Enveloping Superalgebra and Orthogonal Polynomials in Discrete Indeterminate

Abstract

Let A be an associative simple (central) superalgebra over ℂ and L an invariant linear functional on it (trace). Let be an antiautomorphism of A such that (a t)t = (−1)p(a) a, where p(a) is the parity of a, and let L(a t) = L(a). Then A admits a nondegenerate supersymmetric invariant bilinear form a, b = L(ab t). For , where m is any maximal ideal of , Leites and I have constructed orthogonal basis in A whose elements turned out to be, essentially, Chebyshev (Hahn) polynomials in one discrete variable. Here I take for any maximal ideal m and apply a similar procedure. As a result we obtain either Hahn polynomials over ℂ[τ], where τ 2 ∈ ℂ, or a particular case of Meixner polynomials, or — when A=Mat(n+1| n) — dual Hahn polynomials of even degree, or their (hopefully, new) analogs of odd degree. Observe that the nondegenerate bilinear forms we consider for orthogonality are, as a rule, not sign definite.