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 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 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[]
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Distributions on an algebraic group[]
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]
Examples[]
Additive group[]
Let 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 In
0 = (tn).
Multiplicative group[]
Let 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
- 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 .
- 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 In
1 is stable under G and thus G acts on (k[G]/In
1)* 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[]
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Notes[]
References[]
- J. C. Jantzen, Representations of Algebraic groups, Pure and Applied Mathematics, vol. 131, Boston, etc., 1987 (Academic).
- 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.
- Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713
Further reading[]
- Linear algebraic groups and their Lie algebras by Daniel Miller Fall 2014
- Algebraic geometry