A convolution of distributions is again a distribution and thus they form the Hopf algebra on G, denoted by Dist(G), which contains the Lie algebra Lie(G) associated to G. Over a field of characteristic zero, Cartier's theorem says that Dist(G) is isomorphic to the universal enveloping algebra of the Lie algebra of G and thus the construction gives no new information.
Let k be an algebraically closed field and G a linear algebraic group (that is, affine algebraic group) over k. By definition, Lie(G) is the Lie algebra of all derivations of k[G] that commute with the left action of G. As in the Lie group case, it can be identified with the tangent space to G at the identity element.
The finite dual of A is the space of linear functionals on A with kernels containing left ideals of finite codimensions.
Concretely, it can be viewed as the space of matrix coefficients.
Let X = Spec A be an affine scheme over a field k and let Ix be the kernel of the restriction map
By definition, a distribution f supported at x'' is a k-linear functional on A such that
for some n. (Note: the definition is still valid if k is an arbitrary ring.)
Now, if G is an algebraic group over k, we let Dist(G) be the set of all distributions on G supported at the identity element (often just called distributions on G).
If f, g are in it, we define the product of f and g, demoted by f * g, to be the linear functional where Δ is the comultiplication that is the homomorphism induced by the multiplication
) and thus Dist(G) is an associative algebra, as the set is closed under the muplication by the formula: It is also unital with the unity that is the linear functional
Thus, a tangent vector amounts to a linear functional on I1 that has no constant term and kills the square of I1 and the formula (*) implies
be the Lie algebra of G. Then, by the universal property, the inclusion
be the additive group; i.e., G(R) = R for any k-algebra R. As a variety G is the affine line; i.e., the coordinate ring is k[t] and In0 = (tn).
be the multiplicative group; i.e., G(R) = R* for any k-algebra R. The coordinate ring of G is k[t, t−1] (since G is really GL1(k).)
There is an equivalence of categories between the category of finite algebraic groups and the category of finite-dimensional cocommutative Hopf algebras given by mapping G to k[G]*, the dual of the coordinate ring of G. Note that Dist(G) is a (Hopf) subalgebra of k[G]*.