调和多项式经典定理的超对称类比
Supersymmetyric Analogues of the Classical Theorem on Harmonic Polynomials
摘要：
经典调和分析认为齐次调和多项式空间（Laplace方程的解）是相应的正交李群（代数）的不可约模，整个多项式代数是调和多项式生成的不变多项式上的自由模。本文首先建立了李超代数$gl(n|m)$的两参数$\mbb{Z}^2$ -分次超对称振子推广.然后我们将结果推广到李超代数$osp(2n|2m)$和$osp(2n+1|2m)$的两参数$\mbb{Z}$ -阶次超对称振子.
英文摘要：
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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