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Closed subsets of root systems and regular subalgebras. (English) Zbl 07262043
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:
GAP; SLA; images
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References:
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