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.
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 [Citation10,Citation12].
4 Because, trace of this matrix equals (up to multiplication by a constant) the projection of element onto the subspace (see [Citation15]) that is zero.