EXTENSIONS OF FINITE QUANTUM GROUPS BY FINITE GROUPS

Abstract. Let G be a connected, simply connected complex semi-simple Lie group with Lie algebra g, Cartan matrix C and symmetrized Cartan matrix CD. Let ` ≥ 3 be an odd integer, relatively prime to detCD. Given an embedding σ of a finite group Γ on G and a primitive `-th root of unity , we construct a central extension Aσ of the function algebra CΓ by the dual of the Frobenius-Lusztig kernel u(g); Aσ is a quotient of the quantized coordinate algebra O(G) and dimAσ = |Γ|`dim g. If G is simple and σ(Γ) is not central in G, then we obtain an infinite family of pairwise non-isomorphic Hopf algebras which are non-semisimple, non-pointed and their duals are also non-pointed. This generalizes the result obtained by E. Müller [M3] for SL2(C). Nevertheless, it follows from results in [Mk] that these Hopf algebras are cocycle deformations of each other. 1