Definition
Let X = Spec A be an affine scheme over a field k and let Ix be the kernel of the restriction map , the residue field of x. 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 . The multiplication turns out to be associative (use ) 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 , 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 . 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 is still a tangent vector.
Let be the Lie algebra of G. Then, by the universal property, the inclusion induces the algebra homomorphism:
When the base field k has characteristic zero, this homomorphism is an isomorphism.[1]