Bettina Eick ; Tobias Moede - Computing subalgebras and $\mathbb{Z}_2$-gradings of simple Lie algebras over finite fields

cm:10193 - Communications in Mathematics, October 28, 2022, Volume 30 (2022), Issue 2 (Special Issue: CIMPA School "Nonassociative Algebras and Its Applications", Madagascar 2021) - https://doi.org/10.46298/cm.10193
Computing subalgebras and $\mathbb{Z}_2$-gradings of simple Lie algebras over finite fields

Authors: Bettina Eick ; Tobias Moede

    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).


    Volume: Volume 30 (2022), Issue 2 (Special Issue: CIMPA School "Nonassociative Algebras and Its Applications", Madagascar 2021)
    Published on: October 28, 2022
    Accepted on: October 26, 2022
    Submitted on: October 24, 2022
    Keywords: Mathematics - Rings and Algebras,17B05, 17B50, 17B70, 17-08

    Share

    Consultation statistics

    This page has been seen 32 times.
    This article's PDF has been downloaded 21 times.