Valery N. Tolstoy, Moscow State University, Russia
e-mail:  tolstoy@anna19.npi.msu.su

Projection Operator Method for Quantum Groups

Abstract

A method of projection operators is developed for quantum groups. Here the term ``quantum groups" means \$q\$-deformed universal enveloping algebras of the Kac-Moody (super)algebras of finite growth (these (super)algebras include all finite-dimensional simple Lie algebras and classical superalgebras, infinite-dimensional affine (loop) Lie algebras and superalgebras). At first we describe combinatorial structure of root systems of the Kac-Moody (super)algebras of finite growth. The result is used for construction of a \$q\$-analog of  the Cartan-Weyl basis and for description of explicit structure of the universal R-matrices and the extremal projectors.

Then we consider some applications of the extremal projectors. In particular, we apply them for explicit description of reduction algebra over quantum algebras \$U_q(su(n))\$ and for construction of a \$q\$-analog of the Gel'fand-Tsetlin basis for \$U_q(su(n))\$. We also use the projector operator method for construction of  the theory of Clebsch-Gordan coefficients for quantum algebras \$U_q(su(2))\$ and \$U_q(su(3))\$. In particular, we obtain a very compact general formula for the canonical \$U_q(su(3))\supset U_q(su(2))\$ Clebsch-Gordan coefficients in terms of the \$U_q(su(2))\$ Wigner 3nj-symbols which are connected with the basic hyperheometric series. Finally we introduce some generalizations of extremal projector for the case of the quantum algebra \$U_q(su(2))\$ and consider their applications. (See also latex, dvi, ps, pdf files of this abstract.)

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