Properly twisted groups and their algebras

TL;DR

This paper introduces a twist property for group algebras that ensures specific involution and inner product properties, with examples including Cayley-Dickson and Clifford algebras.

Contribution

It develops a new twist property framework that imparts algebraic and inner product structures to twisted group algebras, exemplified by Cayley-Dickson and Clifford algebras.

Findings

Established a twist property that guarantees involution and inner product relations.

Provided examples of algebras satisfying the twist property, such as Cayley-Dickson and Clifford algebras.

Enhanced understanding of algebraic structures through the twist property.

Abstract

A twist property is developed which imparts certain properties on the twisted group algebra. These include an involution * satisfying (xy)*=y*x* and an inner product satisfying <xy,z> = <x,zy*> and <xy,z>=<y,x*z>. Examples of twisted group algebras having this property are the Cayley-Dickson algebras and Clifford algebras.