Categories of bistochastic measures and representations of infinite- dimensional groups. (Russian) Zbl 0755.58010

The following groups are considered: the group of automorphisms of a Lebesgue space with measure (finite or \(\sigma\)-finite), the group of measurable functions with values in a Lie group, and the group of diffeomorphisms of a manifold. The author shows that the theory of representations of all enumerated groups is closely connected with the theory of representations of a certain category, which is called in this paper “the category of \(G\)-polymorphisms”. The objects of this category are the spaces with measure and the morphisms from \(M\) to \(N\) are the probability measures on \(M\times N\times G\) where \(G\) is a fixed Lie group. For some of the mentioned infinite dimensional groups \(\mathcal G\) the author shows that every representation of the group \(\mathcal G\) is canonical extended to a representation of a certain category of \(G\)-polymorphisms. For the group of automorphisms of a space with measure this permits to obtain the classification of all unitary representations.


58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
20C99 Representation theory of groups