Casimir operators for Lie superalgebras

TL;DR

This paper describes Casimir operators for various Lie superalgebras, explores their behavior under deformation and quantization, and states conjectures for cases where explicit operators are not yet known.

Contribution

It extends the description of Casimir operators to remaining Lie superalgebras and explores their deformation and quantization properties, including new conjectures.

Findings

Casimir operators are explicitly described for certain Lie superalgebras.

Under deformation, Casimir operators for po(0|2n) relate to those for gl(2^{n-1}|2^{n-1}).

Casimir operators for vect(0|m) are only constants for m>2.

Abstract

Casimir operators -- the generators of the center of the enveloping algebra -- are described for simple or close to them ``classical'' finite dimensional Lie superalgebras with nondegenerate symmetric even bilinear form in Sergeev A., The invariant polynomials on simple Lie superalgebras. Represent. Theory 3 (1999), 250--280; math-RT/9810111 and for the ``queer'' series in Sergeev A., The centre of enveloping algebra for Lie superalgebra Q(n, C). Lett. Math. Phys. 7, no. 3, 1983, 177--179. Here we consider the remaining cases, and state conjectures proved for small values of parameter. Under deformation (quantization) the Poisson Lie superalgebra po(0|2n) on purely odd superspace turns into gl(2^{n-1}|2^{n-1}) and, conjecturally, the lowest terms of the Taylor series expansion with respect to the deformation parameter (Planck's constant) of the Casimir operators for gl(2^{n-1}|2^{n-1}) are the Casimir operators for po(0|2n). Similarly, quantization sends po(0|2n-1) into q(2^{n-1}) and the above procedure makes Casimir operators for q(2^{n-1}) into same for po(0|2n-1). Casimir operators for the Lie superalgebra vect(0|m) of vector fields on purely odd superspace are only constants for m>2. Conjecturally, same is true for the Lie superalgebra svect(0|m) of divergence free vector fields, and its deform, for m>3. Invariant polynomials on po(0|2n-1) are also described. They do not correspond to Casimir operators.