The Spin group in superspace

JOURNAL OF GEOMETRY AND PHYSICS

卷:163

Article

De Schepper, Hennie¹  Adan, Ali Guzman^1,2  Sommen, Frank¹ 

[1] Univ Ghent, Fac Engn & Architecture, Dept Elect & Informat Syst, Clifford Res Grp, Krijgslaan 281, B-9000 Ghent, Belgium

[2] Res Fdn Flanders FWO, Flanders, Belgium

关键词: Spin groups;    Symplectic groups;    Clifford analysis;    Bivectors;    Superspace;   

DOI  :  10.1016/j.geomphys.2020.104094

来源: Elsevier

【 摘 要 】

There are two well-known ways of describing elements of the rotation group SO(m). First, according to the Cartan-Dieudonne theorem, every rotation matrix can be written as an even number of reflections. And second, they can also be expressed as the exponential of some anti-symmetric matrix. In this paper, we study similar descriptions of a group of rotations SO0 in the superspace setting. This group can be seen as the action of the functor of points of the orthosymplectic supergroup OSp(m vertical bar 2n) on a Grassmann algebra. While still being connected, the group SO0 is thus no longer compact. As a consequence, it cannot be fully described by just one action of the exponential map on its Lie algebra. Instead, we obtain an Iwasawa-type decomposition for this group in terms of three exponentials acting on three direct summands of the corresponding Lie algebra of supermatrices. At the same time, SO0 strictly contains the group generated by super-vector reflections. Therefore, its Lie algebra is isomorphic to a certain extension of the algebra of superbivectors. This means that the Spin group in this setting has to be seen as the group generated by the exponentials of the so-called extended superbivectors in order to cover SO0. We also study the actions of this Spin group on supervectors and provide a proper subset of it that is a double cover of SO0. Finally, we show that every fractional Fourier transform in n bosonic dimensions can be seen as an element of this spin group. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free