q-Onsager algebra

Pascal Baseilhac and Stefan Kolb recently introduced the Lusztig automorphism L of the q-Onsager algebra Oq. 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 Aq associated with Oq; (iv) we describe what happens when a finite-dimensional irreducible Oq-module is twisted via L.