On the double crossed product of weak Hopf algebras

TL;DR

This paper establishes conditions under which the weak wreath product of two weak bialgebras or weak Hopf algebras inherits a weak bialgebra or weak Hopf algebra structure, unifying known examples like the Drinfel'd double.

Contribution

It provides sufficient conditions for the weak wreath product to form a weak bialgebra or weak Hopf algebra, extending the theory to known constructions like the Drinfel'd double.

Findings

Weak wreath product becomes a weak bialgebra under certain conditions.

Conditions ensure the structure extends to weak Hopf algebras.

Includes known examples such as the Drinfel'd double.

Abstract

Given a weak distributive law between algebras underlying two weak bialgebras, we present sufficient conditions under which the corresponding weak wreath product algebra becomes a weak bialgebra with respect to the tensor product coalgebra structure. When the weak bialgebras are weak Hopf algebras, then the same conditions are shown to imply that the weak wreath product becomes a weak Hopf algebra, too. Our sufficient conditions are capable to describe most known examples, (in particular the Drinfel'd double of a weak Hopf algebra).