The Askey--Wilson algebras, the Lie algebra \mathfrakso₃, and their fermionic realizations

2025/11/13 by Hau-Wen Huang, Huang, Hau-Wen
Mathematics · #16S30 #16S35 #81S05 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2511.10290

openalex publication_date 2025/11/13 · openalex created_date 2025/11/15 · openalex updated_date 2026/07/30

Abstract

This paper establishes a comprehensive algebraic framework linking the Lie algebra \mathfrakso₃ to the Askey--Wilson algebras. First, we provide a manifestly symmetric reformulation of the algebra homomorphism from the universal Racah algebra \Re to U(\mathfraksl₂) by exploiting a Lie algebra isomorphism between \mathfraksl₂ and \mathfrakso₃. This perspective facilitates a natural extension to the quantum setting, where we construct an explicit algebra homomorphism from the universal Askey--Wilson algebra \triangle_q⁴ to the nonstandard quantum algebra U_q^′(\mathfrakso₃). By viewing the finite-dimensional irreducible U_q^′(\mathfrakso₃)-modules of classical type as \triangle_q⁴-modules, we demonstrate that the decomposition patterns perfectly parallel the branching rules of U(\mathfrakso₃) over \Re. Furthermore, we extend this correspondence to the fermionic setting by establishing algebra isomorphisms between the skew group rings over U(\mathfrakso₃) and U_q'(\mathfrakso₃) and their associated anticommutator spin algebras. Collectively, these results provide a unified correspondence that bridges the gap between integrable algebraic structures, quantum groups, and their fermionic analogues.

Citations

• Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into U(\mathfraksl₂)
• An imperceptible connection between the Clebsch--Gordan coefficients of U_q(\mathfraksl₂) and the Terwilliger algebras of Grassmann graphs
• Higher Rank Askey-Wilson Algebras as Skein Algebras
• Entanglement of Free Fermions on Hamming Graphs
• Leonard pairs, spin models, and distance-regular graphs
• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity
• The Casimir elements of the Racah algebra
• Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle
• Finite-dimensional irreducible modules of the universal Askey-Wilson algebra

Related

• The Racah algebra and \mathfraksl_n
• The exterior algebra and `Spin' of an orthogonal g-module
• The universal enveloping algebra of \mathfraksl₂ and the Racah algebra
• Dirac geometry I: Commutative algebra
• Extension of a Borel subalgebra symmetry into the sl(2) loop algebra symmetry for the twisted XXZ spin chain at roots of unity and the Onsager algebra