Computing subalgebras and \(\mathbb{Z}_2\)-gradings of simple Lie algebras over finite fields. (English) Zbl 07897667
Summary: 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).
MSC:
| 17-08 | Computational methods for problems pertaining to nonassociative rings and algebras |
| 17B05 | Structure theory for Lie algebras and superalgebras |
| 17B50 | Modular Lie (super)algebras |
| 17B70 | Graded Lie (super)algebras |
Software:
GAPReferences:
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