nLab distribution on an affine group scheme



The notion of distributions – in the sense of linear functionals on some algebra of functions – can to some extent be adapted from smooth manifolds to schemes.

The problem is however to control the singular behaviour. A useful idea is to restrict singularities to a closed subscheme given by some ideal sheaf. The simplest case are distributions on a kk-affine scheme XX over a field, supported at a point pp. For a group scheme, one assumes that the point is the neutral element of the group and the algebra of distributions becomes a Hopf algebra.


Let XX be a kk-affine scheme XX (where kk is a field). For nn a nonnegative integer distributions on XX supported at pp of order n\leq n are linear functionals k[X]kk[X]\to k on the ring of regular functions k[X]k[X] which vanish on (n+1)(n+1)-st power of the ideal I p={fk[X]|f(p)=0}I_p = \{ f\in k[X] | f(p) = 0\} of regular functions vanishing at pp.

If X=GX = G is a kk-affine group scheme by the Hopf algebra of distributions we mean the algebra of distributions supported at the unit of the group with its natural Hopf algebra structure.

Largely equivalent notion, though usually differently defined is of a hyperalgebra of an affine algebraic group.


  • Pierre Cartier, A primer of Hopf algebras, Frontiers in number theory, physics, and geometry II, 537–615, Springer 2007.; preprint IH'ES 2006/40 pdf

  • J. C. Jantzen, Representations of algebraic groups, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)

Last revised on August 22, 2023 at 14:40:54. See the history of this page for a list of all contributions to it.