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de Graaf, Willem A.

Exploring Lie theory with \(\mathsf{GAP}\). (English) Zbl 1526.17032

Detinko, Alla (ed.) et al., Computational aspects of discrete subgroups of Lie groups. Virtual conference, Institute for Computational and Experimental Research in Mathematics, ICERM, Providence, Rhode Island, USA June 14–18, 2021. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 783, 27-46 (2023).
Summary: We illustrate the Lie theoretic capabilities of the computational algebra system \(\mathsf{GAP}4\) by reporting on results on nilpotent orbits of simple Lie algebras that have been obtained using computations in that system. Concerning reachable elements in simple Lie algebras we show by computational means that the simple Lie algebras of exceptional type have the Panyushev property. We computationally prove two propositions on the dimension of the abelianization of the centralizer of a nilpotent element in simple Lie algebras of exceptional type. Finally we obtain the closure ordering of the orbits in the null cone of the spinor representation of the group \(\mathrm{Spin}_{13}(\mathbb{C})\). All input and output of the relevant \(\mathsf{GAP}\) sessions is given.
For the entire collection see [Zbl 1511.20004].

MSC:

17B45 Lie algebras of linear algebraic groups
17B08 Coadjoint orbits; nilpotent varieties
20G05 Representation theory for linear algebraic groups
17-08 Computational methods for problems pertaining to nonassociative rings and algebras

Keywords:

Lie groups; Lie algebras; nilpotent orbits; computational methods

Software:

NoCK; GAP; CoReLG; Sophus; LiePRing; FPLSA; SLA; QuaGroup; LieAlgDB; LieRing; HNC
PDFBibTeX XMLCite
\textit{W. A. de Graaf}, Contemp. Math. 783, 27--46 (2023; Zbl 1526.17032)
Full Text: DOI arXiv

References:

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