## 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 element [Formula presented] is called the Casimir element of U(sl_{2}). Let Δ:U(sl_{2})→U(sl_{2})⊗U(sl_{2}) denote the comultiplication of U(sl_{2}). The universal Hahn algebra H is a unital associative algebra over C generated by A,B,C and the relations assert that [A,B]=C and each of [C,A]+2A^{2}+B,[B,C]+4BA+2C is central in H. Inspired by the Clebsch–Gordan coefficients of U(sl_{2}), we discover an algebra homomorphism ♮:H→U(sl_{2})⊗U(sl_{2}) that maps [Formula presented] By pulling back via ♮ any U(sl_{2})⊗U(sl_{2})-module can be considered as an H-module. For any integer n≥0 there exists a unique (n+1)-dimensional irreducible U(sl_{2})-module L_{n} up to isomorphism. We study the decomposition of the H-module L_{m}⊗L_{n} for any integers m,n≥0. We link these results to the Terwilliger algebras of Johnson graphs. We express the dimensions of the Terwilliger algebras of Johnson graphs in terms of binomial coefficients.

Original language | English |
---|---|

Article number | 105833 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 203 |

DOIs | |

State | Published - Apr 2024 |

## Keywords

- Clebsch–Gordan coefficients
- Hahn polynomials
- Johnson graphs
- Terwilliger algebras

## Fingerprint

Dive into the research topics of 'The Clebsch–Gordan coefficients of U(sl_{2}) and the Terwilliger algebras of Johnson graphs'. Together they form a unique fingerprint.