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 k \to k 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.

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 $A \to k(x)$, the residue field of ''x''. By definition, a ''distribution'' ''f'' supported at ''x'''' is a ''k''-linear functional on ''A'' such that $f(I_x^n) = 0$ 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 \overset\to k \otimes k \overset\to k \otimes k = k
where Δ is the comultiplication that is the homomorphism induced by the multiplication $G \times G \to G$. The multiplication turns out to be associative (use $1 \otimes \Delta \circ \Delta = \Delta \otimes 1 \circ \Delta$) and thus Dist(''G'') is an associative algebra, as the set is closed under the muplication by the formula:
:(*) $\Delta(I_1^n) \subset \sum_^n I_1^r \otimes I^_1.$
It is also unital with the unity that is the linear functional k \to k, \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 $I_1/I_1^2$. 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 , g= f * g - g * f is still a tangent vector.

Let $\mathfrak = \operatorname(G)$ be the Lie algebra of ''G''. Then, by the universal property, the inclusion $\mathfrak \hookrightarrow \operatorname(G)$ induces the algebra homomorphism:
:$U(\mathfrak) \to \operatorname(G).$
When the base field ''k'' has characteristic zero, this homomorphism is an isomorphism.

Examples

Additive group

Let $G = \mathbb_a$ 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'' = (''t''^''n'').

Multiplicative group

Let $G = \mathbb_m$ 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'').)

Correspondence

* For any closed subgroups ''H'', K'' of ''G'', if ''k'' is perfect and ''H'' is irreducible, then
::$H \subset K \Leftrightarrow \operatorname(H) \subset \operatorname(K).$
* 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 $\mathfrak$.
* 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''sup>*, the dual of the coordinate ring of ''G''. Note that Dist(''G'') is a (Hopf) subalgebra of ''k'' 'G''sup>*.

Relation to Lie group–Lie algebra correspondence

Notes

References

*
* 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
*
* {{Citation , last1=Springer , first1=Tonny A. , title=Linear algebraic groups , publisher=Birkhäuser Boston , location=Boston, MA , edition=2nd , series=Progress in Mathematics , isbn=978-0-8176-4021-7 , mr=1642713 , year=1998 , volume=9

Further reading

Linear algebraic groups and their Lie algebras
by Daniel Miller Fall 2014

Algebraic geometry