Weyl group extension of quantized current algebras

In this paper, we extend the Drinfeld current realization of quantum affine algebras Uq(ĝ) and of the Yangians in several directions: we construct current operators for non-simple roots of g, define a new braid group action in terms of the current operators and describe the universal R-matrix for the corresponding “Drinfeld” comultiplication in the form of infinite product and in the form of certain integrals over current operators. 0 Introduction In the theory of simple Lie algebras, the theory of affine Lie algebras ĝ distinguishs themselves from others due to the realization of the affine Lie algebras in current operators (or loop realizations). The current operator is an operator define as x(z) = Σn∈Zxnz −n, where xn is an operator in the algebra and z is a formal variable. One can associate to each root of g a current operator xα(z), which gives a complete description of the whole Lie algebra ĝ. The advantage of the current realization of affine Lie algebras lies in the fact that the formal variable z, which we can also treated as a number in C, allows us to connect the algebraic structures of affine Lie algebra with analytic, geometric and topological structures, which is manifest best in the theory of Knizhinik-Zamolochikov equations. Current realization of quantum affine algebras was first given by Drinfeld after a few years of the discovery of the theory of quantum groups [Dr2]. The connection of the Drinfeld realization with the first realization of quantum groups given by Jimbo and Drinfeld and with other type of realizations is also highly nontrivial [Be][KT2][KT3][DF]. Drinfeld current realization of Uq(ĝ) has since played extremely important role in the theory of affine quantum algebras. However from the structure point of view, Drinfeld realization is not complete in the sense that in this realization only the current operators for the simple roots are given. Also recently, it has gradually become clear that there is a possibility to develop a theory of Drinfeld realization as a theory of quantized algebras of currents completely parallel to the theory of quantum groups. This is the problem we want to address in this paper. The receipt of contructing the new current operator follows from a simple idea in the theory of the affine Lie algebras, which is to use analytic properties of the matrix coefficients of current operators. Let α, β and α+ β be positive roots of g. In the affine Lie algebra ĝ, we have [Xα(z),Xβ(w)] = δ(z/w)Xα+β (z), E-mail: [email protected] E-mail: [email protected]