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).
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 kG] 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.
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.
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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]
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 kt] and In
0 = (tn).
Let be the multiplicative group; i.e., G(R) = R* for any k-algebra R. The coordinate ring of G is kt, t−1] (since G is really GL1(k).)
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 kG*, the dual of the coordinate ring of G. Note that Dist(G) is a (Hopf) subalgebra of kG*.
![]() | This section needs expansion. You can help by
adding to it. (January 2019) |
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).
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 kG] 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.
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.
![]() | This section needs expansion. You can help by
adding to it. (January 2019) |
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]
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 kt] and In
0 = (tn).
Let be the multiplicative group; i.e., G(R) = R* for any k-algebra R. The coordinate ring of G is kt, t−1] (since G is really GL1(k).)
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 kG*, the dual of the coordinate ring of G. Note that Dist(G) is a (Hopf) subalgebra of kG*.
![]() | This section needs expansion. You can help by
adding to it. (January 2019) |