nLab distribution on an affine group scheme

Idea

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 $k$-affine scheme $X$ over a field, supported at a point $p$. 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.

Definition

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

If $X = G$ is a $k$-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.

Literature

• 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)

category: algebraic geometry