{"docId":10193,"paperId":10193,"url":"https:\/\/cm.episciences.org\/10193","doi":"10.46298\/cm.10193","journalName":"Communications in Mathematics","issn":"1804-1388","eissn":"2336-1298","volume":[{"vid":622,"name":"Volume 30 (2022), Issue 2 (Special Issue: CIMPA School \"Nonassociative Algebras and Its Applications\", Madagascar 2021)"}],"section":[],"repositoryName":"arXiv","repositoryIdentifier":"2205.03155","repositoryVersion":2,"repositoryLink":"https:\/\/arxiv.org\/abs\/2205.03155v2","dateSubmitted":"2022-10-24 16:54:06","dateAccepted":"2022-10-26 12:34:36","datePublished":"2022-10-28 11:32:03","titles":["Computing subalgebras and $\\mathbb{Z}_2$-gradings of simple Lie algebras over finite fields"],"authors":["Eick, Bettina","Moede, Tobias"],"abstracts":["This paper introduces two new algorithms for Lie algebras over finite fields and applies them to the investigate the known simple Lie algebras of dimension at most $20$ over the field $\\mathbb{F}_2$ with two elements. The first algorithm is a new approach towards the construction of $\\mathbb{Z}_2$-gradings of a Lie algebra over a finite field of characteristic $2$. Using this, we observe that each of the known simple Lie algebras of dimension at most $20$ over $\\mathbb{F}_2$ has a $\\mathbb{Z}_2$-grading and we determine the associated simple Lie superalgebras. The second algorithm allows us to compute all subalgebras of a Lie algebra over a finite field. We apply this to compute the subalgebras, the maximal subalgebras and the simple subquotients of the known simple Lie algebras of dimension at most $16$ over $\\mathbb{F}_2$ (with the exception of the $15$-dimensional Zassenhaus algebra)."],"keywords":["Mathematics - Rings and Algebras","17B05, 17B50, 17B70, 17-08"]}