A connection behind the Terwilliger algebras of H(D,2) and [Formula presented]

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 element [Formula presented] 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 of α=[C,A]+2A²+B,β=[B,C]+4BA+2C is central in H. The distinguished central element Ω=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 sends [Formula presented] 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 L_n up to isomorphism. We show that the H-module L_n (n≥1) is a direct sum of two non-isomorphic irreducible H-modules.