Let \(\mathrm{Alg}_k\) denote the category of commutative (associative) \(k\)-algebras
An affine group is a group object in the category of representable functors \(\mathrm{Alg}_k \longrightarrow \mathrm{Set}\)
When \(\mathbf{G}\) is represented by a finitely generated \(k\)-algebra, it is called an affine algebraic group
From now on “algebraic group” will mean “affine algebraic group”
Representation has some notational cost
Taming symbols, an algebra for aftermath
.
Affinity, I think, can't be completely lost
Because injective homomorphisms of Hopf algebras are automatically faithfully flat.
Taming symbols, I felt I was going places
\((g,x) \mapsto gxg^{-1}\) the action of \(\mathbf{G}\) on itself by conjugation,
in turn, this gives a map on the tangent spaces.
Days are faithfully flat, like an Hopf injection.
The conjugations have shuffled my tenses
-Sets of derivations, augmentation ideal-
I'm looking for a map of the tangent spaces
Example 3.2 If \(\mathbf{G} = \mathbf{GL}\)
Define the ideal by the exact sequence
\(0 \rightarrow I \rightarrow \mathcal{O}(\mathbf{G}) \rightarrow k \rightarrow 0\), and see what it entails.
In the linear case, perhaps it will make sense,
I wish you had explained this to me in detail.