Extension of a quantized enveloping algebra by a Hopf algebra

Abstract

Suppose that H is a Hopf algebra, and $$ \mathfrak{g} $$ is a generalized Kac-Moody algebra with Cartan matrix A = (a ij )I × I, where I is an index set and is equal to either {1, 2,...,n} or the natural number set ℕ. Let f, g be two mappings from I to G(H), the set of group-like elements of H, such that the multiplication of elements in the set {f(i), g(i)|i ∈, I} is commutative. Then we define a Hopf algebra H ⊗ U q ( $$ \mathfrak{g} $$ ), where U q ( $$ \mathfrak{g} $$ ) is the quantized enveloping algebra of $$ \mathfrak{g} $$ .