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

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

2002-02-18

arxiv.org/abs/math/0202182v1

Duplij S., Wess J. (eds.) Noncommutative structures in mathematics and physics, Proc. NATO Advanced Research Workshop, Kiev, 2

Mathematics

Representation Theory

12p., Latex

Scientific paper

Let A be an associative complex algebra and L an invariant linear functional on it (trace). Let i be an involutive antiautomorphism of A such that L(i(a))=L(a) for any a in A. Then A admits a symmetric invariant bilinear form (a, b)=L(a i(b)). For A=U(sl(2))/m, where m is any maximal ideal of U(sl(2)), Leites and I have constructed orthogonal basis whose elements turned out to be, essentially, Chebyshev and Hahn polynomials in one discrete variable. Here I take A=U(gl(3))/m for the maximal ideals m which annihilate irreducible highest weight gl(3)-modules of particular form (generalizations of symmetric powers of the identity representation). In this way we obtain multivariable analogs of Hahn polynomials.

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