Twisted tensor products of $K^n$ with $K^m$

TL;DR

This paper classifies all twisting maps between finite-dimensional vector spaces over a field, specifically K^n and K^m, revealing three main families and their algebraic structures, including deformations and isomorphisms to matrix algebras.

Contribution

It introduces three new families of twisting maps between K^m and K^n and constructs all such maps for K^3 with K^3, expanding understanding of algebraic deformations.

Findings

Identified three families of twisting maps of K^m with K^n.

Constructed all twisting maps of K^3 with K^3.

Connected one family to truncated quiver algebras and others to matrix algebra deformations.

Abstract

We find three families of twisting maps of K^m with K^n. One of them is related to truncated quiver algebras, the second one consists of deformations of the first and the third one requires m=n and yields algebras isomorphic to M_n(K). Using these families and some exceptional cases we construct all twisting maps of K^3 with K^3.