265 名前：mathbb {C} _{G}}{\displaystyle \mathbb {C} _{G}} (over {\displaystyle \mathbb {C} }\mathbb{C} ) into the category of finite dimensional Hopf algebras, so that the Pontryagin duality functor {\displaystyle G\mapsto {\widehat {G}}}{\displaystyle G\mapsto {\widehat {G}}} turns into the operation {\displaystyle H\mapsto H^{*}}{\displaystyle H\mapsto H^{*}} of taking the dual vector space (which is a duality functor in the category of finite dimensional Hopf algebras).[20]
In 1973 Leonid I. Vainerman, George I. Kac, Michel Enock, and Jean-Marie Schwartz built a general theory of this type for all locally compact groups.[21] From the 1980s the research in this area was resumed after the discovery of quantum groups, to which the constructed theories began to be actively transferred.[22] These theories are formulated in the language of C*-algebras, or Von Neumann algebras, and one of its variants is the recent theory of locally compact quantum groups.[23][22]
One of the drawbacks of these general theories, however, is that in them the objects generalizing the concept of group are not Hopf algebras in the usual algebraic sense.[20] This deficiency can be corrected (for some classes of groups) within the framework of duality theories constructed on the basis of the notion of envelope of topological algebra.[24] (引用終り) 以上 []
