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Construction of Symplectic Quadratic Lie Algebras from Poisson Algebras

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Algebra, Geometry and Mathematical Physics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 85))

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Abstract

We introduce the notion of quadratic (resp. symplectic quadratic) Poisson algebras and we show how one can construct new interesting quadratic (resp. symplectic quadratic) Lie algebras from quadratic (resp. symplectic quadratic) Poisson algebras. Finally, we give inductive descriptions of symplectic quadratic Poisson algebras.

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References

  1. Aubert, A.: Structures affines et pseudo-métriques invariantes à gauche sur des groupes de Lie. Thesis, Université Montpellier II (1996)

    Google Scholar 

  2. Bajo, I., Benayadi, S., Medina, A.: Symplectic structures on quadratic Lie algebras. J. Algebra 316(1), 174–188 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baklouti, A., Benayadi, S.: Symmetric symplectic associative commutative algebras and related Lie algebras. Algebra colloq. 18(1), 973–986 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benayadi, S.: Structures de certaines algèbres de Lie quadratiques. Commun. Algebra 23(10), 3867–3887 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bordemann, M.: Nondegenerate invariant bilinear forms on nonassociative algebras. Acta Math. Univ. Com. LXVIP(2), 151–201 (1997)

    Google Scholar 

  6. Dardié, J.-M., Medina, A.: Double extension symplectique d’un group de Lie symplectique. Adv. Math. 117(2), 208–227 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diatta, A., Medina, A.: Classical Yang-Baxter equation and left invariant affine geometry on Lie groups. Manuscr. Math. 114(4), 477–486 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Favre, G., Santaroubane, L.: Symmetric invariant non-degenerate bilinear form on a Lie algebra. J. Algebra 105, 451–464 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Figueroa-O’Farrill, J.M., Stanciu, S.: On the structure of symmetric self-dual Lie algebras. J. Math. Phys. 37(8), 4121–4134 (1996)

    Google Scholar 

  10. Hofmann, K.H., Keith, V.S.: Invariant quadratic forms on finite dimensional Lie algebras. Bull. Austral. Math. Soc. 33, 21–36 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jacobson, N.: A note on automorphisms and derivations of Lie algebras. Proc. Amer. Math. Soc. 6, 33–39 (1955)

    Article  MathSciNet  Google Scholar 

  12. Lu, C.: Finite-dimensional solvable Lie algebras with nondegenerate invariant bilinear forms. J. Algebra 311, 178–201 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Medina, A., Revoy, Ph.: Algèbres de Lie et produit scalaire invariant. Ann. Scient. Ec. Norm. Sup. 4ème série 18, 556–561 (1985)

    Google Scholar 

  14. O’Neill B.: Semi-riemannian geometry with applications to relativity. Academic Press, New York (1983)

    Google Scholar 

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Author information

Authors and Affiliations

  1. IECL, CNRS UMR 7502, Université de Lorraine, Ile du Saulcy, 57045, Metz cedex, France

    Saïd Benayadi

Authors
  1. Saïd Benayadi
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Correspondence to Saïd Benayadi .

Editor information

Editors and Affiliations

  1. Informatique et Applications, Université of Haute Alsace, Laboratoire de Mathématiques, Mulhouse, France

    Abdenacer Makhlouf

  2. Tallinn University of Technology, Dept. Mathematics, Tallinn, Estonia

    Eugen Paal

  3. Culture and Communication (UKK) Division of Applied Mathematics, Mälardalen University, The School of Education, Västerås, Sweden

    Sergei D. Silvestrov

  4. Dept. Mathematical Sciences, University of Göteborg, Göteborg, Sweden

    Alexander Stolin

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Benayadi, S. (2014). Construction of Symplectic Quadratic Lie Algebras from Poisson Algebras. In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A. (eds) Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics & Statistics, vol 85. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55361-5_8

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