Abstract
Let G be a basic classical Lie superalgebra except A(n, n) and D(2, 1, α) over the complex number field ℂ. Using existence of a non-degenerate invariant bilinear form and root space decomposition, we prove that every 2-local automorphism on G is an automorphism. Furthermore, we give an example of a 2-local automorphism which is not an automorphism on a subalgebra of Lie superalgebra spl(3, 3).
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Ayupov, Sh., Arzikulov, F: 2-Local derivations on semi-finite von Neumann algebras. Glasg. Math. J., 56(1), 9–12 (2014)
Ayupov, Sh., Kudaybergenov, K.: 2-Local derivations on von Neumann algebras. Positivity, 19(3), 445–455 (2015)
Ayupov, Sh., Kudaybergenov, K.: 2-Local automorphisms on finite-dimensional Lie algebras. Linear Algebra Appl., 507, 121–131 (2016)
Ayupov, Sh., Kudaybergenov, K., Rakhimov, I.: 2-Local derivations on finite-dimensional Lie algebras. Linear Algebra Appl., 474, 1–11 (2015)
Burgos, M., Fernández-Polo, F., Garcés, J., et al.: 2-Local triple homomorphisms on von Neumann algebras and JBW*-triples. J. Math. Anal. Appl., 426(1), 43–63 (2015)
Chen, L., Huang, L., Lu, F.: 2-Local Lie isomorphism of operator algebras on Banach spaces. Studia Math., 229(1), 1–11 (2015)
Chen, Z., Wang, D.: Nonlinear maps satisfying derivability on standard parabolic subalgebras of finitedimensional simple Lie algebras. Linear Multilinear Algebra, 59(3), 261–270 (2011)
Chen, Z., Wang, D.: 2-Local automorphisms of finite-dimensional simple Lie algebras. Linear Algebra Appl., 486, 335–344 (2015)
Chen, H., Wang, Y., Nan, J.: Local superderivations on the basic classical Lie superalgebras. Algebra Colloq., 24(4), 673–684 (2018)
Grantcharov, D., Pianzola, A.: Automorphisms and twisted loop algebras of finite-dimensional simple Lie superalgebras. Int. Math. Res. Not., 73, 3937–3962 (2004)
Kac, V. G.: Lie superalgebras. Adv. Math., 26(1), 8–96 (1977)
Musson, I.: Lie Superalgebras and Enveloping Algebras. Graduate Studies in Mathematics, 131. Providence, RI, 2012
Šemrl, P.: Local automorphisms and derivations on B(H). Proc. Amer. Math. Soc., 125(9), 2677–2680 (1997)
Serganova, V.: Automorphisms of simple Lie superalgebras. Izv. Akad. Nauk SSSR Ser. Mat., 48(3), 585–598 (1984)
Sun, H., Han, Q.: Lie algebras and Lie superalgebras and their applications in Physics (in Chinese), Peking University, Beijing, 1995
Wang, Y., Chen, H., Nan, J.: 2-Local superderivations on basic classical Lie superalgebras. J. Math. Res. Appl., 37(5), 527–534 (2017)
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The second author is supported by the National Natural Science Foundation of China (Grant No. 11471090)
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Yu, L., Wang, Y., Chen, H.X. et al. 2-Local Automorphisms on Basic Classical Lie Superalgebras. Acta. Math. Sin.-English Ser. 35, 427–437 (2019). https://doi.org/10.1007/s10114-018-7519-6
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DOI: https://doi.org/10.1007/s10114-018-7519-6