Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Inner ideals of Lie algebras of skew elements of prime rings with involution


Authors: Jose Brox, Antonio Fernández López and Miguel Gómez Lozano
Journal: Proc. Amer. Math. Soc. 144 (2016), 2741-2751
MSC (2010): Primary 17B60; Secondary 16W10, 17C50
DOI: https://doi.org/10.1090/proc/12903
Published electronically: March 22, 2016
MathSciNet review: 3487211
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we extend the Lie inner ideal structure of simple Artinian rings with involution, initiated by Benkart and completed by Benkart and Fernández López, to centrally closed prime rings with involution of characteristic not $ 2$, $ 3$ or $ 5$. New Lie inner ideals (which we call special) occur when making this extension. We also give a purely algebraic description of the so-called Clifford inner ideals, which had only been described in geometric terms.


References [Enhancements On Off] (What's this?)

  • [1] A. A. Baranov, Finitary simple Lie algebras, J. Algebra 219 (1999), no. 1, 299-329. MR 1707673 (2000f:17011), https://doi.org/10.1006/jabr.1999.7856
  • [2] K. I. Beidar, W. S. Martindale and A. V. Mikhalev, Rings with Generalized Identities, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995.
  • [3] Georgia Benkart, The Lie inner ideal structure of associative rings, J. Algebra 43 (1976), no. 2, 561-584. MR 0435149 (55 #8110)
  • [4] Georgia Benkart, On inner ideals and ad-nilpotent elements of Lie algebras, Trans. Amer. Math. Soc. 232 (1977), 61-81. MR 0466242 (57 #6122)
  • [5] Georgia Benkart and Antonio Fernández López, The Lie inner ideal structure of associative rings revisited, Comm. Algebra 37 (2009), no. 11, 3833-3850. MR 2573222 (2011b:16127), https://doi.org/10.1080/00927870802545729
  • [6] Arjeh M. Cohen, Gábor Ivanyos, and Dan Roozemond, Simple Lie algebras having extremal elements, Indag. Math. (N.S.) 19 (2008), no. 2, 177-188. MR 2489325 (2010d:17012), https://doi.org/10.1016/S0019-3577(09)00003-2
  • [7] John R. Faulkner, On the geometry of inner ideals, J. Algebra 26 (1973), 1-9. MR 0367002 (51 #3247)
  • [8] Antonio Fernández López, Lie inner ideals are nearly Jordan inner ideals, Proc. Amer. Math. Soc. 142 (2014), no. 3, 795-804. MR 3148514, https://doi.org/10.1090/S0002-9939-2013-11809-5
  • [9] Antonio Fernández López and Eulalia García Rus, Inner ideals in quadratic Jordan algebras of infinite capacity, Int. J. Math. Game Theory Algebra 9 (1999), no. 1, 35-54. MR 1697472 (2000f:17044)
  • [10] Antonio Fernández López, Esther García, and Miguel Gómez Lozano, Inner ideals of finitary simple Lie algebras, J. Lie Theory 16 (2006), no. 1, 97-114. MR 2196417 (2006m:17020)
  • [11] Antonio Fernández López, Esther García, Miguel Gómez Lozano, and Erhard Neher, A construction of gradings of Lie algebras, Int. Math. Res. Not. IMRN 16 (2007), Art. ID rnm051, 34. MR 2353091 (2008j:17057), https://doi.org/10.1093/imrn/rnm051
  • [12] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294 (99i:16001)
  • [13] Ottmar Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR 0444721 (56 #3071)
  • [14] W. S. Martindale III and C. Robert Miers, On the iterates of derivations of prime rings, Pacific J. Math. 104 (1983), no. 1, 179-190. MR 683736 (84c:16033)
  • [15] Kevin McCrimmon, Inner ideals in quadratic Jordan algebras, Trans. Amer. Math. Soc. 159 (1971), 445-468. MR 0279145 (43 #4871)
  • [16] A. A. Premet, Lie algebras without strong degeneration, Mat. Sb. (N.S.) 129(171) (1986), no. 1, 140-153 (Russian). MR 830100 (87g:17018)
  • [17] A. A. Premet, Inner ideals in modular Lie algebras, Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 5 (1986), 11-15, 123 (Russian, with English summary). MR 876665 (88d:17013)
  • [18] Alexander Premet and Helmut Strade, Simple Lie algebras of small characteristic. I. Sandwich elements, J. Algebra 189 (1997), no. 2, 419-480. MR 1438184 (98e:17029), https://doi.org/10.1006/jabr.1996.6861
  • [19] Oleg N. Smirnov, Simple associative algebras with finite $ \mathbf {Z}$-grading, J. Algebra 196 (1997), no. 1, 171-184. MR 1474168 (98i:16045), https://doi.org/10.1006/jabr.1997.7087
  • [20] Oleg N. Smirnov, Finite $ {\bf Z}$-gradings of Lie algebras and symplectic involutions, J. Algebra 218 (1999), no. 1, 246-275. MR 1704686 (2000f:17034), https://doi.org/10.1006/jabr.1999.7880
  • [21] E. I. Zelmanov, Lie algebras with finite gradation, Mat. Sb. (N.S.) 124(166) (1984), no. 3, 353-392 (Russian). MR 752226 (86d:17016)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 17B60, 16W10, 17C50

Retrieve articles in all journals with MSC (2010): 17B60, 16W10, 17C50


Additional Information

Jose Brox
Affiliation: Departamento de álgebra, Geometría y Topología, Universidad de Málaga, 29071, Málaga, Spain
Email: brox@agt.cie.uma.es

Antonio Fernández López
Affiliation: Departamento de álgebra, Geometría y Topología, Universidad de Málaga, 29071, Málaga, Spain
Email: emalfer@uma.es

Miguel Gómez Lozano
Affiliation: Departamento de álgebra, Geometría y Topología, Universidad de Málaga, 29071, Málaga, Spain
Email: magomez@agt.cie.uma.es

DOI: https://doi.org/10.1090/proc/12903
Keywords: Inner ideals, Lie algebras, associative algebras with involution
Received by editor(s): April 23, 2014
Published electronically: March 22, 2016
Additional Notes: The first author was supported by the Spanish MEC through the FPU grant AP2009-4848, and by the Junta de Andalucía FQM264.
The second author was supported by the Spanish MEC and Fondos FEDER, MTM2010-19482.
The third author was supported by the Spanish MEC and Fondos FEDER, MTM2010-19482, and by the Junta de Andalucía FQM264.
Dedicated: Dedicated to Professor W. S. Martindale, 3rd.
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society