Distribution on a linear algebraic group explained

In algebraic geometry, given a linear algebraic group G over a field k, a distribution on it is a linear functional

k[G]\tok

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

A\tok(x)

, the residue field of x. By definition, a distribution f supported at x'' is a k-linear functional on A such that

	n)
f(I
	x

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

k[G]\overset{\Delta}\tok[G] ⊗ k[G]\overset{f ⊗ g}\tok ⊗ k=k

where Δ is the comultiplication that is the homomorphism induced by the multiplication

G x G\toG

. The multiplication turns out to be associative (use

1 ⊗ \Delta\circ\Delta=\Delta ⊗ 1\circ\Delta

) and thus Dist(G) is an associative algebra, as the set is closed under the muplication by the formula:

(*)

	n)
\Delta(I
	1

\subset

	n
\sum
	r=0

	r
I
	1

⊗

	n-r
I
	1.

It is also unital with the unity that is the linear functional

k[G]\tok,\phi\mapsto\phi(1)

, 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

I_1/I

	2

	1

. 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

[f,g]=f*g-g*f

is still a tangent vector.

Let

ak{g}=\operatorname{Lie}(G)

be the Lie algebra of G. Then, by the universal property, the inclusion

ak{g}\hookrightarrow\operatorname{Dist}(G)

induces the algebra homomorphism:

U(ak{g})\to\operatorname{Dist}(G).

When the base field k has characteristic zero, this homomorphism is an isomorphism.

Examples

Additive group

Let

G=G_a

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 = (tⁿ).

Multiplicative group

Let

G=G_m

be the multiplicative group; i.e., G(R) = R^* for any k-algebra R. The coordinate ring of G is k[''t'', ''t''<sup>−1</sup>] (since G is really GL₁(k).)

Correspondence

For any closed subgroups H, K of G, if k is perfect and H is irreducible, then

H\subsetK\Leftrightarrow\operatorname{Dist}(H)\subset\operatorname{Dist}(K).

If V is a G-module (that is a representation of G), then it admits a natural structure of Dist(G)-module, which in turns gives the module structure over

ak{g}

Any action G on an affine algebraic variety X induces the representation of G on the coordinate ring k[''G'']. In particular, the conjugation action of G induces the action of G on k[''G'']. One can show I is stable under G and thus G acts on (k[''G'']/I)^* and whence on its union Dist(G). The resulting action is called the adjoint action of G.

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'']^*.

Relation to Lie group–Lie algebra correspondence

See main article: Lie group–Lie algebra correspondence.

References

Book: Jantzen . Jens Carsten . Representations of Algebraic Groups . 1987 . Boston . Academic Press . 978-0-12-380245-3 . Pure and Applied Mathematics . 131.
Milne, iAG: Algebraic Groups: An introduction to the theory of algebraic group schemes over fields
Claudio Procesi, Lie groups: An approach through invariants and representations, Springer, Universitext 2006
Book: Mukai , S. . 2002. An introduction to invariants and moduli. Cambridge Studies in Advanced Mathematics. 81. 978-0-521-80906-1.