Krein duality, positive 2-algebras, and dilation of comultiplications (To the centenary of Mark G.Krein)

The Krein--Tannaka duality for compact groups was a generalization the Pontryagin--Van Kampen duality for locally compact abelian groups and a remote predecessor of the theory of tensor categories. It is less known that it found applications in algebraic combinatorics (``Krein algebras''). Later, this duality was substantially extended: in \cite{V}, the notion of {\it involutive algebras in positive vector duality} was introduced. In this paper, we reformulate the notions of this theory using the language of bialgebras (and Hopf algebras) and introduce the class of involutive bialgebras and positive 2-algebras. The main goal of the paper is to give a precise statement of a new problem, which we consider as one of the main problems in this field, concerning the existence of dilations (embeddings) of positive 2-algebras into involutive bialgebras, or, in other words, the problem of describing subobjects in involutive bialgebras. We define two types of subobjects in the category of bialgebras, strict and nonstrict ones, and consider the corresponding embeddings (dilations) of positive 2-algebras into bialgebras. The difference between the two types of dilations is illustrated by the example of bicommutative positive 2-algebras (commutative hypergroups). The most interesting instance of our problem concerns dilations of the Hecke algebra $H_n(q)$. It seems that in this case strict dilations may exist only for $q=p^k$ (with $p$ a prime); it is not known whether nonstrict dilations exist for other $q$. We also show that the class of finite-dimensional involutive semisimple bialgebras coincides with the class of semigroup algebras of finite inverse semigroups.