The Universal Askey-Wilson Algebra and the Equitable Presentation of U q (sl 2 )

2011/07/31 by Paul Terwilliger · 2 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Associative algebra #Casimir element #Cellular algebra #Center (category theory) #Combinatorics #Connection (principal bundle) #Discrete mathematics #Division algebra #Element (criminal law) #Field (mathematics) #Geometry #Mathematics #Nonlinear Waves and Solitons #Physics #Pure mathematics #Quantum #Root of unity #Subalgebra #math.QA #math.RA #msc:33D45 #msc:33D80

paper · pdf · doi:10.3842/sigma.2011.099

published as SIGMA 7 (2011), 099, 26 pages · misprint in Lemma 7.1 is corrected

openalex publication_date 2011/10/25 · arxiv created 2012/03/16 · arxiv updated 2012/03/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

Let F denote a field, and fix a nonzero q F such that q 4 = 1. The universal Askey-Wilson algebra is the associative F-algebra = q defined by generators and relations in the following way. The generators are A, B, C. The relations assert that each of

Citations

• Classical and quantum orthogonal polynomials in one variable
• “Hidden symmetry” of Askey-Wilson polynomials
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other
• Two Linear Transformations each Tridiagonal with Respect to an Eigenbasis of the other; Comments on the Parameter Array
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• PSL 2 (Z) = Z 2 ∗ Z 3
• Classical and Quantum Orthogonal Polynomials in One Variable

Cited by

• Verma modules and finite-dimensional irreducible modules of the universal Askey–Wilson algebra at roots of unity
• Bispectrality of the Complementary Bannai-Ito Polynomials

Related

• A higher rank extension of the Askey-Wilson Algebra
• The quantum algebra U_q(sl₂) and its equitable presentation
• Multivariate Askey-Wilson functions and overlap coefficients
• Askey-Wilson polynomials and the quantum SU(2) group: survey and applications
• A note on the O_q(sl₂) algebra