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Generalized Casimir operators for loop Lie superalgebras
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 27 January 2025, pp. 1-17
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- Article
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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.
