ENVELOPING ALGEBRA U(gl(3)) AND ORTHOGONAL POLYNOMIALS IN SEVERAL DISCRETE INDETERMINATES
نویسنده
چکیده
Let A be an associative algebra over C and L an invariant linear functional on it (trace). Let ω be an involutive antiautomorphism of A such that L(ω(a)) = L(a) for any a ∈ A. Then A admits a symmetric invariant bilinear form 〈a, b〉 = L(aω(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 whis way we obtain multivariable analogs of Hahn polynomials. This paper appeared in: Duplij S., Wess J. (eds.) Noncommutative structures in mathematics and physics, Proc. NATO Advanced Reserch Workshop, Kiev, 2000. Kluwer, 2001, 113–124; I just want to make it more accessible.
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