de Graaf, W. A.; Vinberg, E. B.; Yakimova, O. S. An effective method to compute closure ordering for nilpotent orbits of \(\theta \)-representations. (English) Zbl 1317.17011 J. Algebra 371, 38-62 (2012). Summary: We develop an algorithm for computing the closure of a given nilpotent \(G_{0}\)-orbit in \(\mathfrak g_{1}\), where \(\mathfrak g_{1}\) and \(G_{0}\) are coming from a \(\mathbb Z\) or a \(\mathbb Z/m\mathbb Z\)-grading \(\mathfrak g=\oplus \mathfrak g_{i}\) of a simple complex Lie algebra \(\mathfrak g\). Cited in 5 Documents MSC: 17B08 Coadjoint orbits; nilpotent varieties 17-08 Computational methods for problems pertaining to nonassociative rings and algebras 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras Keywords:simple Lie algebras; finite order automorphisms; nilpotent orbits; closure Software:Magma; SLA; GAP PDFBibTeX XMLCite \textit{W. A. de Graaf} et al., J. Algebra 371, 38--62 (2012; Zbl 1317.17011) Full Text: DOI arXiv References: [1] Antonyan, L. V., Classification of four-vectors of an eight-dimensional space, Trudy Sem. Vektor. Tenzor. Anal., 20, 144-161 (1981) · Zbl 0467.15018 [2] Antonyan, L. V.; Èlashvili, A. G., Classification of spinors of dimension sixteen, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR, 70, 5-23 (1982) · Zbl 0519.17006 [3] Barbasch, D.; Sepanski, M. R., Closure ordering and the Kostant-Sekiguchi correspondence, Proc. Amer. Math. Soc., 126, 1, 311-317 (1998) · Zbl 0896.22004 [4] Beynon, W. M.; Spaltenstein, N., Green functions of finite Chevalley groups of type \(E_n (n = 6, 7, 8)\), J. Algebra, 88, 584-614 (1984) · Zbl 0539.20025 [5] Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system. I. The user language, Computational Algebra and Number Theory. Computational Algebra and Number Theory, London, 1993. Computational Algebra and Number Theory. Computational Algebra and Number Theory, London, 1993, J. Symbolic Comput., 24, 3-4, 235-265 (1997) · Zbl 0898.68039 [6] Collingwood, D. H.; McGovern, W. M., Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold Mathematics Series (1993), Van Nostrand Reinhold Co.: Van Nostrand Reinhold Co. New York · Zbl 0972.17008 [7] Cox, D.; Little, J.; OʼShea, D., Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (1992), Springer-Verlag: Springer-Verlag New York, Heidelberg, Berlin [8] Đoković, Dragomir Ž., The closure diagrams for nilpotent orbits of real forms of \(F_4\) and \(G_2\), J. Lie Theory, 10, 2, 491-510 (2000) · Zbl 0974.17010 [9] Đoković, Dragomir Ž., The closure diagram for nilpotent orbits of the real form EIX of \(E_8\), Asian J. Math., 5, 3, 561-584 (2001) · Zbl 1033.17012 [10] Đoković, Dragomir Ž., The closure diagram for nilpotent orbits of the split real form of \(E_7\), Represent. Theory, 5, 284-316 (2001), (electronic) · Zbl 1050.17007 [11] Đoković, Dragomir Ž., The closure diagrams for nilpotent orbits of real forms of \(E_6\), J. Lie Theory, 11, 2, 381-413 (2001) · Zbl 1049.17006 [12] Đoković, Dragomir Ž., The closure diagrams for nilpotent orbits of the real forms E VI and E VII of \(E_7\), Represent. Theory, 5, 17-42 (2001), (electronic) · Zbl 1031.17004 [13] Đoković, Dragomir Ž., Correction to: “The closure diagrams for nilpotent orbits of the real forms E VI and E VII of <mml:math altimg=”si17.gif“>E7” [Represent. Theory 5 (2001) 17-42], Represent. Theory, 5, 503 (2001), (electronic) [14] Đoković, Dragomir Ž., The closure diagram for nilpotent orbits of the split real form of \(E_8\), Cent. Eur. J. Math., 1, 4, 573-643 (2003), (electronic) · Zbl 1050.17006 [15] Đoković, Dragomir Ž., Corrections for “The closure diagram for nilpotent orbits of the split real form of <mml:math altimg=”si19.gif“>E8” [Cent. Eur. J. Math. 1 (4) (2003) 573-643], Cent. Eur. J. Math., 3, 3, 578-579 (2005), (electronic) · Zbl 1099.17500 [16] Galitski, L. Yu.; Timashev, D. A., On classification of metabelian Lie algebras, J. Lie Theory, 9, 125-156 (1999) · Zbl 0923.17015 [17] GAP - Groups, algorithms, and programming, Version 4.4 (2004) [18] de Graaf, Willem A., Constructing algebraic groups from their Lie algebras, J. Symbolic Comput., 44, 1223-1233 (2009) · Zbl 1180.20038 [19] de Graaf, Willem A., SLA - Computing with simple Lie algebras. A GAP package (2009) [20] de Graaf, Willem A., Computing representatives of nilpotent orbits of \(θ\)-groups, J. Symbolic Comput., 46, 438-458 (2011) · Zbl 1222.17002 [21] Grunewald, Fritz; OʼHalloran, Joyce, Varieties of nilpotent Lie algebras of dimension less than six, J. Algebra, 112, 2, 315-325 (1988) · Zbl 0638.17005 [22] Helgason, Sigurdur, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math., vol. 80 (1978), Academic Press Inc., Harcourt Brace Jovanovich Publishers: Academic Press Inc., Harcourt Brace Jovanovich Publishers New York · Zbl 0451.53038 [23] Kac, V. G., Automorphisms of finite order of semisimple Lie algebras, Funktsional. Anal. i Prilozhen., 3, 3, 94-96 (1969) · Zbl 0274.17002 [24] Kac, V. G., Some remarks on nilpotent orbits, J. Algebra, 64, 1, 190-213 (1980) · Zbl 0431.17007 [25] G. Kempken, Eine Darstellung des Köchers \(\widetilde{A}_K\); G. Kempken, Eine Darstellung des Köchers \(\widetilde{A}_K\) [26] Littelmann, P., An effective method to classify nilpotent orbits, (Algorithms in Algebraic Geometry and Applications. Algorithms in Algebraic Geometry and Applications, Santander, 1994. Algorithms in Algebraic Geometry and Applications. Algorithms in Algebraic Geometry and Applications, Santander, 1994, Progr. Math., vol. 143 (1996), Birkhäuser: Birkhäuser Basel), 255-269 · Zbl 0866.20037 [27] Mizuno, Kenzo, The conjugate classes of unipotent elements of the Chevalley groups \(E_7\) and \(E_8\), Tokyo J. Math., 3, 2, 391-461 (1980) · Zbl 0454.20046 [28] Ohta, T., Orbits, rings of invariants and Weyl groups for classical \(θ\)-groups, Tohoku Math. J., 62, 527-558 (2010) · Zbl 1242.17024 [29] Pervushin, D. D., Invariants and orbits of the standard \((SL_4(C) \times SL_4(C) \times SL_2(C))\)-module, Izv. Ross. Akad. Nauk Ser. Mat.. Izv. Ross. Akad. Nauk Ser. Mat., Izv. Math., 64, 5, 1003-1015 (2000), English translation: · Zbl 0974.22017 [30] Pervushin, D. D., On the closures of the nilpotent orbits of fourth-order matrix pencils, Izv. Ross. Akad. Nauk Ser. Mat.. Izv. Ross. Akad. Nauk Ser. Mat., Izv. Math., 66, 5, 1047-1055 (2002), English translation: · Zbl 1075.14047 [31] Popov, V. L., Two orbits: When is one in the closure of the other?, Mnogomernaya Algebraicheskaya Geometriya. Mnogomernaya Algebraicheskaya Geometriya, Tr. Mat. Inst. Steklova. Mnogomernaya Algebraicheskaya Geometriya. Mnogomernaya Algebraicheskaya Geometriya, Tr. Mat. Inst. Steklova, Proc. Steklov Inst. Math., 264, 1, 146-158 (2009), English translation: · Zbl 1312.14110 [32] Snow, D. M., Weyl group orbits, ACM Trans. Math. Software, 16, 1, 94-108 (1990) · Zbl 0885.22001 [33] Spaltenstein, Nicolas, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Math., vol. 946 (1982), Springer-Verlag: Springer-Verlag Berlin · Zbl 0486.20025 [34] Vergne, M., Cohomologie des algèbres de Lie nilpotentes. Application à lʼétude de la variété des algèbres de Lie nilpotentes, Bull. Soc. Math. France, 98, 81-116 (1970) · Zbl 0244.17011 [35] Vinberg, È. B., The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat.. Izv. Akad. Nauk SSSR Ser. Mat., Math. USSR-Izv., 10, 3, 463-495 (1976), English translation: · Zbl 0371.20041 [36] Vinberg, È. B., Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra, Trudy Sem. Vektor. Tenzor. Anal.. Trudy Sem. Vektor. Tenzor. Anal., Selecta Math. Sov., 6, 15-35 (1987), English translation: · Zbl 0612.17010 [37] Vinberg, È. B.; Gorbatsevich, V. V.; Onishchik, A. L., Lie Groups and Lie Algebras, III, Sovrem. Probl. Mat. Fund. Naprav.. Sovrem. Probl. Mat. Fund. Naprav., Encyclopaedia Math. Sci., vol. 41 (1994), VINITI: VINITI Moskva: Springer: VINITI: VINITI Moskva: Springer Berlin, English translation: · Zbl 0797.22001 [38] Vinberg, È. B.; Popov, V. L., Invariant theory, (Algebraic Geometry, IV. Algebraic Geometry, IV, Itogi Nauki i Tekhniki (1989), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform.: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. Moscow). (Algebraic Geometry, IV. Algebraic Geometry, IV, Itogi Nauki i Tekhniki (1989), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform.: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. Moscow), Encyclopaedia Math. Sci., vol. 55, 137-314 (1994), Springer: Springer Berlin, English translation: [39] Vinberg, È. B.; Èlašvili, A. G., A classification of the three-vectors of nine-dimensional space, Trudy Sem. Vektor. Tenzor. Anal.. Trudy Sem. Vektor. Tenzor. Anal., Selecta Math. Sov., 7, 63-98 (1988), English translation: · Zbl 0648.15021 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.