de Graaf, Willem A. Computing representatives of nilpotent orbits of \(\theta\)-groups. (English) Zbl 1222.17002 J. Symb. Comput. 46, No. 4, 438-458 (2011). Summary: Two algorithms are described for finding representatives of the nilpotent orbits of a \(\theta\)-group, corresponding to a \(\mathbb Z/m\mathbb Z\)-grading of a simple Lie algebra \(\mathfrak g\) over \(\mathbb C\). The first algorithm uses the classification of the nilpotent orbits in \(\mathfrak g\), an idea also used in [D. Ž. Đoković, J. Algebra 112, No. 2, 503–524 (1988; Zbl 0639.17005)]. To get a working algorithm from it, we combine this idea with a new method for computing normal \(\mathfrak{sl}_2\)-triples. The second algorithm is based on Vinberg’s theory of carrier algebras, that reduces the classification of nilpotent orbits to the classification of subalgebras of \(\mathfrak g\) with certain properties. We describe an algorithm for the latter problem, using a method for classifying \(\pi \)-systems. The algorithms have been implemented in the computer algebra system GAP (inside the package SLA). We briefly comment on their performance. At the end of the paper the algorithms are used to study the nilpotent orbits of \(\theta\)-groups, where \(\theta\) is an \(N\)-regular automorphism of a simple Lie algebra of exceptional type. Cited in 11 Documents MSC: 17-08 Computational methods for problems pertaining to nonassociative rings and algebras 17B08 Coadjoint orbits; nilpotent varieties 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras 17B25 Exceptional (super)algebras Keywords:reductive algebraic groups; Lie algebras; orbits; algorithms Citations:Zbl 0639.17005 Software:HNC; CHEVIE; SLA; GAP; Magma PDFBibTeX XMLCite \textit{W. A. de Graaf}, J. Symb. Comput. 46, No. 4, 438--458 (2011; Zbl 1222.17002) Full Text: DOI arXiv References: [1] Antonyan, L.V., 1987. Homogeneous nilpotent elements of periodically graded semisimple Lie algebras. 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