The Spin Group in Superspace
Hennie De Schepper, Al\'i Guzm\'an Ad\'an, Frank Sommen

TL;DR
This paper extends classical descriptions of rotation groups to superspace, introducing a supergroup Spin, its algebraic structure, and applications to fractional Fourier transforms in a novel supergeometric setting.
Contribution
It develops a new Iwasawa-type decomposition for the superspace rotation group and constructs a super Spin group covering it, linking to fractional Fourier transforms.
Findings
Established an Iwasawa-type decomposition for SO₀ in superspace.
Constructed a super Spin group as a double cover of SO₀.
Connected fractional Fourier transforms to elements of the super Spin group.
Abstract
There are two well-known ways of describing elements of the rotation group SO. First, according to the Cartan-Dieudonn\'e 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 SO in the superspace setting. This group can be seen as the action of the functor of points of the orthosymplectic supergroup OSp on a Grassmann algebra. While still being connected, the group SO 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.…
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