On some Lie groups containing spin group in Clifford algebra

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 $n\le 5$. We present classical matrix Lie groups that contain spin group in the case of arbitrary dimension.

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Cited by in corpus (10)

• Clifford algebras and their applications to Lie groups and spinors
• Classification of Lie algebras of specific type in complexified Clifford algebras
• Subgroups of Clifford algebras
• Covariantly constant solutions of the Yang-Mills equations
• On inner automorphisms preserving subspaces of Clifford algebras
• On generalization of Lipschitz groups and spin groups
• On Some Lie Groups in Degenerate Clifford Geometric Algebras
• On SVD and Polar Decomposition in Real and Complexified Clifford Algebras
• Quasi-Clifford algebras, Quadratic forms over ${\mathbb{F}}_2$, and Lie Algebras
• On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras