Projection operator method for quantum groups (Q2760164)

This paper is the text of a series of lectures in which the projection operator method is developed for quantum groups, or more specifically, for all \(q\)-deformed universal enveloping algebras of contragredient Lie (super)algebras of finite growth. This carries over to the quantum setting the standard uses of projection operators for Lie (super)algebras, for example in the description of irreducible modules, computation of Clebsch-Gordan coefficients, construction of Gelfand-Tsetlin bases and so forth. NEWLINENEWLINENEWLINEThe paper can be divided into roughly two halves. The first defines all the basic terms, starting with Lie (super)algebras of Cartan type and their quantization, the quantum Cartan-Weyl basis and on to the universal \(R\)-matrix and extremal projectors. The second half gives some applications of the extremal projector (and its derived general projection operators) in order to compute Clebsch-Gordan and Racah coefficients for \(U_qsl(2)\), \(U_qsu(3)\) as well as derive the explicit action of the Cartan-Weyl generators on the Gelfand-Tsetlin basis for \(U_qu(n)\). NEWLINENEWLINENEWLINEThe paper is mainly self-contained and provides a good introductory survey of the use of projection operators in quantum groups.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00053].