Projection operator method for quantum groups

Abstract

In these lectures we develop the projection operator method for quantum groups. Here the term "quantum groups" means q-deformed universal enveloping algebras of contragredient Lie (super)algebras of finite growth. Contains of the lectures can be divided on two parts. Basis fragments of the first part are: combinatorial structure of root systems, the q-analog of the Cartan-Weyl basis, the extremal projector and the universal R-matrix for any contragredient Lie (super)algebra of finite growth. The explicit expressions for the extremal projectors and the universal R-matrices are ordered products of special q-series depending on noncommutative Cartan-Weyl generators. In second part we consider some applications of the extremal projectors. Here we use the projector operator method to develop the theory of the Clebsch-Gordan coefficients for the quantum algebras U_q(su(2)) and U_q(su(3)). In particular, we give 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. Then we apply the projection operator method for the construction of the q-analog of the Gelfand-Tsetlin basis for U_q(su(n)). Finally using analogy between the extremal projector p(U_q(sl(2))) of the quantum algebra U_q(sl(2)) and the \delta(x)-function we introduce 'adjoint extremal projectors' p^{(n)}(U_q(sl(2)) (n=1,2,...) which are some generalizations of the extremal projector p(U_q(sl(2))), and which are analogies of the derivatives of the \delta(x)-function, \delta^{(n)}(x).