Fundaments of Hermitean Clifford analysis, I: complex structure
- Author
- Fred Brackx (UGent) , Jarolim Bures, Hennie De Schepper (UGent) , David Eelbode (UGent) , Franciscus Sommen (UGent) and Vladimir Soucek
- Organization
- Abstract
- Hermitean Clifford analysis focusses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Monogenicity is expressed here by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a Clifford realisation of the unitary group. In this contribution we present a deeper insight in the transition from the orthogonal setting to the Hermitean one. Starting from the orthogonal Clifford setting, by simply introducing a so-called complex structure J is an element of SO(2n; R), the fundamental elements of the Hermitean setting arise in a quite natural way. Indeed, the corresponding projection operators 1/2 (1 +/- iJ) project the initial basis (e(alpha), alpha = 1, ... , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of Cl-2n(C) into two components, where the SO(2n; R)-elements leaving those two subspaces invariant, commute with the complex structure J. They generate a subgroup which is doubly covered by a subgroup of Spin(2n; R), denoted Spin(J)(2n; R), being isomorphic with the unitary group U(n; C). Finally the two Hermitean Dirac operators are shown to originate as generalized gradients when projecting the gradient on the invariant subspaces mentioned, which actually implies their invariance under the action of Spin(J)(2n; R). The eventual goal is to extend the complex structure J to the whole Clifford algebra Cl-2n(C), in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity.
- Keywords
- complex structure, Hermitean Clifford analysis
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU-425628
- MLA
- Brackx, Fred, et al. “Fundaments of Hermitean Clifford Analysis, I: Complex Structure.” COMPLEX ANALYSIS AND OPERATOR THEORY, vol. 1, no. 3, 2007, pp. 341–65, doi:10.1007/s11785-007-0010-5.
- APA
- Brackx, F., Bures, J., De Schepper, H., Eelbode, D., Sommen, F., & Soucek, V. (2007). Fundaments of Hermitean Clifford analysis, I: complex structure. COMPLEX ANALYSIS AND OPERATOR THEORY, 1(3), 341–365. https://doi.org/10.1007/s11785-007-0010-5
- Chicago author-date
- Brackx, Fred, Jarolim Bures, Hennie De Schepper, David Eelbode, Franciscus Sommen, and Vladimir Soucek. 2007. “Fundaments of Hermitean Clifford Analysis, I: Complex Structure.” COMPLEX ANALYSIS AND OPERATOR THEORY 1 (3): 341–65. https://doi.org/10.1007/s11785-007-0010-5.
- Chicago author-date (all authors)
- Brackx, Fred, Jarolim Bures, Hennie De Schepper, David Eelbode, Franciscus Sommen, and Vladimir Soucek. 2007. “Fundaments of Hermitean Clifford Analysis, I: Complex Structure.” COMPLEX ANALYSIS AND OPERATOR THEORY 1 (3): 341–365. doi:10.1007/s11785-007-0010-5.
- Vancouver
- 1.Brackx F, Bures J, De Schepper H, Eelbode D, Sommen F, Soucek V. Fundaments of Hermitean Clifford analysis, I: complex structure. COMPLEX ANALYSIS AND OPERATOR THEORY. 2007;1(3):341–65.
- IEEE
- [1]F. Brackx, J. Bures, H. De Schepper, D. Eelbode, F. Sommen, and V. Soucek, “Fundaments of Hermitean Clifford analysis, I: complex structure,” COMPLEX ANALYSIS AND OPERATOR THEORY, vol. 1, no. 3, pp. 341–365, 2007.
@article{425628,
abstract = {{Hermitean Clifford analysis focusses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Monogenicity is expressed here by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a Clifford realisation of the unitary group. In this contribution we present a deeper insight in the transition from the orthogonal setting to the Hermitean one. Starting from the orthogonal Clifford setting, by simply introducing a so-called complex structure J is an element of SO(2n; R), the fundamental elements of the Hermitean setting arise in a quite natural way. Indeed, the corresponding projection operators 1/2 (1 +/- iJ) project the initial basis (e(alpha), alpha = 1, ... , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of Cl-2n(C) into two components, where the SO(2n; R)-elements leaving those two subspaces invariant, commute with the complex structure J. They generate a subgroup which is doubly covered by a subgroup of Spin(2n; R), denoted Spin(J)(2n; R), being isomorphic with the unitary group U(n; C). Finally the two Hermitean Dirac operators are shown to originate as generalized gradients when projecting the gradient on the invariant subspaces mentioned, which actually implies their invariance under the action of Spin(J)(2n; R). The eventual goal is to extend the complex structure J to the whole Clifford algebra Cl-2n(C), in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity.}},
author = {{Brackx, Fred and Bures, Jarolim and De Schepper, Hennie and Eelbode, David and Sommen, Franciscus and Soucek, Vladimir}},
issn = {{1661-8254}},
journal = {{COMPLEX ANALYSIS AND OPERATOR THEORY}},
keywords = {{complex structure,Hermitean Clifford analysis}},
language = {{eng}},
number = {{3}},
pages = {{341--365}},
title = {{Fundaments of Hermitean Clifford analysis, I: complex structure}},
url = {{http://doi.org/10.1007/s11785-007-0010-5}},
volume = {{1}},
year = {{2007}},
}
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