Supersymmetyric Analogues of the Classical Theorem on Harmonic Polynomials

Details Supersymmetyric Analogues of the Classical Theorem on Harmonic Polynomials Supersymmetyric Analogues of the Classical Theorem on Harmonic Polynomials

2012-02-08

arxiv.org/abs/1202.1673v1

Mathematics

Representation Theory

35pages. This an extension and replacement of the first author's article arXiv:1001.3474v1[Math.RT]

Scientific paper

Classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free module over the invariant polynomials generated by harmonic polynomials. In this paper, we first establish two-parameter $\mbb{Z}^2$-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebra $gl(n|m)$. Then we extend the result to two-parameter $\mbb{Z}$-graded supersymmetric oscillator generalizations of the above theorem for the Lie superalgebras $osp(2n|2m)$ and $osp(2n+1|2m)$.

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