Distribution on a linear algebraic group

In algebraic geometry, given a linear algebraic group "G" over a field "k", a distribution on it is a linear functional formula_1 satisfying some support condition. 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. In the positive characteristic case, the algebra can be used as a substitute for the Lie group–Lie algebra correspondence and its variant for algebraic groups in the characteristic zero ; for example, this approach taken in .
Construction.
The Lie algebra of a linear algebraic group.
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.
Enveloping algebra.
There is the following general construction for a Hopf algebra. Let "A" be a Hopf algebra. 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.
Distributions on an algebraic group.
Definition.
Let "X" = Spec "A" be an affine scheme over a field "k" and let "I""x" be the kernel of the restriction map formula_2, the residue field of "x". By definition, a "distribution" "f" supported at "x"" is a "k"-linear functional on "A" such that formula_3 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 formula_5. The multiplication turns out to be associative (use formula_6) 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 formula_8, the Dirac's delta measure.
The Lie algebra Lie("G") sits inside Dist("G"). Indeed, by definition, Lie("G") is the tangent space to "G" at the identity element 1; i.e., the dual space of formula_9. Thus, a tangent vector amounts to a linear functional on "I"1 that has no constant term and kills the square of "I"1 and the formula (*) implies formula_10 is still a tangent vector.
Let formula_11 be the Lie algebra of "G". Then, by the universal property, the inclusion formula_12 induces the algebra homomorphism:
When the base field "k" has characteristic zero, this homomorphism is an isomorphism.
Examples.
Additive group.
Let formula_14 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 "I" = ("t""n").
Multiplicative group.
Let formula_15 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 "GL"1("k").)
The case of finite algebraic groups.
Let "G" be an algebraic group that is "finite" as a group scheme; for example, any finite group may be viewed as a finite algebraic group. 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"]*.