Techniques of graphs in the study of the structure of graded modules. (English) Zbl 1414.05308
Summary: We associate an adequate graph to any pair \((\mathcal{V}, \mathcal{W})\) where \(\mathcal{V}\) is a graded module over a graded linear space \(\mathcal{W}\), in such a way that allows us to study the inner algebraic structure of \((\mathcal{V}, \mathcal{W})\). In particular, the homogeneous indecomposability, the homogeneous semisimplicity and the homogeneous simplicity of \((\mathcal{V}, \mathcal{W})\) are characterized in terms of this graph. Some applications to the theory of non-associative algebras and modules over algebras are also provided.
MSC:
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 17C70 | Super structures |
| 17C20 | Simple, semisimple Jordan algebras |
| 17C55 | Finite-dimensional structures of Jordan algebras |
References:
| [1] | Bahturin, Y. A.; Goze, M.; Remm, E., Group gradings on filiform Lie algebras, Comm. Algebra, 44, 1, 40-62 (2016) · Zbl 1400.17005 |
| [2] | Bahturin, Y. A.; Shestakov, I. P.; Zaicev, M., Gradings on simple Jordan and Lie algebras, J. Algebra, 283, 2, 849-868 (2005) · Zbl 1066.17018 |
| [3] | Cabrera, Y.; Siles, M.; Velasco, M. V., Evolution algebras of arbitrary dimension and their decompositions, Linear Algebra Appl., 495, 122-162 (2016) · Zbl 1395.17079 |
| [4] | Calderón, A. J.; Draper, C. C.; Martín, C., Gradings on the Kac superalgebra, J. Algebra, 324, 12, 3249-3261 (2010) · Zbl 1226.17027 |
| [5] | Draper, C., Models on the Lie algebra \(f_4\), Linear Algebra Appl., 428, 11-12, 2813-2839 (2008) · Zbl 1140.17006 |
| [6] | Elduque, A.; Kochetov, M., Gradings on the Lie algebra \(D_4\) revisited, J. Algebra, 441, 441-474 (2015) · Zbl 1394.17064 |
| [7] | Elduque, A.; Labra, A., Evolution algebras and graphs, J. Algebra Appl., 14, 7, 10 (2015), 1550103 · Zbl 1356.17028 |
| [8] | Havlíček, M.; Patera, J.; Pelantová, E., On Lie gradings III, Linear Algebra Appl., 314, 1-47 (2000) · Zbl 1017.17027 |
| [9] | Humphreys, J. E., Introduction to Lie algebras and representation theory, (Graduate Texts in Mathematics, vol. 9 (1978), Springer-Verlag: Springer-Verlag New York-Berlin), Second printing, revised · Zbl 0447.17001 |
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.
