The algebra A_{\hbar,\eta}(\hat{g}) and Infinite Hopf family of algebras

Abstract

New deformed affine algebras A_{\hbar,\eta}(\hat{g}) are defined for any simply-laced classical Lie algebra g, which are generalizations of the algebra A_{\hbar,\eta}(\hat{sl_2}) recently proposed by Khoroshkin, Lebedev and Pakuliak (KLP). Unlike the work of KLP, we associate to the new algebras the structure of an infinite Hopf family of algebras in contrast to the one containing only finite number of algebras introduced by KLP. Bosonic representation for A_{\hbar,\eta}(\hat{g}) at level 1 is obtained, and it is shown that, by repeated application of Drinfeld-like comultiplications, a realization of A_{\hbar,\eta}(\hat{g}) at any positive integer level can be obtained. For the special case of g=sl_{r+1}, (r+1)-dimensional evaluation representation is given. The corresponding intertwining operators are defined and the intertwining relations are also derived explicitly.