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Closed subsets of root systems and regular subalgebras. (English) Zbl 1467.17005

A closed subset of a root system \(\Phi\) is a subset \(T\) such that if \(\alpha,\beta\in T\) and \(\alpha+\beta\in\Phi\) then \(\alpha+\beta\in T\). Closed subsets of root system are integral to the classification of regular semisimple subalgebras of the exceptional Lie algebras, they also find applications to classification of the reflection subgroups of finite and affine Weyl groups, and appear in the theory of Chevalley groups.
The paper describes an algorithm for classifying the closed subsets of a root system, up to conjugation by the associated Weyl group. The algorithm is implemented in the language of the computer algebra system GAP4. The paper discusses the implementation and gives runtimes on some sample inputs, and shows how to obtain the regular subalgebras corresponding to a given closed set for the root systems of rank \(3\). The version of this paper on the arXiv has tables describing the subalgebras that were obtained.

MSC:

17B22 Root systems
17-08 Computational methods for problems pertaining to nonassociative rings and algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
17B30 Solvable, nilpotent (super)algebras
17B05 Structure theory for Lie algebras and superalgebras

Software:

SLA; GAP; images
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Betten, A., How fast can we compute orbits of groups?, (Davenport, J. H.; Kauers, M.; Labahn, G.; Urban, J., Mathematical Software. Mathematical Software, ICMS 2018. Mathematical Software. Mathematical Software, ICMS 2018, LNCS, vol. 10931 (2018)), 62-70 · Zbl 1395.05192
[2] Borevich, Z. I., Description of the subgroups of the general linear group that contain the group of diagonal matrices, (Rings and Modules. Rings and Modules, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 64 (1976)), 12-29, (Russian) · Zbl 0364.20056
[3] Borevich, Z. I., On the question of the enumeration of finite topologies, (Modules and Representations. Modules and Representations, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 71 (1977)), 47-65, (Russian) · Zbl 0426.54014
[4] Bourbaki, N., Groupes et Algèbres de Lie, Chapitres 4, 5 et 6 (1968), Hermann: Hermann Paris · Zbl 0483.22001
[5] Brinkmann, G.; McKay, B. D., Posets on up to 16 points, Order, 19, 2, 147-179 (2002) · Zbl 1006.06003
[6] Đoković, D.Ž.; Check, P.; Hée, J.-Y., On closed subsets of root systems, Can. Math. Bull., 37, 338-345 (1994) · Zbl 0808.17015
[7] Douglas, A.; de Graaf, W. A., Closed subsets of root systems and regular subalgebras (2019)
[8] Douglas, A.; Repka, J., The Levi decomposable subalgebras of \(C_2\), J. Math. Phys., 56, Article 051703 pp. (2015) · Zbl 1394.17023
[9] Douglas, A.; Repka, J., The subalgebras of \(\mathfrak{so}(4, \mathbb{C})\), Commun. Algebra, 44, 12, 5269-5286 (2016) · Zbl 1403.17005
[10] Douglas, A.; Repka, J., A classification of the subalgebras of \(A_2\), J. Pure Appl. Algebra, 220, 6, 2389-2413 (2016) · Zbl 1377.17007
[11] Douglas, A.; Repka, J., The subalgebras of the rank two symplectic Lie algebra, Linear Algebra Appl., 527, 303-348 (2017) · Zbl 1419.17016
[12] Dyer, M. J.; Lehrer, G. I., Reflection subgroups of finite and affine Weyl groups, Trans. Am. Math. Soc., 363, 5971-6005 (2011) · Zbl 1243.20051
[13] Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras, Mat. Sb. N.S., 30(72), 2, 349-462 (1952), (Russian) · Zbl 0048.01701
[14] GAP - groups, algorithms, and programming (2018), Version 4.9.2
[15] de Graaf, W. A., Lie Algebras: Theory and Algorithms (2000), North Holland Mathematical Library, Elsevier: North Holland Mathematical Library, Elsevier Amsterdam · Zbl 1122.17300
[16] de Graaf, W. A., Computation with Linear Algebraic Groups (2017), CRC Press: CRC Press Boca Raton · Zbl 1518.14001
[17] de Graaf, W. A., SLA, computing with simple Lie algebras (2019), Version 1.5.2 (Refereed GAP package)
[18] Harebov, A. L.; Vavilov, H. A., On the lattice of subgroups of Chevalley groups containing a split maximal torus, Commun. Algebra, 24, 1, 109-133 (1996) · Zbl 0857.20023
[19] Humphreys, J. E., Introduction to Lie Algebras and Representation Theory (1972), Springer-Verlag: Springer-Verlag New York · Zbl 0254.17004
[20] Jefferson, C.; Jonauskyte, E.; Pfeiffer, M.; Waldecker, R., Images - minimal and canonical images, a GAP package (2018), Version 1.1.0
[21] Jefferson, C.; Jonauskyte, E.; Pfeiffer, M.; Waldecker, R., Minimal and canonical images, J. Algebra, 521, 481-506 (2019) · Zbl 1439.20001
[22] Lorente, M.; Gruber, B., Classification of semisimple subalgebras of simple Lie algebras, J. Math. Phys., 13, 1639-1663 (1972) · Zbl 0241.17006
[23] Malyshev, F. M., Decompositions of root systems, Math. Notes Acad. Sci. USSR, 27, 418-421 (1980) · Zbl 0456.17007
[24] Mayanskiy, E., The subalgebras of \(G_2\)
[25] Seress, Ákos, Permutation Group Algorithms, Cambridge Tracts in Mathematics, vol. 152 (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1028.20002
[26] Sopkina, E. A., On the sum of roots of a closed set, J. Math. Sci., 124, 1, 4832-4836 (2004) · Zbl 1080.17006
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