Journal of Geometry and Symmetry in Physics

On Some Lie Groups Containing Spin Group in Clifford Algebra

Dmitry Shirokov

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.

Article information

Source
J. Geom. Symmetry Phys., Volume 42 (2016), 73-94.

Dates
First available in Project Euclid: 31 May 2017

https://projecteuclid.org/euclid.jgsp/1496196024

Digital Object Identifier
doi:10.7546/jgsp-42-2016-73-94

Mathematical Reviews number (MathSciNet)
MR3586444

Zentralblatt MATH identifier
1379.15016

Citation

Shirokov, Dmitry. On Some Lie Groups Containing Spin Group in Clifford Algebra. J. Geom. Symmetry Phys. 42 (2016), 73--94. doi:10.7546/jgsp-42-2016-73-94. https://projecteuclid.org/euclid.jgsp/1496196024