Supersymmetyric Analogues of the Classical Theorem on Harmonic Polynomials

TL;DR

This paper extends classical harmonic polynomial theorems to supersymmetric oscillator models for Lie superalgebras, revealing new irreducibility and module structures in supersymmetric harmonic analysis.

Contribution

It introduces two-parameter graded supersymmetric oscillator generalizations of classical harmonic polynomial theorems for various Lie superalgebras, expanding the scope of harmonic analysis.

Findings

Established supersymmetric oscillator generalizations for $gl(n|m)$.

Extended results to $osp(2n|2m)$ and $osp(2n+1|2m)$.

Demonstrated irreducibility of supersymmetric harmonic modules.

Abstract

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 $osp(2n+1|2m)$.