A skew group ring of \mathbb Z/2\mathbb Z over U(\mathfraksl₂), Leonard triples and odd graphs

2025/10/27 by Huang, Hau-Wen, Lee, Chin-Yen
#05E10 #05E30 #16S35 #33D45 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2510.23173

Abstract

We employ a skew group ring of \mathbb Z/2\mathbb Z over U(\mathfraksl₂) to construct modules over the universal Bannai--Ito algebra. In addition, we give the conditions under which the defining generators act as Leonard triples on the resulting modules. As a combinatorial realization, we establish an algebra homomorphism from the universal Bannai--Ito algebra onto the Terwilliger algebra of an odd graph. This homomorphism provides a unified description of Leonard triples on all irreducible modules over the Terwilliger algebra.

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
• The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and U_q(sl₂)
• Dual polar graphs, the quantum algebra U_q(sl₂), and Leonard systems of dual q-Krawtchouk type
• The classification of Leonard triples of QRacah type
• Introduction to representation theory
• Leonard pairs and the Askey-Wilson relations
• Two linear transformations each tri-diagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form

Related

• Hypercubes, Leonard triples and the anticommutator spin algebra
• Totally Bipartite/ABipartite Leonard pairs and Leonard triples of Bannai/Ito type
• Terwilliger algebra of Odd graphs
• The Terwilliger algebra of the Odd graph revisited from the viewpoint of group action
• On Skein Algebras And Sl₂(C)-Character Varieties