Preantipodes for dual-quasi bialgebras

TL;DR

This paper investigates the structure of dual quasi-bialgebras, establishing that the existence of a specific map called a preantipode characterizes when a dual quasi-bialgebra satisfies a fundamental theorem similar to dual quasi-Hopf algebras.

Contribution

It introduces the concept of a preantipode and proves its role in characterizing dual quasi-bialgebras that fulfill a fundamental theorem.

Findings

Existence of a preantipode is equivalent to the structure theorem for dual quasi-bialgebras.

Not all dual quasi-bialgebras with a preantipode are dual quasi-Hopf algebras.

Provides a new criterion for the structure theorem in the theory of dual quasi-bialgebras.

Abstract

It is known that a dual quasi-bialgebra with antipode $H$, i.e. a dual quasi-Hopf algebra, fulfils a fundamental theorem for right dual quasi-Hopf $H$-bicomodules. The converse in general is not true. We prove that, for a dual quasi-bialgebra $H$, the structure theorem amounts to the existence of a suitable map $S:H\rightarrow H$ that we call a preantipode of $H$.