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Strongly non-degenerate Lie algebras

Authors: Francesc Perera and Mercedes Siles Molina
Journal: Proc. Amer. Math. Soc. 136 (2008), 4115-4124
MSC (2000): Primary 17B60; Secondary 16W25
Published electronically: July 23, 2008
MathSciNet review: 2431022
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Abstract: Let $ A$ be a semiprime $ 2$- and $ 3$-torsion free non-commutative associative algebra. We show that the Lie algebra $ \mathcal{D}\mathrm{er}(A)$ of (associative) derivations of $ A$ is strongly non-degenerate, which is a strong form of semiprimeness for Lie algebras, under some additional restrictions on the center of $ A$. This result follows from a description of the quadratic annihilator of a general Lie algebra inside appropriate Lie overalgebras. Similar results are obtained for an associative algebra $ A$ with involution and the Lie algebra $ \mathrm{SDer}(A)$ of involution preserving derivations of $ A$.

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  • 1. G. Benkart, The Lie inner ideal structure of associative rings, J. Algebra 43 (1976), 561-584. MR 0435149 (55:8110)
  • 2. M. Brešar, F. Perera, J. Sánchez Ortega, M. Siles Molina, Computing the maximal algebra of quotients of a Lie algebra, Forum Math., to appear.
  • 3. M. Cabrera, Ideals which memorize the extended centroid, J. Algebra Appl. 1 (2002), 281-288. MR 1932152 (2003f:17001)
  • 4. M. Cabrera, J. Sánchez Ortega, Lie quotients for skew Lie algebras, to appear in Algebra Colloq.
  • 5. C. Draper Fontanals, A. Fernández López, E. García, M. Gómez Lozano, The socle of a nondegenerate Lie algebra, J. Algebra 280 (2004), 635-654. MR 2089256 (2005i:17007)
  • 6. I. N. Herstein, Topics in Ring Theory, The University of Chicago Press, 1969. MR 0271135 (42:6018)
  • 7. I. N. Herstein, On the Lie structure of an associative ring, J. Algebra 14 (1970), 561-571. MR 0255610 (41:270)
  • 8. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. MR 0323842 (48:2197)
  • 9. D. A. Jordan, The Lie ring of symmetric derivations of a ring with involution, J. Austral. Math. Soc. 29 (1980), 153-161. MR 566283 (81d:16025)
  • 10. C. R. Jordan, D. A. Jordan, Lie rings of derivations of associative rings, J. London Math. Soc. 17 (1978), 33-41. MR 0472927 (57:12609)
  • 11. A. I. Kostrikin, Around Burnside, Springer-Verlag, Berlin-Heidelberg, 1990. MR 1075416 (91i:20038)
  • 12. C. Lanski, Lie structure in semi-prime rings with involution, Comm. Alg. 4 (1976), 731-746. MR 0412226 (54:353)
  • 13. C. Martínez, The ring of fractions of a Jordan algebra, J. Algebra 237 (2001), 798-812. MR 1816717 (2002d:17035)
  • 14. F. Perera, M. Siles Molina, Associative and Lie algebras of quotients, Publ. Mat. 52 (2008), 129-149.
  • 15. M. Siles Molina, Algebras of quotients of Lie algebras, J. Pure Appl. Algebra 188 (2004), 175-188. MR 2030813 (2004m:17003)
  • 16. O. Smirnov, Finite $ \mathbb{Z}$-gradings of Lie algebras and symplectic involutions, J. Algebra 218 (1999), 246-275. MR 1704686 (2000f:17034)
  • 17. E. Zelmanov, Lie algebras with an algebraic adjoint representation. Math. USSR Sb. 49(2) (1984), 537-552.

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Additional Information

Francesc Perera
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

Mercedes Siles Molina
Affiliation: Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Málaga, Spain

Received by editor(s): April 13, 2007
Received by editor(s) in revised form: September 26, 2007
Published electronically: July 23, 2008
Additional Notes: The first author was partially supported by the DGI MEC-FEDER through Project MTM2005-00934 and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.
The second author was partially supported by the MEC and Fondos FEDER jointly through project MTM2004-06580-C02-02 and by the Junta de Andalucía PAI, projects FQM-336 and FQM-1215.
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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