U(sl₂) AND THE TERWILLIGER ALGEBRAS (Research on finite groups, algebraic combinatorics, and vertex algebras)

Abstract

The universal enveloping algebra U(sl₂) of sl₂ is a unital associative algebra over ℂ generated by E, F, H subject to the relations [H, E] =2E, [H, F] = -2F, [E, F]=H. In 2002, Junie T. Go showed that the Terwilliger algebra of H(D, 2) is a homomorphic image of U(sl₂). Firstly, I will present a connection of the even subalgebra of U(sl₂) with the Terwilliger algebra of ½H(D, 2). Secondly, I will show how the Clebsch-Gordan rule of U(sl₂) is related to the Terwilliger algebra of H(D, q). Thirdly, I will give an algebraic connection between the Clebsch-Gordan coefficients of U(sl₂) and the Terwilliger algebra of J(D, k). The first part is a joint work with Chia-Yi Wen.