Distinguish from the notion of a group of affine transformations (of an affine space) or the (general) affine group.
An affine group scheme is a group object (hence a group scheme) internal to the category of affine schemes.
If $k$ is a commutative ring, the category of affine group $k$-schemes is opposite to the category of commutative Hopf $k$-algebras.
By an affine algebraic group, one additionally assumes that the underlying scheme is over an algebraic variety (in particular reducible) and over a field. The narrower notion then agrees with the notion of a linear algebraic group.
Waterhouse12.1 The Lie algebra of the affine group $k$-scheme $X$ is the Lie algebra of left-invariant $k$-linear derivations $T:\mathcal{O}(X)\to\mathcal{O}(X)$ of the Hopf algebra of regular functions of $X$, which are also morphisms of right comodules: $\Delta\circ T = (id\otimes T)\circ\Delta:\mathcal{O}(X)\to\mathcal{O}(X)\otimes\mathcal{O}(X)$ (one says also that $T$ is left-invariant).
This Lie algebra is also isomorphic to the Lie algebra of $k$-linear $k$-valued derivations of $\mathcal{O}(X)$.
Comprehensive textbooks include
James S. Milne, Basic theory of affine group schemes, pdf
William C. Waterhouse, Introduction to affine group schemes, Graduate texts in mathematics 66, 1979
Last revised on August 22, 2023 at 17:09:14. See the history of this page for a list of all contributions to it.