compare it with the twist studied in Chapter \[chap: thcrtwist\]. In §\[subsec: gradedskewclifford\] we twist a graded skew Clifford algebra, while the main portion of the thesis ends by briefly addressing the ungraded situation – we study twists of a universal enveloping algebra and its homogenisation in §\[subsec: homenvelopalg\].

Appendix \[app: calc\] contains any calculations that have been omitted from the main thesis, while Appendix \[app: comp\] contains any computer code used to prove earlier results.

Basic conventions and notation {#sec: notation}
------------------------------

In this short section we state our basic conventions with regard to cocycle twists and also some more general notation.

### Basic conventions {#subsec: conventions}

The conventions that follow will hold throughout this thesis apart from certain occurrences where deviation from them will be stated explicitly.

We assume that $0 \in {\ensuremath{\mathbb{N}}}$ for grading purposes and work over a base field $k$ which is assumed to be algebraically closed. Any algebra $A$ will be an associative $k$-algebra with identity that is finitely generated (f.g.) as an algebra. By $G$ we will denote a finite *abelian* group with identity element $e$. Furthermore, we will assume that the characteristic of the base field does not divide the order of 