Extremal projector and universal R-matrix for quantized contragredient Lie (super)algebras

Abstract

Two basic elements of the representation theory of quantized finite-dimensional contragredient Lie (super)algebras g (U q(g)) are presented. These are the universal R-matrix to be an interwining operator, and the extremal projector which gives a powerful method for decomposition of representations. Properties of Cartan-Weyl basis for U q(g) are discussed. Some Taylor extension of U q(g) and U q(g) ⊗ U q(g) are introduced in terms of this basis. The extremal projector p and the universal R-matrix R are described as unique elements of these extensions. Explicit formulae for p and R are given.