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Generalized Casimir operators for loop Lie superalgebras

Part of: Special varieties Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  27 January 2025

Abhishek Das  and
Santosha Pattanayak
Abhishek Das
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India abhidas20@iitk.ac.in
Santosha Pattanayak*
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India abhidas20@iitk.ac.in
*
e-mail: santosha@iitk.ac.in
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Abstract

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.

Keywords

Lie superalgebrasCasimir operatorsloop superalgebrasuniversal enveloping algebra

MSC classification

Primary: 17B10: Representations, algebraic theory (weights) 17B35: Universal enveloping (super)algebras 17B65: Infinite-dimensional Lie (super)algebras
Secondary: 14M30: Supervarieties
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 17
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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