Some new simple Lie algebras in characteristic 2
In a previous paper [Linear Algebra Appl.\ 382, 195--209 (2004; Zbl 1045.17001)] the author describes effective algorithms for computing the automorphism groups of finite-dimensional solvable Lie algebras over finite fields and for testing isomorphisms between these algebras. The first part of the paper under review contains deterministic and probabilistic algorithms for the same tasks but now for not necessarily solvable Lie algebras based on certain nice presentations of the algebras under consideration. The algorithms are particularly effective for simple or almost simple Lie algebras and therefore complement the ones in the previous paper. Recently, the classification of finite-dimensional simple Lie algebras has been completed for algebraically closed fields of characteristic \(p>3\). It is known from a wealth of examples that a classification in characteristic \(2\) or \(3\) will look very different and so any ideas on how such classifications might look like would be helpful. \textit{M. Vaughan-Lee} [LMS J.\ Comput.\ Math.\ 9, 174--192 electronic only (2006; Zbl 1162.17009)] classified the simple Lie algebras of dimension at most \(9\) over the field \(\mathbb{F}_2\) with two elements. In the second part of her paper the author shows how the isomorphism test from the first part can be used in a computer-based randomized search for new simple Lie algebras of dimension up to \(20\) over \(\mathbb{F}_2\). It is pointed out that the author's approach does not yield a complete classification but that it enabled the author to find two simple Lie algebras in dimension \(15\) and four simple Lie algebras in dimension \(16\) that were previously unknown. The new simple Lie algebras in dimension \(15\) arose as constituents of subalgebras of a simple Lie algebra found by \textit{I.\ Kaplansky} thirty years ago [Some simple Lie algebras of characteristic 2, in: Lie algebras and related topics, Proc.\ Conf., New Brunswick/USA 1981, Lect.\ Notes Math.\ 933, 127--129 (1982; Zbl 0493.17005)] and the new simple Lie algebras in dimension \(16\) arose as constituents of subalgebras of \(\mathfrak{sl}_8 (\mathbb{F}_2)\). Neither of these Lie algebras is restricted. The paper concludes by listing the isomorphism classes of all known simple Lie algebras of dimension at most \(20\) (together with certain numerical invariants as the dimension of their derivation algebras and the order of their automorphism groups) and a description of explicit generators for the new simple Lie algebras.
scientific article; zbMATH DE number 5770524