Rad HAZU, Matematičke znanosti, Vol. 28 (2024), 339-352.

ON BASES OF g-INVARIANT ENDOMORPHISM ALGEBRAS

Jing-Song Huang and Yufeng Zhao

Department of Mathematics, Peking University, Beijing, China
e-mail: Zhaoyufeng@math.pku.edu.cn

Abstract.   Let g be a complex simple Lie algebra. Let Z(g) be the center of the universal enveloping algebra U(g). Let V_λ be the finite-dimensional irreducible g-module with highest weight λ. Our main result is a criterion of the existence of Z(g)-bases for the g-invariant endomorphism algebra R_λ=: Hom_g(End V_λ,U(g)). Then we obtain a Clifford algebra analogue, namely a criterion of the existence C(g)^g-bases for R_λ^C =: Hom_g(End V_λ,C(g)). We also describe a criterion of the existence of bases generated by powers of the Casimir element for R_{λ,ν} =: Hom_g(End V_λ, End V_ν).

2020 Mathematics Subject Classification.   22E47, 22E46.

Key words and phrases.   Simple Lie algebra, Casimir operator, invariant endomorphism algebra.

DOI: https://doi.org/10.21857/y6zolb4v6m

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