2 grammar maven!

I will offer a sketch of an argument, and maybe someone who knows what a stack is can make it happen for real. There is probably a non-stacky deformation theory of commutative Hopf algebras, but I don't know what it looks like.

Deforming G as a group should be the same as deforming BG as a plain old geometric object. Pulling back a point in BG along a cover by a point is very roughly taking a based loop space, and the deformed loop space comes with the deformed composition law. Similarly, deforming a representation of G should be the same as deforming a sheaf on BG.

I'm going to assume G is smooth. Then the tangent complex of BG mapping to a point is just the sheaf Ad, concentrated in degree 1. If we boldly assume that deformation theory of/on stacks works just like deformations of/on schemes, but maybe with some degree shifts, we should get the answers you want. For deforming G in particular, there is a canonical class in H^2(BG, Ad[-1]) that classifies obstructions, and if that vanishes, H^1(BG, Ad[-1]) classifies deformations and H^0(BG, Ad[-1]) classifies automorphisms of a deformation. When deforming the sheaf V, one usually sees the sheaf End(V) written as coefficients.

Olsson wrote a paper on deformations of representable morphisms of stacks, and while the morphism BG -> S isn't representable, one might benefit from asking the author for additional details if one were, say, working in the same building as himhe.

1

I will offer a sketch of an argument, and maybe someone who knows what a stack is can make it happen for real. There is probably a non-stacky deformation theory of commutative Hopf algebras, but I don't know what it looks like.

Deforming G as a group should be the same as deforming BG as a plain old geometric object. Pulling back a point in BG along a cover by a point is very roughly taking a based loop space, and the deformed loop space comes with the deformed composition law. Similarly, deforming a representation of G should be the same as deforming a sheaf on BG.

I'm going to assume G is smooth. Then the tangent complex of BG mapping to a point is just the sheaf Ad, concentrated in degree 1. If we boldly assume that deformation theory of/on stacks works just like deformations of/on schemes, but maybe with some degree shifts, we should get the answers you want. For deforming G in particular, there is a canonical class in H^2(BG, Ad[-1]) that classifies obstructions, and if that vanishes, H^1(BG, Ad[-1]) classifies deformations and H^0(BG, Ad[-1]) classifies automorphisms of a deformation. When deforming the sheaf V, one usually sees the sheaf End(V) written as coefficients.

Olsson wrote a paper on deformations of representable morphisms of stacks, and while the morphism BG -> S isn't representable, one might benefit from asking the author for additional details if one were, say, working in the same building as him.