The q-Onsager Algebra and the Universal Askey-Wilson Algebra

Abstract

Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the q-Onsager algebra O q . They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey-Wilson algebra ∆ q . There is a natural algebra homomorphism : O q → ∆ q . We apply to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.

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

“…(i) We invoke Proposition 3.9. For notational convenience we identify the algebras U + q and U, via the isomorphism ♮ from below (15). Let Ω denote the set of alternating words listed in (71).…”

“…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.…”

A Casimir Element Inexpressible as a Lie Polynomial

“…These expressions resemble the usual Lie bracket in the associative algebra, which is [U, V ] := U V − V U . Some specific examples are the quantum group U q (sl 2 ) in its "equitable presentation" [16], the Fairlie-Odesskii algebra U q (so 3 ) in its usual presentation [10,11,12,15], the parametric family of Askey-Wilson algebras [18,19], and the parametric family of q-deformed Heisenberg algebras [13,14] whose defining relations are q-deformations of the Heisenberg-Weyl relation.…”

“…Thus, even if the defining relations of an associative algebra involve deformed commutation relations, it is still possible to compute Lie polynomials in the generators of the algebra. This notion first appeared in [18,Problem 12.14] for the universal Askey-Wilson algebra. It was found that the defining relations in this algebra and This work was partially supported by a grant from De La Salle University, Manila, and also by a grant from the Commission for Developing Countries of the International Mathematical Union (IMU-CDC).…”

Higher Rank Relations for the Askey-Wilson and q-Bannai-Ito Algebra

“…Further applications arise in the theory of special functions [4,23], tridiagonal and Leonard pairs [24,27], superintegrable quantum systems [3] and the reflection equation [2]. Its use to quantum mechanics is further emphasized by its identification as a quotient of the q-Onsager algebra [26], which originates from statistical mechanics. Furthermore, AW (3) has arisen as the spherical subalgebra of the double affine Hecke algebra of type (C ∨ 1 , C 1 ) [22,20].…”