Twisting finite-dimensional modules for the $q$-Onsager algebra $\mathcal O_q$ via the Lusztig automorphism

Abstract

The q-Onsager algebra O q is defined by two generators A, A * and two relations, called the q-Dolan/Grady relations. Recently P. Baseilhac and S. Kolb found an automorphism L of O q , that fixes A and sends A * to a linear combination of A * , A 2 A * , AA * A, A * A 2 . Let V denote an irreducible O q -module of finite dimension at least two, on which each of A, A * is diagonalizable. It is known that A, A * act on V as a tridiagonal pair of q-Racah type, giving access to four familiar elements K, B, K ↓ , B ↓ in End(V ) that are used to compare the eigenspace decompositions for A, A * on V . We display an invertiblewhere a is a certain scalar that is used to describe the eigenvalues of A on V . We use the conjugation results to compare the eigenspace decompositions for A, A * , L ±1 (A * ) on V . In this comparison we use the notion of an equitable triple; this is a 3-tuple of elements in End(V ) such that any two satisfy a q-Weyl relation. Our comparison involves eight equitable triples. One of them is aA − a 2 K, M −1 , K where M = (aK − a −1 B)(a − a −1 ) −1 . The map M appears in earlier work of S. Bockting-Conrad concerning the double lowering operator ψ of a tridiagonal pair.

Tridiagonal pairs of $q$-Racah type and the $q$-tetrahedron algebra

“…See [29] for a comprehensive description of W in the context of spin models, distanceregular graphs, and spin Leonard pairs. We also remark that W 2 is closely related to the Lusztig automorphism of the q-Onsager algebra [3,35]; indeed W 2 = H where H is from [36,Section 3]. In the present paper, we will obtain a number of identities involving W ±1 ; for example…”

“…Definition 9.1. (See [36,Definition 7.3].) An equitable triple on V is a 3-tuple X, Y, Z of invertible elements in End(V ) such that…”

Leonard pairs, spin models, and distance-regular graphs

“…For example, the spin model concept motivated the notions of a spin Leonard pair [12], a modular Leonard triple [13], and the pseudo intertwiners for a Leonard triple of q-Racah type [47]. In [48,Theorem 8.6] and [51,Section 3], an algebraic analog of a spin model is used to describe the Lusztig automorphism of the q-Onsager algebra. Given a tridiagonal pair of q-Racah type, an algebraic analog of a spin model is used in [49, p. 3] to turn the underlying vector space into a module for the q-tetrahedron algebra.…”

The compact presentation for the alternating central extension of the $q$-Onsager algebra

“…In [9], two automorphisms of O q are introduced that resemble the Lusztig automorphisms of U q ( sl 2 ); see also [27,30]. These automorphisms are used in [9] to obtain a PBW basis for O q , and they are used in [33] to describe the Bockting double lowering operator [12,13] of a tridiagonal pair. A Drinfeld type presentation of O q is obtained in [21], and this is used in [22] to realize O q as an ιHall algebra of the projective line.…”