Symplectic twistor operator and its solution space on 2

Marie Dostálová; Petr Somberg

Archivum Mathematicum (2013)

  • Volume: 049, Issue: 3, page 161-185
  • ISSN: 0044-8753

Abstract

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We introduce the symplectic twistor operator T s in symplectic spin geometry of real dimension two, as a symplectic analogue of the Dolbeault operator in complex spin geometry of complex dimension 1. Based on the techniques of the metaplectic Howe duality and algebraic Weyl algebra, we compute the space of its solutions on 2 .

How to cite

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Dostálová, Marie, and Somberg, Petr. "Symplectic twistor operator and its solution space on ${\mathbb {R}}^2$." Archivum Mathematicum 049.3 (2013): 161-185. <http://eudml.org/doc/260662>.

@article{Dostálová2013,
abstract = {We introduce the symplectic twistor operator $T_s$ in symplectic spin geometry of real dimension two, as a symplectic analogue of the Dolbeault operator in complex spin geometry of complex dimension 1. Based on the techniques of the metaplectic Howe duality and algebraic Weyl algebra, we compute the space of its solutions on $\{\mathbb \{R\}\}^2$.},
author = {Dostálová, Marie, Somberg, Petr},
journal = {Archivum Mathematicum},
keywords = {symplectic spin geometry; metaplectic Howe duality; symplectic twistor operator; symplectic Dirac operator; symplectic spin geometry; metaplectic Howe duality; symplectic twistor operator; symplectic Dirac operator},
language = {eng},
number = {3},
pages = {161-185},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Symplectic twistor operator and its solution space on $\{\mathbb \{R\}\}^2$},
url = {http://eudml.org/doc/260662},
volume = {049},
year = {2013},
}

TY - JOUR
AU - Dostálová, Marie
AU - Somberg, Petr
TI - Symplectic twistor operator and its solution space on ${\mathbb {R}}^2$
JO - Archivum Mathematicum
PY - 2013
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 049
IS - 3
SP - 161
EP - 185
AB - We introduce the symplectic twistor operator $T_s$ in symplectic spin geometry of real dimension two, as a symplectic analogue of the Dolbeault operator in complex spin geometry of complex dimension 1. Based on the techniques of the metaplectic Howe duality and algebraic Weyl algebra, we compute the space of its solutions on ${\mathbb {R}}^2$.
LA - eng
KW - symplectic spin geometry; metaplectic Howe duality; symplectic twistor operator; symplectic Dirac operator; symplectic spin geometry; metaplectic Howe duality; symplectic twistor operator; symplectic Dirac operator
UR - http://eudml.org/doc/260662
ER -

References

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  1. Baum, H., Friedrich, T., Kath., I., Gruenewald, F., Twistors and Killing Spinors on Riemannian Manifolds, B.G. Teubner, 1991. (1991) 
  2. Bie, H. De, Somberg, P., Souček, V., The Howe Duality and Polynomial Solutions for the Symplectic Dirac Operator, Archive http://arxiv.org/pdf/1002.1053v1.pdf. 
  3. Britten, D. J., Lemire, F. W., 10.1090/S0002-9947-99-02338-7, Trans. Amer. Math. Soc. 351 (1999), 3413–3431. (1999) Zbl0930.17005MR1615943DOI10.1090/S0002-9947-99-02338-7
  4. Crumeyrolle, A., Orthogonal and Symplectic Clifford Algebras: Spinor Structures, Springer Netherlands, 2009. (2009) MR1044769
  5. Dostálová, M., Somberg, P., Symplectic twistor operator and its solution space on 2 n , Complex Analysis and Operator Theory 4 (2013). (2013) MR3144180
  6. Friedrich, T., Dirac Operators in Riemannian Geometry, AMS, 2000. (2000) Zbl0949.58032MR1777332
  7. Fulton, W., Harris, J., Representation Theory: A First Course (Graduate Texts in Mathematics/Readings in Mathematics), Springer, 1991. (1991) MR1153249
  8. Habermann, K., Habermann, L., Introduction to symplectic Dirac operators, Lecture Notes in Mathematics, 1887, Springer-Verlag, Berlin, 2006. (2006) Zbl1102.53032MR2252919
  9. Kadlcakova, L., Contact Symplectic Geometry in Parabolic Invariant Theory and Symplectic Dirac Operator, Dissertation Thesis, Mathematical Institute of Charles University, Prague, 2002. Zbl1039.58017MR1890440
  10. Kostant, B., Symplectic Spinors, Rome Symposia XIV (1974), 139–152. (1974) Zbl0321.58015MR0400304

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