Harmonic polynomials in Grassmann algebras (Q2719423)

This paper addresses the representations of the classical simple Lie algebras \(sp(2n,C)\) and \(so(m,C)\) in the Grassmann algebras. For each algebra an analogue of the Laplace operator \(\Delta\) is considered which is interchangeable with the action of the corresponding algebra in the Grassmann algebra. The author follows the Constant's terminology and the solutions of the equation \( \Delta f = 0 \) are called \(Sp\)-harmonic polynomials or \(SO\)-harmonic polynomials, respectively. In the space \(H\) of such polynomials an orthogonal basis is constructed. Besides, the translators [\textit{D.~P.~Zhelobenko}, ``Representations of Reductive Lie Algebras'', Nauka, Moscow (1993; Zbl 0842.17007)] are used, i.e. operators translating the solutions of the equation \(\Delta f = 0\) again into solutions of this equation.