Classification of Lie algebras of specific type in complexified Clifford algebras

Abstract

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These 16 Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical matrix Lie algebras in the cases of arbitrary dimension and signature. We present 16 Lie groups: one Lie group for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups.

Keywords:

• Clifford algebra
• Lie algebra
• quaternion type
• Lie group
• spin group

AMS Subject Classifications:

• 15A66
• 22E60

Acknowledgements

The author thanks the Referees for their constructive remarks and comments.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 One can easily obtain these expressions using twice Binomial Theorem: and .

2 Here and below we omit the sign of the direct sum to simplify notation: , , , etc.

3 The pseudo-Hermitian conjugation of Clifford algebra elements is related to the pseudo-unitary matrix groups as Hermitian conjugation is related to the unitary groups, see [10,12].

4 Because, trace of this matrix equals (up to multiplication by a constant) the projection of element onto the subspace (see [15]) that is zero.

Additional information

Funding

The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2017–2018 [grant number 17-01-0009]; by the Russian Academic Excellence Project ‘5-100’.