## Abstract

The universal enveloping algebra U(sl_{2}) of sl_{2} 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(sl_{2}). 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]+2A^{2}+B,β=[B,C]+4BA+2C is central in H. The distinguished central element Ω=4ABA+B^{2}−C^{2}−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(sl_{2}) 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(sl_{2})-module can be regarded as an H-module. For each integer n≥0 there exists a unique (n+1)-dimensional irreducible U(sl_{2})-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.

Original language | English |
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Pages (from-to) | 456-479 |

Number of pages | 24 |

Journal | Journal of Algebra |

Volume | 634 |

DOIs | |

State | Published - 15 Nov 2023 |

## Keywords

- Askey–Wilson relations
- Halved cubes
- Hypercubes
- Lie algebras
- Terwilliger algebras