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Weight modules over split Lie algebras. (English) Zbl 1259.17022

Summary: We study the structure of weight modules \(V\) with restrictions neither on the dimension nor on the base field, over split Lie algebras \(L\). We show that if \(L\) is perfect and \(V\) satisfies \(LV = V\) and \(\mathcal{Z}(V)=0\), then
\[ L=\oplus_{i\in I}I_i \quad \text{and} \quad V=\oplus_{j\in J}V_j \]
where each \(I_i\) is an ideal of \(L\) satisfying \([I_i, I_k] = 0\) if \(i\neq k\) and each \(V_j\) is a (weight) submodule of \(V\) in such a way that for any \(j\in J\) there exists a unique \(i\in I\) such that \(I_i V_j\neq 0\), with \(V_j\) a weight module over \(I_i\). Under certain conditions, it is shown that the above decomposition of \(V\) is by means of the family of its minimal submodules, each one being a simple (weight) submodule.

MSC:

17B65 Infinite-dimensional Lie (super)algebras
17B81 Applications of Lie (super)algebras to physics, etc.
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References:

[1] Benamor H., Lett. Math. Phys. 18 pp 307–
[2] DOI: 10.1007/PL00004314 · Zbl 0884.17004 · doi:10.1007/PL00004314
[3] DOI: 10.1063/1.2834919 · Zbl 1153.81327 · doi:10.1063/1.2834919
[4] DOI: 10.1090/S0002-9947-99-02338-7 · Zbl 0930.17005 · doi:10.1090/S0002-9947-99-02338-7
[5] Calderón A. J., Proc. Indian Acad. Sci. Math. Sci. 118 pp 351–
[6] DOI: 10.1007/s12044-010-0021-4 · Zbl 1209.17004 · doi:10.1007/s12044-010-0021-4
[7] Calderón A. J., Commun. Algebra 38 pp 28–
[8] DOI: 10.1063/1.3464265 · Zbl 1311.17006 · doi:10.1063/1.3464265
[9] DOI: 10.1023/A:1026640918026 · Zbl 0905.06006 · doi:10.1023/A:1026640918026
[10] Draper C., Forum Math. 22 pp 863–
[11] Fialowski A., SIGMA 2 pp 1–
[12] DOI: 10.1016/0370-2693(91)91137-K · doi:10.1016/0370-2693(91)91137-K
[13] DOI: 10.1088/1751-8113/43/39/392002 · Zbl 1213.47043 · doi:10.1088/1751-8113/43/39/392002
[14] DOI: 10.1063/1.3220609 · Zbl 1259.17023 · doi:10.1063/1.3220609
[15] DOI: 10.1006/jabr.1999.7978 · Zbl 1027.17019 · doi:10.1006/jabr.1999.7978
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