论文标题
计算子词法和$ \ mathbb {z} _2 $ - 在有限字段上简单谎言代数
Computing subalgebras and $\mathbb{Z}_2$-gradings of simple Lie algebras over finite fields
论文作者
论文摘要
本文在有限字段上介绍了两种用于Lie代数的新算法，并将其应用于$ \ Mathbb {f} _2 $带有两个元素的$ 20 $的已知简单谎言代数，最多为20美元。第一种算法是在有限的特征$ 2 $的有限字段上建造$ \ mathbb {z} _2 $ gradings的新方法。使用此情况，我们观察到，每一个已知的简单谎言尺寸为$ 20 $，超过$ \ mathbb {f} _2 $具有$ \ mathbb {z} _2 $ - grading，我们确定相关的简单谎言lie superalgebras。第二个算法使我们能够在有限场上计算Lie代数的所有子代数。我们将其应用于计算尺寸的最大亚位bras和简单的尺寸代数的简单亚质子，最多超过$ \ m \ m \ m \ mathbb {f} _2 $（除$ 15 $ dimensional-dimensional-dimensional zassenhaus algebra）。
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).
