A Primer on Twists in the Noncommutative Realm Focusing on Algebra, Representation Theory, and Geometry

TL;DR

This paper reviews various twisting techniques in noncommutative algebra, focusing on automorphism twists, 2-cocycle twists, and twisted tensor products, highlighting their properties and interconnections.

Contribution

It provides a clear, example-rich overview of twisting methods in noncommutative algebra, connecting automorphism twists, cocycle twists, and tensor product twists.

Findings

Zhang twists relate to 2-cocycle twists of bialgebras

Classification of twisted tensor products is outlined

Twisted Segre products are examined in detail

Abstract

We review several techniques that twist an algebra's multiplicative structure. We first consider twists by an automorphism, also known as Zhang twists, and we relate them to 2-cocycle twists of certain bialgebras. We then outline the classification and properties of twisted tensor products, and we examine twisted Segre products. Our exposition emphasizes clarity over generality, providing a wealth of interconnecting examples.