The Clebsch–Gordan coefficients of U(sl2) and the Terwilliger algebras of Johnson graphs

研究成果: 雜誌貢獻期刊論文同行評審

摘要

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 element [Formula presented] is called the Casimir element of U(sl2). Let Δ:U(sl2)→U(sl2)⊗U(sl2) denote the comultiplication of U(sl2). 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]+2A2+B,[B,C]+4BA+2C is central in H. Inspired by the Clebsch–Gordan coefficients of U(sl2), we discover an algebra homomorphism ♮:H→U(sl2)⊗U(sl2) that maps [Formula presented] By pulling back via ♮ any U(sl2)⊗U(sl2)-module can be considered as an H-module. For any integer n≥0 there exists a unique (n+1)-dimensional irreducible U(sl2)-module Ln up to isomorphism. We study the decomposition of the H-module Lm⊗Ln 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.

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文章編號105833
期刊Journal of Combinatorial Theory. Series A
203
DOIs
出版狀態已出版 - 4月 2024

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