Abstract
In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras—subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras—graded central simple algebras or graded central simple algebras with involution.
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Notes
The area of interest of the author of this paper is mostly the applications of some specific algebras (as Clifford algebras) in physics, in particular, in the field theory. Therefore, let us present results only for the particular case of Clifford algebras in this paper. One can reformulate the statements for the more general cases of GCSAs or GCSAsWI, if the need arises for some purpose. We accompanied the statements of this paper with footnotes, including those received from the reviewer, about their possible reformulation for the more general case.
According to the reviewer, this fact is well-known. We present this statement for the sake of completeness. The new results for the groups \(\Gamma ^k\) are presented in Sect. 7.
As one of the anonymous reviewers of this paper noted, this statement can be reformulated for the more general case of the graded central simple algebras (GCSAs).
As one of the anonymous reviewers of this paper noted, the groups \(\mathrm{A}, \mathrm{B}, \mathrm{Q}\) can be defined in the more general case of the graded central simple algebras with involution (GCSAsWI) and the corresponding statements can be reformulated for this more general case.
This operation coincides with the trace of the corresponding matrix representation up to the scalar, which is the dimension of this representation (see [13]).
We present Lie algebras for all Lie groups considered in this paper for the sake of completeness. Some of them are known.
As one of the anonymous reviewers noted, the dimensions of the four subspaces \(\mathrm{C}^{\overline{k}}\), \(k=0, 1, 2, 3\) in the case of more general algebras (GCSAsWI) are also known from other works.
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Acknowledgements
The study presented in this paper was stimulated by discussions with Prof. A. Odzijewicz and other participants during the International Conference on Mathematical Methods in Physics (Morocco, Marrakesh, 2019). The author is grateful to the organizers and the participants of this conference. The author is grateful to the anonymous reviewers (especially to the second reviewer, who pointed out the feasibility of some of the results of this paper not only for the Clifford algebras but for more general algebras, as well as for other important comments) for their careful reading of the paper and helpful comments on how to improve the presentation. This work is supported by the grant of the President of the Russian Federation (Project MK-404.2020.1).
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This article is part of the Topical Collection on ICMMP, April 1–6, 2019, Marrakech, Morocco, edited by Zouhair Mouayn.
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Shirokov, D. On Inner Automorphisms Preserving Fixed Subspaces of Clifford Algebras. Adv. Appl. Clifford Algebras 31, 30 (2021). https://doi.org/10.1007/s00006-021-01135-6
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DOI: https://doi.org/10.1007/s00006-021-01135-6