Different methods for the study of obstructions in the schemes of Jacobi

Roger Carles[1]; M. Carmen Márquez[2]

  • [1] Université de Poitiers Laboratoire de Mathématiques et Applications UMR 6086 du CNRS 8692 Futuroscope Chasseneuil (France)
  • [2] Universidad de Sevilla Departamento de Geometría y Topología Apdo. 1160 41080-Sevilla (Spain)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 2, page 453-490
  • ISSN: 0373-0956

Abstract

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In this paper the problem of obstructions in Lie algebra deformations is studied from four different points of view. First, we illustrate the method of local ring, an alternative to Gerstenhaber’s method for Lie deformations. We draw parallels between both methods showing that an obstruction class corresponds to a nilpotent local parameter of a versal deformation of the law in the scheme of Jacobi. Then, an elimination process in the global ring, which defines the scheme, allows us to obtain nilpotent elements and to describe the global method. Finally, the obstruction problem is studied in the geometry defined by generators and relations. Under certain conditions, we prove that subschemes of grassmannians of T -invariant ideals of a free Lie algebra ( T being a torus of derivations), after quotient by an action group, are the same as those defined from Jacobi polynomials after a similar quotient.

How to cite

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Carles, Roger, and Márquez, M. Carmen. "Different methods for the study of obstructions in the schemes of Jacobi." Annales de l’institut Fourier 61.2 (2011): 453-490. <http://eudml.org/doc/219680>.

@article{Carles2011,
abstract = {In this paper the problem of obstructions in Lie algebra deformations is studied from four different points of view. First, we illustrate the method of local ring, an alternative to Gerstenhaber’s method for Lie deformations. We draw parallels between both methods showing that an obstruction class corresponds to a nilpotent local parameter of a versal deformation of the law in the scheme of Jacobi. Then, an elimination process in the global ring, which defines the scheme, allows us to obtain nilpotent elements and to describe the global method. Finally, the obstruction problem is studied in the geometry defined by generators and relations. Under certain conditions, we prove that subschemes of grassmannians of $T$-invariant ideals of a free Lie algebra ($T$ being a torus of derivations), after quotient by an action group, are the same as those defined from Jacobi polynomials after a similar quotient.},
affiliation = {Université de Poitiers Laboratoire de Mathématiques et Applications UMR 6086 du CNRS 8692 Futuroscope Chasseneuil (France); Universidad de Sevilla Departamento de Geometría y Topología Apdo. 1160 41080-Sevilla (Spain)},
author = {Carles, Roger, Márquez, M. Carmen},
journal = {Annales de l’institut Fourier},
keywords = {Deformation; obstruction; free Lie algebra; Lie algebra; deformation; local ring method; rigidity},
language = {eng},
number = {2},
pages = {453-490},
publisher = {Association des Annales de l’institut Fourier},
title = {Different methods for the study of obstructions in the schemes of Jacobi},
url = {http://eudml.org/doc/219680},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Carles, Roger
AU - Márquez, M. Carmen
TI - Different methods for the study of obstructions in the schemes of Jacobi
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 2
SP - 453
EP - 490
AB - In this paper the problem of obstructions in Lie algebra deformations is studied from four different points of view. First, we illustrate the method of local ring, an alternative to Gerstenhaber’s method for Lie deformations. We draw parallels between both methods showing that an obstruction class corresponds to a nilpotent local parameter of a versal deformation of the law in the scheme of Jacobi. Then, an elimination process in the global ring, which defines the scheme, allows us to obtain nilpotent elements and to describe the global method. Finally, the obstruction problem is studied in the geometry defined by generators and relations. Under certain conditions, we prove that subschemes of grassmannians of $T$-invariant ideals of a free Lie algebra ($T$ being a torus of derivations), after quotient by an action group, are the same as those defined from Jacobi polynomials after a similar quotient.
LA - eng
KW - Deformation; obstruction; free Lie algebra; Lie algebra; deformation; local ring method; rigidity
UR - http://eudml.org/doc/219680
ER -

References

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  9. A. Fialowski, Deformations of Lie algebras, Math. USSR Sbornik 127(169) (1985), 476-482 Zbl0583.17010MR806511
  10. A. Fialowski, An example of formal deformations of Lie algebras, NATO Conference on Deformation Theory of Algebras and Applications, Il Ciocco, Italy, 1986, Proceedings (1988), 375-401, Kluwer, Dordrecht Zbl0663.17009MR981622
  11. A. Fialowski, D. Fuchs, Construction of miniversal deformations of Lie algebras, J. Funct. Analysis 161 (1999), 76-110 Zbl0944.17015MR1670210
  12. M. Gerstenhaber, On the deformations of rings and algebras, Ann. of Math. 79 (1964), 59-103 Zbl0123.03101MR171807
  13. A. Nijenhuis, R. W. Richardson, Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc. 72 (1966), 1-29 Zbl0136.30502MR195995
  14. G. Rauch, Remarque sur les constantes de structure des C -algèbres de Lie de dimension finie, C. R. Acad. Sc. Paris 266 (1968), 330-332 Zbl0157.07603MR232809
  15. R. W. Richardson, On the rigidity of semi-direct products of Lie algebras, Pac. J. Math. 22 (1967), 339-344 Zbl0166.30301MR212061

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