nLab hyperalgebra

Overview

Hyperalgebra of an affine algebraic group $G$ is the finite dual of the Hopf algebra of representative functions of (irreducible component containing the unit element in) $G$. It can be interpreted as (and is sometimes called) the algebra of distributions supported at unit. This algebra comes with a natural filtration. In characteristic $0$ it coincides (by L. Schwarz’s theorem) with the universal enveloping algebra of the Lie algebra of $G$, but it is much bigger in positive characteristic. It can also be obtained by base change from the Kostant’s integral form of the universal enveloping algebra of the complex Lie algebra associated to $G$.

Literature

See also closely related entry distribution on an affine algebraic group.

Some books on algebraic groups and on Hopf algebras have chapters dedicated to this topic e.g.

• M. Sweedler, Hopf algebras, Benjamin, NY 1969
• C. Jantzen, Representations of algebraic groups, chapter 7
• Η. Yanagihara, Theory of Hopf algebras attached to group schemes, Springer Lecture Notes in Mathematics 614 (1977)

Articles:

• J. Sullivan, Simply connected groups, the hyperalgebra, and Verma’s conjecture, Amer. J. Math. 100 (1978) 1015-1019.
• Mitsuhiro Takeuchi, Tangent coalgebras and hyperalgebras I, Japan. J. Math. 42 (1974) 1-143 pdf; On coverings and hyperalgebras of affine algebraic groups, Trans. AMS

  211 (1975), 249-275; A hyperalgebraic proof of the isomorphism and isogeny theorems for reductive groups, J. Algebra 85 (1983), 179-196; Generators and relations for the hyperalgebras of reductive groups, doi

• W. J. Haboush, Central differential operators of split semisimple groups over fields of positive characteristic, In: Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, Springer Lecture Notes in Mathematics 795, pp 35-85 doi
• Edward Cline, Brian Parshall, Leonard Scott, Cohomology, hyperalgebras, and representations, Journal of Algebra 63:1, 98-123 (1980) doi

MathOverflow: which-is-the-correct-universal-enveloping-algebra-in-positive-characteristic

A quantum version at root of unity is proposed in

• Iván Angiono, A quantum version of the algebra of distributions of $SL(2)$, arxiv/1607.04869

and another approach is in

• W. Chin, L. Krop, Quantized hyperalgebra of rank 1, pdf; Spectra of quantized hyperalgebras, pdf