A connection behind the Terwilliger algebras of H(D,2) and [formula omitted

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. The distinguished central elementΛ=EF+FE+H22 is called the Casimir element of U(sl2). The universal Hahn algebra H is a unital associative algebra over C with generators A,B,C and the relations assert that [A,B]=C and each ofα=[C,A]+2A2+B,β=[B,C]+4BA+2C is central in H. The distinguished central elementΩ=4ABA+B2−C2−2βA+2(1−α)B is called the Casimir element of H. By investigating the relationship between the Terwilliger algebras of the hypercube and its halved graph, we discover the algebra homomorphism ♮:H→U(sl2) that sendsA↦H4,B↦E2+F2+Λ−14−H28,C↦E2−F24. We determine the image of ♮ and show that the kernel of ♮ is the two-sided ideal of H generated by β and 16Ω−24α+3. By pulling back via ♮ each U(sl2)-module can be regarded as an H-module. For each integer n≥0 there exists a unique (n+1)-dimensional irreducible U(sl2)-module Ln up to isomorphism. We show that the H-module Ln (n≥1) is a direct sum of two non-isomorphic irreducible H-modules.