### Electronic Research Archive

2021, Issue 5: 2945-2957. doi: 10.3934/era.2021020
Special Issues

# Structure of sympathetic Lie superalgebras

• 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.

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

### Related Papers:

• 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.

 [1] 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. (1988) 306: 523-525. [2] Certaines propriétés d'une classe d'algèbres de Lie qui généralisent les algèbres de Lie semi-simples. Ann. Fac. Sci. Toulouse Math. (1991) 12: 29-35. [3] Structure of perfect Lie algebras without center and outer derivations. Ann. Fac. Sci. Toulouse Math. (1996) 5: 203-231. [4] On complete Lie superalgebras. Commun. Korean Math. Soc. (1996) 11: 323-334. [5] N. Jacobson, Lie Algebras, Willey New York, 1962. [6] Some complete Lie algebras. J. Algebra (1996) 186: 807-817. [7] Lie superalgebras. Advances in Math. (1977) 26: 8-96. [8] Characteristically nilpotent algebras. Canadian J. Math. (1971) 23: 222-235. [9] M. Scheunert, The Theory of Lie Superalgebra, Lecture notes in mathematics 716, Springer-verlag Berlin Heidelberg New-York, 1979. [10] Derivation algebras of centerless perfect Lie algebras are complete. J. Algebra (2005) 285: 508-515.
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