Calderón Martín, Antonio J.; Sánchez-Delgado, José M. Weight modules over split Lie algebras. (English) Zbl 1259.17022 Mod. Phys. Lett. A 28, No. 5, Paper No. 1350008, 9 p. (2013). 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. Cited in 1 Document MSC: 17B65 Infinite-dimensional Lie (super)algebras 17B81 Applications of Lie (super)algebras to physics, etc. Keywords:infinite dimensional Lie module; infinite dimensional split Lie algebra; structure theory PDFBibTeX XMLCite \textit{A. J. Calderón Martín} and \textit{J. M. Sánchez-Delgado}, Mod. Phys. Lett. A 28, No. 5, Paper No. 1350008, 9 p. (2013; Zbl 1259.17022) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.