Krein Duality, Positive 2-Algebras, and Dilation of Comultiplications

The Krein–Tannaka duality for compact groups was a generalization of 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 [A. M. Vershik, Zap. Nauchn. Semin. LOMI, 29, 1972, 147–154], the notion of 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 in involutive bialgebras, or, in other words, the problem of describing subobjects of involutive bialgebras; we define two types of subobjects of bialgebras, strict and nonstrict ones. The dilation problem is illustrated by the example of the Hecke algebra, which is viewed as a positive involutive $2$-algebra. We consider in detail only the simplest situation and classify two-dimensional Hecke algebras for various values of the parameter $q$, demonstrating the difference between the two types of dilations. We also prove that the class of finite-dimensional involutive semisimple bialgebras coincides with the class of semigroup algebras of finite inverse semigroups.