Structure of sympathetic Lie superalgebras

Yusi Fan; Chenrui Yao; Liangyun Chen; Yusi Fan; Chenrui Yao; Liangyun Chen

doi:10.3934/era.2021020

Electronic Research Archive

2021, Volume 29, Issue 5: 2945-2957. 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

Received: 01 July 2020 Revised: 01 February 2021 Published: 15 March 2021
Primary: 17A30; Secondary: 16E40

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.
- Lie superalgebras,
- sympathetic,
- perfect,
- decomposition,
- derivations
Citation: Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras[J]. Electronic Research Archive, 2021, 29(5): 2945-2957. doi: 10.3934/era.2021020

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Abstract

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.

References

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