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 ⊗ ^f _g U _q ($\mathfrak{g}$), where U _q ($\mathfrak{g}$) is the quantized enveloping algebra of $\mathfrak{g}$.