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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.

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
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