Casimir operators for Lie superalgebras

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.