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Lie superautomorphisms on associative algebras


Authors: Yuri Bahturin and Matej Bresar
Journal: Proc. Amer. Math. Soc. 138 (2010), 417-425
MSC (2000): Primary 17B40; Secondary 16W10, 16R50, 17B60
DOI: https://doi.org/10.1090/S0002-9939-09-10136-3
Published electronically: October 1, 2009
MathSciNet review: 2557159
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Abstract: The results on Lie homomorphisms of associative algebras are extended to certain associative superalgebras. It is shown that under appropriate conditions a Lie superautomorphism of $ A=A_0\oplus A_1$ is a sum of a superautomorphism or the negative of a superantiautomorphism and a central map. In particular we consider the situation when $ A$ is a central simple algebra and its $ \mathbb{Z}_2$-grading is induced by an idempotent.


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  • 1. Yu. Bahturin, M. Brešar, Lie gradings on associative algebras, J. Algebra 321 (2009), 264-283. MR 2469360
  • 2. Yu. Bahturin, M. Brešar, I. Shestakov, Jordan gradings on associative algebras, Algebr. Represent. Theory, to appear.
  • 3. Yu. Bahturin, D. Fischman, S. Montgomery, On the generalized Lie structure of associative algebras, Israel J. Math. 96 (1996), 27-48. MR 1432725 (98d:16048)
  • 4. Yu. Bahturin, O. H. Kegel, Universal sums of abelian subalgebras, Comm. Algebra 23 (1995), 2975-2990. MR 1332159 (96g:16033)
  • 5. K. I. Beidar, M. Brešar, M. A. Chebotar, Functional identities revised: The fractional and the strong degree, Comm. Algebra 30 (2002), 935-969. MR 1883042 (2002k:16048)
  • 6. K. I. Beidar, M. Brešar, M. A. Chebotar, Jordan superhomomorphisms, Comm. Algebra 31 (2003), 633-644. MR 1968917 (2004a:16060)
  • 7. G. Benkart, X. Xu, K. Zhao, Classical Lie superalgebras over simple associative algebras, Proc. London Math. Soc. (3) 92 (2006), 581-600. MR 2223537 (2007c:17006)
  • 8. M. Brešar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), 525-546. MR 1069746 (93d:16044)
  • 9. M. Brešar, M. A. Chebotar, W. S. Martindale III, Functional Identities, Birkhäuser Verlag, 2007. MR 2332350 (2008j:16068)
  • 10. C. Gómez-Ambrosi, J. Laliena, I. P. Shestakov, On the Lie structure of the skew elements of a prime superalgebra with superinvolution, Comm. Algebra 28 (2000), 3277-3291. MR 1765316 (2001g:17007)
  • 11. C. Gómez-Ambrosi, I. P. Shestakov, On the Lie structure of the skew elements of a simple superalgebra with superinvolution, J. Algebra 208 (1998), 43-71. MR 1643975 (99j:17032)
  • 12. I. N. Herstein, Topics in Ring Theory, University of Chicago Press, 1969. MR 0271135 (42:6018)
  • 13. J. Laliena, S. Sacristán, Lie structure in semiprime superalgebras with superinvolution, J. Algebra 315 (2007), 751-760. MR 2351892 (2008j:16123)
  • 14. W. S. Martindale III, Lie isomorphisms of prime rings, Trans. Amer. Math. Soc. 142 (1969), 437-455. MR 0251077 (40:4308)
  • 15. F. Montaner, On the Lie structure of associative superalgebras, Comm. Algebra 26 (1998), 2337-2349. MR 1626634 (99g:16055)
  • 16. S. Montgomery, Constructing simple Lie superalgebras from associative graded algebras, J. Algebra 195 (1997), 558-579. MR 1469640 (99g:17043)
  • 17. Y. Wang, Functional identities and Lie superhomomorphisms on prime superalgebras, Comm. Algebra, to appear.
  • 18. K. Zhao, Simple Lie color algebras from graded associative algebras, J. Algebra 269 (2003), 439-455. MR 2015286 (2004m:17044)

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Additional Information

Yuri Bahturin
Affiliation: Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Newfoundland, A1C5S7, Canada
Email: yuri@math.mun.ca

Matej Bresar
Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia – and – Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor, Slovenia
Email: matej.bresar@fmf.uni-lj.si

DOI: https://doi.org/10.1090/S0002-9939-09-10136-3
Received by editor(s): March 30, 2009
Published electronically: October 1, 2009
Additional Notes: The first author was partially supported by NSERC grant #227060-04 and URP grant, Memorial University of Newfoundland.
The second author was partially supported by ARRS grant #P1-0288.
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2009 American Mathematical Society

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