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

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.


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
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