The Lusztig automorphism of the q-Onsager algebra

Abstract

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

An Infinite-Dimensional q-Module Obtained from the q-Shuffle Algebra for Affine sl₂

“…The algebra O q originated in algebraic combinatorics [24,Lemma 5.4], [26] and statistical mechanics [1,Section 1], [2,Section 2]. Research on O q is presently active in both areas; see [10,15,16,17,25,27,28,29,30,31,32] and [1,2,3,4,5,6,7,8,9,11,12]. The algebra O q is defined by two generators, subject to the q-Dolan/Grady relations [29,Definition 4.1].…”

Totally bipartite tridiagonal pairs

“…The Onsager algebra (see [37,38,57,129]) is the tridiagonal algebra for the case β = 2, γ = 0, γ * = 0, = 0, * = 0. The q-Onsager algebra (see [16,19,23,151,153]) is the tridiagonal algebra for the case β = q 2 + q −2 , γ = 0, γ * = 0, = 0, * = 0. The Askey-Wilson algebra (see [166]) is defined by two generators subject to the Askey-Wilson relations (19.139), (19.140).…”

The alternating PBW basis for the positive part of Uq(sl^2)

“…The following results can be obtained using Lemma 5.9 and induction on n. The proofs are straightforward and omitted. [2], [13], [14], [15], [16], [18] might be helpful in this direction. Problem 13.4.…”