Invariant Hopf $2$-cocycles for affine algebraic groups

TL;DR

This paper extends the theory of invariant second cohomology groups from finite groups to affine algebraic groups, providing new computations and properties for connected affine algebraic groups over algebraically closed fields of characteristic zero.

Contribution

It generalizes the invariant cohomology theory to affine algebraic groups and proves the bijectivity of a key map for connected groups, enabling explicit calculations of the cohomology group.

Findings

H^2_{inv}(G) is computable for connected affine algebraic groups

The cohomology group is commutative for these groups

The map Θ is bijective in this setting

Abstract

We generalize the theory of the second invariant cohomology group $H^2_{\rm inv}(G)$ for finite groups $G$, developed in [Da2,Da3,GK], to the case of affine algebraic groups $G$, using the methods of [EG1,EG2,G]. In particular, we show that for connected affine algebraic groups $G$ over an algebraically closed field of characteristic $0$, the map $\Theta$ from [GK] is bijective (unlike for some finite groups, as shown in [GK]). This allows us to compute $H^2_{\rm inv}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown in [GK]).