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

2018/01/31 by Paul Terwilliger · 3 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Cellular algebra #Filtered algebra #Mathematics #Physics #Pure mathematics #math.QA

paper · pdf · doi:10.3842/sigma.2018.044

published as SIGMA 14 (2018), 044, 18 pages

arxiv created 2018/05/07 · openalex publication_date 2018/05/07 · arxiv updated 2018/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• The q-Onsager algebra and the positive part of U_q(\widehat\mathfraksl₂)
• “Hidden symmetry” of Askey-Wilson polynomials
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other
• How to sharpen a tridiagonal pair
• Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition
• PSL 2 (Z) = Z 2 ∗ Z 3
• A classification of sharp tridiagonal pairs
• The Modular Tree of Pythagoras
• Chebyshev Polynomials

Cited by

• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity
• FRT presentation of classical Askey–Wilson algebras
• Using the quantum torus to investigate the q-Onsager algebra