A theory of locally convex Hopf algebras

Abstract

Using the completed inductive, projective and injective tensor products of Grothendieck for locally convex topological vector spaces, we develop a systematic theory of locally convex Hopf algebras with an emphasis on Pontryagin-type dualities. We describe how classical Hopf algebras, real and complex Lie groups, as well as compact and discrete quantum groups, can all give rise to natural examples of this theory in a variety of different ways. We also show that the space of all continuous functions on a topological group $ G $ whose topological structures are compactly generated has an $ \varepsilon $-Hopf algebra structure, and we can recover $ G $ fully as a topological group from this locally convex Hopf algebra. The latter is done via a generalization of Gelfand duality, which is of its own interest. Certain projective and inductive limits are also considered in this framework, and it is shown that how this can lead to examples seemingly outside of the framework of locally compact quantum groups in the sense of Kustermans-Vaes. As an illustration, we propose a version of the infinite quantum permutation group $ S^{+}_{\infty} $, the free orthogonal group $ O^{+}_{\infty} $, and the free unitary group $ U^{+}_{\infty} $ as certain strict inductive limits, all of which still retain a nice duality. Combined with our duality theory, this may be seen as an alternative tentative approach to the Kac program of developing a Pontryagin-type duality to a wider class, while at the same time, we include many more interesting examples of classical and quantum groups.