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doi: 10.3934/era.2021020
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Structure of sympathetic Lie superalgebras

School of Mathematics and Statistics, Northeast Normal University, Changchun 130034, China

* Corresponding author: chenly640@nenu.edu.cn

Received  July 2020 Revised  February 2021 Early access March 2021

Fund Project: Supported by NNSF of China (Nos. 11771069 and 12071405)

Sympathetic Lie superalgebras are defined and some classical properties of sympathetic Lie superalgebras are given. Among the main results, we prove that any Lie superalgebra $ L $ contains a maximal sympathetic graded ideal and we obtain some properties about sympathetic decomposition. More specifically, we study a general sympathetic Lie superalgebra $ L $ with graded ideals $ I $, $ J $ and $ S $ such that $ L = I\oplus J $ and $ L/S $ is a sympathetic Lie superalgebra, and we obtain some properties of $ L/S $. Furthermore, under certain assumptions on $ L $ we prove that the derivation algebra $ \mathrm{Der}(L) $ is sympathetic and that if in addition $ L $ is indecomposable, then $ \mathrm{Der}(L) $ is simply sympathetic.

Citation: Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, doi: 10.3934/era.2021020
References:
[1]

E. Angelopoulos, Algèbres de Lie $\mathfrak{g}$ satisfaisant $[\mathfrak{g}, \mathfrak{g}] = \mathfrak{g}$, $\text{Der}\mathfrak{g} = \text{ad}\mathfrak{g}$, (French) C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 523-525.   Google Scholar

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S. Benayadi, Certaines propriétés d'une classe d'algèbres de Lie qui généralisent les algèbres de Lie semi-simples, Ann. Fac. Sci. Toulouse Math., 12 (1991), 29-35.  doi: 10.5802/afst.717.  Google Scholar

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S. Benayadi, Structure of perfect Lie algebras without center and outer derivations, Ann. Fac. Sci. Toulouse Math., 5 (1996), 203-231.  doi: 10.5802/afst.828.  Google Scholar

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J.-H. Chun and J.-S. Lee, On complete Lie superalgebras, Commun. Korean Math. Soc., 11 (1996), 323-334.   Google Scholar

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N. Jacobson, Lie Algebras, Willey New York, 1962.  Google Scholar

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C. P. JiangD. J. Meng and S. Q. Zhang, Some complete Lie algebras, J. Algebra, 186 (1996), 807-817.  doi: 10.1006/jabr.1996.0396.  Google Scholar

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V. G. Kac, Lie superalgebras, Advances in Math., 26 (1977), 8-96.  doi: 10.1016/0001-8708(77)90017-2.  Google Scholar

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T. S. Ravisankar, Characteristically nilpotent algebras, Canadian J. Math., 23 (1971), 222-235.  doi: 10.4153/CJM-1971-022-2.  Google Scholar

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M. Scheunert, The Theory of Lie Superalgebra, Lecture notes in mathematics 716, Springer-verlag Berlin Heidelberg New-York, 1979.  Google Scholar

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Y. Su and L. Zhu, Derivation algebras of centerless perfect Lie algebras are complete, J. Algebra, 285 (2005), 508-515.  doi: 10.1016/j.jalgebra.2004.09.033.  Google Scholar

show all references

References:
[1]

E. Angelopoulos, Algèbres de Lie $\mathfrak{g}$ satisfaisant $[\mathfrak{g}, \mathfrak{g}] = \mathfrak{g}$, $\text{Der}\mathfrak{g} = \text{ad}\mathfrak{g}$, (French) C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 523-525.   Google Scholar

[2]

S. Benayadi, Certaines propriétés d'une classe d'algèbres de Lie qui généralisent les algèbres de Lie semi-simples, Ann. Fac. Sci. Toulouse Math., 12 (1991), 29-35.  doi: 10.5802/afst.717.  Google Scholar

[3]

S. Benayadi, Structure of perfect Lie algebras without center and outer derivations, Ann. Fac. Sci. Toulouse Math., 5 (1996), 203-231.  doi: 10.5802/afst.828.  Google Scholar

[4]

J.-H. Chun and J.-S. Lee, On complete Lie superalgebras, Commun. Korean Math. Soc., 11 (1996), 323-334.   Google Scholar

[5]

N. Jacobson, Lie Algebras, Willey New York, 1962.  Google Scholar

[6]

C. P. JiangD. J. Meng and S. Q. Zhang, Some complete Lie algebras, J. Algebra, 186 (1996), 807-817.  doi: 10.1006/jabr.1996.0396.  Google Scholar

[7]

V. G. Kac, Lie superalgebras, Advances in Math., 26 (1977), 8-96.  doi: 10.1016/0001-8708(77)90017-2.  Google Scholar

[8]

T. S. Ravisankar, Characteristically nilpotent algebras, Canadian J. Math., 23 (1971), 222-235.  doi: 10.4153/CJM-1971-022-2.  Google Scholar

[9]

M. Scheunert, The Theory of Lie Superalgebra, Lecture notes in mathematics 716, Springer-verlag Berlin Heidelberg New-York, 1979.  Google Scholar

[10]

Y. Su and L. Zhu, Derivation algebras of centerless perfect Lie algebras are complete, J. Algebra, 285 (2005), 508-515.  doi: 10.1016/j.jalgebra.2004.09.033.  Google Scholar

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