Czechoslovak Mathematical Journal, Vol. 68, No. 3, pp. 661-675, 2018


Local superderivations on Lie superalgebra $\mathfrak{q}(n)$

Haixian Chen, Ying Wang

Received November 20, 2016.   First published March 2, 2018.

Abstract:  Let $\mathfrak{q}(n)$ be a simple strange Lie superalgebra over the complex field $\mathbb{C}$. In a paper by A. Ayupov, K. Kudaybergenov (2016), the authors studied the local derivations on semi-simple Lie algebras over $\mathbb{C}$ and showed the difference between the properties of local derivations on semi-simple and nilpotent Lie algebras. We know that Lie superalgebras are a generalization of Lie algebras and the properties of some Lie superalgebras are similar to those of semi-simple Lie algebras, but $\mathfrak{p}(n)$ is an exception. In this paper, we introduce the definition of the local superderivation on $\mathfrak{q}(n)$, give the structures and properties of the local superderivations of $\mathfrak{q}(n)$, and prove that every local superderivation on $\mathfrak{q}(n)$, $n>3$, is a superderivation.
Keywords:  simple Lie superalgebra; superderivation; local superderivation
Classification MSC:  16W55, 17B20, 17B40


References:
[1] S. Albeverio, S. A. Ayupov, K. K. Kudaybergenov, B. O. Nurjanov: Local derivations on algebras of measurable operators. Commun. Contemp. Math. 13 (2011), 643-657. DOI 10.1142/S0219199711004270 | MR 2826440 | Zbl 1230.46056
[2] R. Alizadeh, M. J. Bitarafan: Local derivations of full matrix rings. Acta Math. Hung. 145 (2015), 433-439. DOI 10.1007/s10474-014-0460-y | MR 3325800 | Zbl 1363.17003
[3] S. Ayupov, K. Kudaybergenov: Local derivations on finite-dimensional Lie algebras. Linear Algebra Appl. 493 (2016), 381-398. DOI 10.1016/j.laa.2015.11.034 | MR 3452744 | Zbl 06536636
[4] S. Ayupov, K. Kudaybergenov, B. Nurjanov, A. Alauadinov: Local and 2-local derivations on noncommutative Arens algebras. Math. Slovaca 64 (2014), 423-432. DOI 10.2478/s12175-014-0215-9 | MR 3201356 | Zbl 1349.46071
[5] V. G. Kac: Lie superalgebras. Adv. Math. 26 (1977), 8-96. DOI 10.1016/0001-8708(77)90017-2 | MR 0486011 | Zbl 0366.17012
[6] R. V. Kadison: Local derivations. J. Algebra 130 (1990), 494-509. DOI 10.1016/0021-8693(90)90095-6 | MR 1051316 | Zbl 0751.46041
[7] F. Mukhamedov, K. Kudaybergenov: Local derivations on subalgebras of $\tau$-measurable operators with respect to semi-finite von Neumann algebras. Mediterr. J. Math. 12 (2015), 1009-1017. DOI 10.1007/s00009-014-0447-5 | MR 3376827 | Zbl 1321.47089
[8] I. M. Musson: Lie Superalgebras and Enveloping Algebras. Graduate Studies in Mathematics 131, American Mathematical Society, Providence (2012). DOI 10.1090/gsm/131 | MR 2906817 | Zbl 1255.17001
[9] A. Nowicki, I. Nowosad: Local derivations of subrings of matrix rings. Acta Math. Hung. 105 (2004), 145-150. DOI 10.1023/B:AMHU.0000045539.32024.db | MR 2093937 | Zbl 1070.16035
[10] M. Scheunert: The Theory of Lie Superalgebras. An Introduction. Lecture Notes in Mathematics 716, Springer, Berlin (1979). DOI 10.1007/bfb0070929 | MR 0537441 | Zbl 0407.17001
[11] J.-H. Zhang, G.-X. Ji, H.-X. Cao: Local derivations of nest subalgebras of von Neumann algebras. Linear Algebra Appl. 392 (2004), 61-69. DOI 10.1016/j.laa.2004.05.015 | MR 2095907 | Zbl 1067.46063

Affiliations:   Ying Wang (corresponding author), Haixian Chen, School of Mathematical Sciences, Dalian University of Technology, No.2 Linggong Road, Ganjingzi District, Dalian City, Liaoning Province, P.R.China, 116024, e-mail: wangying@dlut.edu.cn, chenhx2012@mail.dlut.edu.cn


 
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