Hopf algebras of smooth functions on compact Lie groups. (English) Zbl 1051.16021

Farkas, Eva C.

Commentat. Math. Univ. Carol. 41, No. 4, 651-661 (2000).

Let \(G\) be a Lie group and let \(C^\infty(G)\) be the algebra of smooth functions on \(G\). A \(C^\infty\)-algebra, which, at the same time, satisfies the axioms of a convenient Hopf algebra, is called a \(C^\infty\)-Hopf algebra. The author characterizes those \(C^\infty\)-Hopf algebras which are given by \(C^\infty(G)\) for some Lie group \(G\), obtaining thereby an anti-isomorphism between the category of compact Lie groups and the category of convenient Hopf algebras.

Reviewer: Luboš Pick (Praha)

MSC:

16W30	Hopf algebras (associative rings and algebras) (MSC2000)
22E20	General properties and structure of other Lie groups
46E25	Rings and algebras of continuous, differentiable or analytic functions

Keywords:

\(C^\infty\)-Hopf algebras; compact Lie groups; dualities; categories of Hopf algebras