Abstract
We show that the two Dirac operators arising in Hermitian Clifford analysis are identical to standard differential operators arising in several complex variables. We also show that the maximal subgroup that preserves these operators are generated by translations, dilations and actions of the unitary n-group. So the operators are not invariant under Kelvin inversion. We also show that the Dirac operators constructed via two by two matrices in Hermitian Clifford analysis correspond to standard Dirac operators in euclidean space. In order to develop Hermitian Clifford analysis in a different direction we introduce a sub elliptic Dirac operator acting on sections of a bundle over odd dimensional spheres. The particular case of the three sphere is examined in detail. We conclude by indicating how this construction could extend to other CR manifolds.
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References
Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3(Suppl. 1), 3–38 (1964)
Brackx, F., De Schepper, H., Sommen, F.: The hermitian clifford analysis toolbox. Adv. Appl. Clifford Alg. 18, 451–487 (2008)
Brackx, F., De Knock, B., De Schepper, H., Sommen, F.: On Cauchy and Martinelli-Bochner integral formulae in Hermitian Clifford analysis. Bull. Braz. Math. Soc. (N.S.) 40(3), 395416 (2009)
Brackx, F., De Schepper, H., Eelbode, D., Soucek, V.: The Howe dual pair in Hermitean Clifford analysis. Rev. Mat. Iberoamericana 26(2), 449–479 (2010)
Boggess, A.: CR Manifolds and the Tangential Cauchy Riemann Complex. CRC Press, Studies in Advanced Mathematics (1991)
D’Angelo, J.P.: Several Complex Variables and the Geometry of Real Hypersurfaces. CRC Press (1993)
Epstein, C.L.: Subelliptic Spin\(_{\mathbb{C}}\) Dirac operators I. Ann. Math. 166, 183–214 (2007)
Eastwood, M.G., Ryan, J.: Aspects of Dirac operators in analysis. Milan J. Math. 75, 91–116 (2007)
Kath, I., Ungermann, O.: Spectra of sub-Dirac operators on certain nilmanifolds. Math. Scand., 117(1) (2015). arXiv:1311.2418
Liu, H., Ryan, J.: Clifford analysis techniques for spherical pde. J. Fourier Anal. Appl. 8(6), 535–564 (2002)
Petit, R.: Spin\(^c\)-structures and Dirac operators on contact manifolds. Differ. Geom. Appl. 22, 229–252 (2005)
Porteous, I.: Clifford Algebras and the Classical Groups. Cambridge University Press (1995)
Rocha-Chávez, R., Shapiro, M., Sommen, F.: Integral Theorems for Functions and Differential Forms in \({\mathbb{C}^m}\), Res. Notes Math., vol. 428, Chapman & Hall/CRC, Boca Raton, FL (2002)
Ryan, J.: Conformally covariant operators in Clifford analysis. Z. Anal. Anwend. 14, 677–704 (1995)
Shirrell, S., Walter, R.: Hermitian Clifford Analysis and Its Connections with Representation Theory. submitted
Acknowledgements
The work of the first and second authors was partially supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project UID/MAT/ 0416/2013.
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Cerejeiras, P., Kähler, U., Ryan, J. (2018). From Hermitean Clifford Analysis to Subelliptic Dirac Operators on Odd Dimensional Spheres and Other CR Manifolds. In: Cerejeiras, P., Nolder, C., Ryan, J., Vanegas Espinoza, C. (eds) Clifford Analysis and Related Topics. CART 2014. Springer Proceedings in Mathematics & Statistics, vol 260. Springer, Cham. https://doi.org/10.1007/978-3-030-00049-3_3
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