A resolution (minimal model) of the PROP for bialgebras

TL;DR

This paper constructs a minimal resolution of the PROP for bialgebras, providing explicit formulas and a framework that captures the deformation theory and homotopy invariants of bialgebras.

Contribution

It introduces a minimal model of the PROP for bialgebras, including explicit differential formulas, and links it to homotopy bialgebras and deformation theory.

Findings

Proved a theorem describing the form of the minimal resolution.

Provided explicit formulas for the differential in the minimal model.

Showed that algebras over the minimal model are strongly homotopy bialgebras.

Abstract

This paper is concerned with a minimal resolution of the PROP for bialgebras. We prove a theorem about the form of this resolution (Theorem 15) and give, in Section 5, a lot of explicit formulas for the differential. Our minimal model contains all information about the deformation theory of bialgebras and related cohomology. Algebras over this minimal model are strongly homotopy bialgebras, that is, homotopy invariant versions of bialgebras.