On some Lie groups containing spin group in Clifford algebra

TL;DR

This paper explores Lie groups within complexified Clifford algebras, establishing isomorphisms with classical matrix groups and analyzing the inclusion of spin groups, especially for dimensions up to five.

Contribution

It provides new isomorphisms between Clifford algebra-based Lie groups and classical matrix groups, extending understanding across arbitrary dimensions and signatures.

Findings

Isomorphisms between Clifford-based groups and matrix groups established.

Spin group is a subgroup of these groups and coincides with one for n ≤ 5.

Classical matrix groups containing spin groups are identified for all dimensions.

Abstract

In this paper we consider some Lie groups in complexified Clifford algebras. Using relations between operations of conjugation in Clifford algebras and matrix operations we prove isomorphisms between these groups and classical matrix groups (symplectic, orthogonal, linear, unitary) in the cases of arbitrary dimension and arbitrary signature. Also we obtain isomorphisms of corresponding Lie algebras which are direct sums of subspaces of quaternion types. Spin group is a subgroup of all considered groups and it coincides with one of them in the cases $n\leq 5$. We present classical matrix Lie groups that contain spin group in the case of arbitrary dimension.