Distribution On A Linear Algebraic Group

Short description: Linear function satisfying a support condition

In algebraic geometry, given a linear algebraic group G over a field k, a distribution on it is a linear functional [math]\displaystyle{ k[G] \to k }[/math] 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 (Jantzen 1987).

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.

The adjoint group of a Lie algebra

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 [math]\displaystyle{ A \to k(x) }[/math], the residue field of x. By definition, a distribution f supported at x'' is a k-linear functional on A such that [math]\displaystyle{ f(I_x^n) = 0 }[/math] 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

[math]\displaystyle{ k[G] \overset{\Delta}\to k[G] \otimes k[G] \overset{f \otimes g}\to k \otimes k = k }[/math]

where Δ is the comultiplication that is the homomorphism induced by the multiplication [math]\displaystyle{ G \times G \to G }[/math]. The multiplication turns out to be associative (use [math]\displaystyle{ 1 \otimes \Delta \circ \Delta = \Delta \otimes 1 \circ \Delta }[/math]) and thus Dist(G) is an associative algebra, as the set is closed under the muplication by the formula:

(*) [math]\displaystyle{ \Delta(I_1^n) \subset \sum_{r=0}^n I_1^r \otimes I^{n-r}_1. }[/math]

It is also unital with the unity that is the linear functional [math]\displaystyle{ k[G] \to k, \phi \mapsto \phi(1) }[/math], 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 [math]\displaystyle{ I_1/I_1^2 }[/math]. 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 [math]\displaystyle{ [f, g] = f * g - g * f }[/math] is still a tangent vector.

Let [math]\displaystyle{ \mathfrak{g} = \operatorname{Lie}(G) }[/math] be the Lie algebra of G. Then, by the universal property, the inclusion [math]\displaystyle{ \mathfrak{g} \hookrightarrow \operatorname{Dist}(G) }[/math] induces the algebra homomorphism:

[math]\displaystyle{ U(\mathfrak{g}) \to \operatorname{Dist}(G). }[/math]

When the base field k has characteristic zero, this homomorphism is an isomorphism.^[1]

Examples

Additive group

Let [math]\displaystyle{ G = \mathbb{G}_a }[/math] 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).

Multiplicative group

Let [math]\displaystyle{ G = \mathbb{G}_m }[/math] 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).)

Correspondence

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

[math]\displaystyle{ H \subset K \Leftrightarrow \operatorname{Dist}(H) \subset \operatorname{Dist}(K). }[/math]

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 [math]\displaystyle{ \mathfrak{g} }[/math].
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 In1 is stable under G and thus G acts on (k[G]/In1)^* 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

Main page: Lie group–Lie algebra correspondence

Notes

↑ Jantzen 1987, Part I, § 7.10.

References

Jantzen, Jens Carsten (1987). Representations of Algebraic Groups. Pure and Applied Mathematics. 131. Boston: Academic Press. ISBN 978-0-12-380245-3.
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
Mukai, S. (2002). An introduction to invariants and moduli. Cambridge Studies in Advanced Mathematics. 81. ISBN 978-0-521-80906-1. http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521809061.
Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7