Lie superautomorphisms on associative algebras

Bahturin, Yuri; Brešar, Matej

doi:10.1090/S0002-9939-09-10136-3

Lie superautomorphisms on associative algebras
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by Yuri Bahturin and Matej Brešar PDF

Proc. Amer. Math. Soc. 138 (2010), 417-425 Request permission

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.

References

Yuri Bahturin and Matej Brešar, Lie gradings on associative algebras, J. Algebra 321 (2009), no. 1, 264–283. MR 2469360, DOI 10.1016/j.jalgebra.2008.08.032
Yu. Bahturin, M. Brešar, I. Shestakov, Jordan gradings on associative algebras, Algebr. Represent. Theory, to appear.
Y. Bahturin, D. Fischman, and S. Montgomery, On the generalized Lie structure of associative algebras. part A, Israel J. Math. 96 (1996), no. part A, 27–48. MR 1432725, DOI 10.1007/BF02785532
Yuri Bahturin and O. H. Kegel, Universal sums of abelian subalgebras, Comm. Algebra 23 (1995), no. 8, 2975–2990. MR 1332159, DOI 10.1080/00927879508825381
K. I. Beidar, M. Brešar, and M. A. Chebotar, Functional identities revised: the fractional and the strong degree, Comm. Algebra 30 (2002), no. 2, 935–969. MR 1883042, DOI 10.1081/AGB-120013192
K. I. Beidar, M. Brešar, and M. A. Chebotar, Jordan superhomomorphisms, Comm. Algebra 31 (2003), no. 2, 633–644. MR 1968917, DOI 10.1081/AGB-120017336
Georgia Benkart, Xiaoping Xu, and Kaiming Zhao, Classical Lie superalgebras over simple associative algebras, Proc. London Math. Soc. (3) 92 (2006), no. 3, 581–600. MR 2223537, DOI 10.1017/S0024611505015583
Matej Brešar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), no. 2, 525–546. MR 1069746, DOI 10.1090/S0002-9947-1993-1069746-X
Matej Brešar, Mikhail A. Chebotar, and Wallace S. Martindale III, Functional identities, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. MR 2332350
Carlos Gómez-Ambrosi, Jesús Laliena, and Ivan P. Shestakov, On the Lie structure of the skew elements of a prime superalgebra with superinvolution, Comm. Algebra 28 (2000), no. 7, 3277–3291. MR 1765316, DOI 10.1080/00927870008827024
Carlos Gómez-Ambrosi and Ivan P. Shestakov, On the Lie structure of the skew elements of a simple superalgebra with superinvolution, J. Algebra 208 (1998), no. 1, 43–71. MR 1643975, DOI 10.1006/jabr.1998.7452
I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0271135
Jesús Laliena and Sara Sacristán, Lie structure in semiprime superalgebras with superinvolution, J. Algebra 315 (2007), no. 2, 751–760. MR 2351892, DOI 10.1016/j.jalgebra.2007.04.005
Wallace S. Martindale III, Lie isomorphisms of prime rings, Trans. Amer. Math. Soc. 142 (1969), 437–455. MR 251077, DOI 10.1090/S0002-9947-1969-0251077-5
Fernando Montaner, On the Lie structure of associative superalgebras, Comm. Algebra 26 (1998), no. 7, 2337–2349. MR 1626634, DOI 10.1080/00927879808826279
S. Montgomery, Constructing simple Lie superalgebras from associative graded algebras, J. Algebra 195 (1997), no. 2, 558–579. MR 1469640, DOI 10.1006/jabr.1997.7050
Y. Wang, Functional identities and Lie superhomomorphisms on prime superalgebras, Comm. Algebra, to appear.
Kaiming Zhao, Simple Lie color algebras from graded associative algebras, J. Algebra 269 (2003), no. 2, 439–455. MR 2015286, DOI 10.1016/S0021-8693(02)00564-1