2002/02/18 by Dimitry Leites, Alexander Sergeev
Mathematics · #math.RT #msc:17B20 #msc:17A70 #msc:17B35 #msc:17B70
published as E. Ivanov E. et. al. (eds.) {\it Supersymmetries and Quantum Symmetries} (SQS'99, 27--31 July, 1999), Dubna, JINR, 2000, 409--411 · 4 p., Latex
arxiv created 2002/02/18 · arxiv updated 2009/11/30
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(2n-1|2n-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(2n-1|2n-1) are the Casimir operators for po(0|2n). Similarly, quantization sends po(0|2n-1) into q(2n-1) and the above procedure makes Casimir operators for q(2n-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.