Generalized Casimir operators for loop Lie superalgebras

Das, Abhishek ; Pattanayak, Santosha

Canadian mathematical bulletin, Tome 68 (2025) no. 3, pp. 744-760

Résumé

Let $\mathfrak{g}$ be the queer superalgebra $\operatorname {\mathfrak{q}}(n)$ over the field of complex numbers $\mathbb C$. For any associative, commutative, and finitely generated $\mathbb C$-algebra A with unity, we consider the loop Lie superalgebra $\mathfrak{g} \otimes A$. We define a class of central operators for $\mathfrak{g} \otimes A$, which generalizes the classical Gelfand invariants. We show that they generate the algebra $U(\mathfrak{g} \otimes A)^{\mathfrak{g}}$. We also show that there are no non-trivial $\mathfrak{g}$-invariants of $U(\mathfrak{g} \otimes A)$ where $\mathfrak{g}=\mathfrak{p}(n)$, the periplectic Lie superalgebra.

Détail

DOI : 10.4153/S0008439525000050

Mots-clés : Lie superalgebras, Casimir operators, loop superalgebras, universal enveloping algebra