Deformation of tensor product (co)algebras via non-(co)normal twists

TL;DR

This paper explores new ways to deform tensor product coalgebras and algebras using non-(co)normal twists, analyzing their structures, equivalences, and providing concrete examples.

Contribution

It introduces a general framework for twisting tensor product (co)algebras, including deformations of the counit, and discusses equivalence classes of such twists.

Findings

Identified classes of deformations of tensor product coalgebras

Defined a notion of equivalence for twists

Provided examples illustrating the deformations

Abstract

We study new coalgebra structures on the tensor product of two coalgebras $C$ and $D$ by twisting the tensor product coalgebra via a twist map $\Psi: C \otimes D \rightarrow D \otimes C$. We deal with the general case in which the counit of the tensor product coalgebra is deformed as well. Some classes of such deformations are analyzed and a notion of equivalence of twists is discussed. We also present the dual deformation of tensor product algebras and provide examples.