Crossed Products by a Coalgebra

TL;DR

This paper introduces a generalized notion of crossed products of algebras by coalgebras, extending existing concepts in Hopf algebra theory, and explores their properties, equivalences, and specific examples like the quantum Euclidean group.

Contribution

It defines coalgebra crossed products, establishes conditions for their equivalence, and applies the concept to the quantum Euclidean group, expanding the framework of algebra-coalgebra interactions.

Findings

Coalgebra crossed products generalize bialgebra crossed products.

Necessary and sufficient conditions for equivalence of coalgebra crossed products.

The two-dimensional quantum Euclidean group is a coalgebra crossed product.

Abstract

We introduce the notion of a crossed product of an algebra by a coalgebra $C$, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right $C$-comodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimensional quantum Euclidean group is a coalgebra crossed product. The paper is completed with an appendix describing the dualisation of construction of coalgebra crossed products.