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

Abstract

We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's and Hahn's q-polynomials and introduce orthogonal polynomials corresponding to Lie superlagebras. We also describe the real forms of gl(N), quasi-finite modules over gl(N), and conditions for unitarity of the quasi-finite modules. Analogs of tensors over gl(N) are also introduced.

25 pages, LaTeX

References in corpus (1)

• Enveloping algebra U(gl(3)) and orthogonal polynomials in several discrete indeterminates

Cited by in corpus (6)

• Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras
• Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them
• Enveloping algebra U(gl(3)) and orthogonal polynomials in several discrete indeterminates
• The Shapovalov determinant for the Poisson superalgebras
• Orthogonalization of graded sets of vectors
• Enveloping superalgebra U(osp(1|2)) and orthogonal polynomials in discrete indeterminate