Invariant Hopf 2-Cocycles for Affine Algebraic Groups

Abstract

Abstract We generalize the theory of the second invariant cohomology group $H^{2}_{\textrm{inv}}(G)$ for finite groups G, developed in [3, 4, 14], to the case of affine algebraic groups G, using the methods of [9, 10, 12]. In particular, we show that for connected affine algebraic groups G over an algebraically closed field of characteristic 0, the map Θ from [14] is bijective (unlike for some finite groups, as shown in [14]). This allows us to compute $H^{2}_{\textrm{inv}}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown in [14]).