Enveloping Superalgebra U(osp(1|2)) and Orthogonal Polynomials in Discrete Indeterminate

DOI

10.2991/jnmp.2001.8.2.7How to use a DOI?

Abstract

Let A be an associative simple (central) superalgebra over C and L an invariant linear functional on it (trace). Let a at be an antiautomorphism of A such that (at )t = (-1)p(a) a, where p(a) is the parity of a, and let L(at ) = L(a). Then A admits a nondegenerate supersymmetric invariant bilinear form a, b = L(abt ). For A = U(sl(2))/m, where m is any maximal ideal of U(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(osp(1|2))/m for any maximal ideal m and apply a similar procedure. As a result we obtain either Hahn polynomials over C[], where 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.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - A. Sergeev
PY  - 2001
DA  - 2001/05/01
TI  - Enveloping Superalgebra U(osp(1|2)) and Orthogonal Polynomials in Discrete Indeterminate
JO  - Journal of Nonlinear Mathematical Physics
SP  - 229
EP  - 255
VL  - 8
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2001.8.2.7
DO  - 10.2991/jnmp.2001.8.2.7
ID  - Sergeev2001
ER  -