## 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.

### MSC:

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