Clifford algebras and their applications to Lie groups and spinors

TL;DR

This paper provides an overview of Clifford algebras, their matrix representations, periodicity, and applications to Lie groups and spinors, including new perspectives on classification and generalizations.

Contribution

It offers new insights into the classification of Lie subalgebras, generalizations of the Pauli theorem, and methods like averaging within Clifford algebras.

Findings

Matrix representations and periodicity of Clifford algebras

Classification of Lie subalgebras in Clifford algebras

Generalizations of the Pauli theorem

Abstract

In these lectures, we discuss some well-known facts about Clifford algebras: matrix representations, Cartan's periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in $n$ dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.