×

zbMATH — the first resource for mathematics

Closed subsets of root systems and regular subalgebras. (English) Zbl 07262043
Summary: We describe an algorithm for classifying the closed subsets of a root system, up to conjugation by the associated Weyl group. Our algorithm is implemented in the language of the computer algebra system GAP4. We discuss the implementation and give runtimes on some sample inputs. The classification of the closed subsets of an irreducible root system is closely related to the classification of the regular subalgebras, up to inner automorphism, of the corresponding simple Lie algebra. We show 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 has tables describing the subalgebras that were obtained.
Reviewer: Reviewer (Berlin)
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
Software:
GAP; images; SLA
PDF BibTeX XML Cite
Full Text: DOI
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 06690823
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.