Enveloping superalgebra U(osp(1|2)) and orthogonal polynomials in discrete indeterminate

Abstract

Let $A$ be an associative simple (central) superalgebra over $\mathbb{C}$ and $L$ an invariant linear functional on it (trace). Let $a\mapsto a^t$ 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 $⟨a,b⟩=L(a{b}^t)$. For $A=U(\mathfrak{sl}(2))/\mathfrak{m}$, where $\mathfrak{m}$ is any maximal ideal of $U(\mathfrak{sl}(2))$, 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 $A=U(\mathfrak{osp}(1|2))/\mathfrak{m}$ for any maximal ideal $\mathfrak{m}$ and apply a similar procedure. As a result we obtain either Hahn polynomials over $\mathbb{C}[\tau ]$, where ${\tau }^2\in \mathbb{C}$, or a particular case of Meixner polynomials, or --- when $A=\mbox{Mat}(n+1|n)$ --- 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.

References in corpus (1)

• Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices

Cited by in corpus (1)

• Invariants and orthogonal $q$-polynomials associated with ${\mathbb{C}}_q(osp(1,2))$