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# Spin actions in Euclidean and Hermitian Clifford analysis in superspace

(2018) 457(1). p.23-50
Author
Organization
Abstract
In [4] we studied the group invariance of the inner product of supervectors as introduced in the framework of Clifford analysis in superspace. The fundamental group SO_0 leaving invariant such an inner product turns out to be an extension of SO(m) x Sp(2n) and gives rise to the definition of the spin group in superspace through the exponential of the so-called extended superbivectors, where the spin group can be seen as a double covering of SO_0 by means of the representation h(s) [x] = sx\bar{s}. In the present paper, we study the invariance of the Dirac operator in superspace under the classical H and L actions of the spin group on superfunctions. In addition, we consider the Hermitian Clifford setting in superspace, where we study the group invariance of the Hermitian inner product of supervectors introduced in [3]. The group of complex supermatrices leaving this inner product invariant constitutes an extension of U(m) X U(n) and is isomorphic to the subset SO_0^J of SO_0 of elements that commute with the complex structure J. The realization of SO_0^J within the spin group is studied together with the invariance under its actions of the super Hermitian Dirac system. It is interesting to note that the spin element leading to the complex structure can be expressed in terms of the n-dimensional Fourier transform.
Keywords
Group invariance, Hermitian Clifford analysis, superspace, Dirac operator

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## Citation

Chicago
De Schepper, Hennie, Ali Guzmán Adán, and Franciscus Sommen. 2018. “Spin Actions in Euclidean and Hermitian Clifford Analysis in Superspace.” Journal of Mathematical Analysis and Applications 457 (1): 23–50.
APA
De Schepper, Hennie, Guzmán Adán, A., & Sommen, F. (2018). Spin actions in Euclidean and Hermitian Clifford analysis in superspace. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 457(1), 23–50.
Vancouver
1.
De Schepper H, Guzmán Adán A, Sommen F. Spin actions in Euclidean and Hermitian Clifford analysis in superspace. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA: Elsevier BV; 2018;457(1):23–50.
MLA
De Schepper, Hennie, Ali Guzmán Adán, and Franciscus Sommen. “Spin Actions in Euclidean and Hermitian Clifford Analysis in Superspace.” JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 457.1 (2018): 23–50. Print.
@article{8551504,
abstract     = {In [4] we studied the group invariance of the inner product of supervectors as introduced in the framework of Clifford analysis in superspace. The fundamental group SO\_0 leaving invariant such an inner product turns out to be an extension of SO(m) x Sp(2n) and gives rise to the definition of the spin group in superspace through the exponential of the so-called extended superbivectors, where the spin group can be seen as a double covering of SO\_0 by means of the representation h(s) [x] = sx{\textbackslash}bar\{s\}. In the present paper, we study the invariance of the Dirac operator in superspace under the classical H and L actions of the spin group on superfunctions. In addition, we consider the Hermitian Clifford setting in superspace, where we study the group invariance of the Hermitian inner product of supervectors introduced in [3]. The group of complex supermatrices leaving this inner product invariant constitutes an extension of U(m) X U(n) and is isomorphic to the subset SO\_0\^{ }J of SO\_0 of elements that commute with the complex structure J. The realization of SO\_0\^{ }J within the spin group is studied together with the invariance under its actions of the super Hermitian Dirac system. It is interesting to note that the spin element leading to the complex structure can be expressed in terms of the n-dimensional Fourier transform.},
author       = {De Schepper, Hennie and Guzm{\'a}n Ad{\'a}n, Ali and Sommen, Franciscus},
issn         = {0022-247X},
journal      = {JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS},
language     = {eng},
number       = {1},
pages        = {23--50},
publisher    = {Elsevier BV},
title        = {Spin actions in Euclidean and Hermitian Clifford analysis in superspace},
url          = {http://dx.doi.org/10.1016/j.jmaa.2017.08.009},
volume       = {457},
year         = {2018},
}


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