Simple Lie algebras and graphs
A finite-dimensional vector space \(V\) over \(GF (2)\) with an alternating bilinear form is called a symplectic space and if \((u,v) = 0\) for all \(v \in V\) implies that \(u = 0\) then \(V\) is called nondegenerate. If \(S\) is a subset of \(V\) consisting of nonzero vectors such that if \(a,b \in S\) and \((a,b) = 1\) imply that \(a + b \in S\), then it is called a \(J\)-system. For each \(J\)-system in a symplectic space \(V\), Kaplansky defined a Lie algebra \({\mathfrak L} (S)\) over \(GF (2)\) and gave a list of infinite families of \(J\)-systems \(S\) in nondegenerate symplectic spaces of cardinals \({1 \over 2} n(n + 1)\), \(4^ n - 1\), and \(2^{n - 1} (2^ n - \varepsilon)\) for \(\varepsilon = \pm 1\) and claimed that the corresponding \({\mathfrak L} (S)\) are central simple Lie algebras. The authors view these algebras in terms of graphs, classify them using a theorem of Shult characterizing graphs with the cotriangle condition and observe a connection with Fischer's theorem on groups generated by 3- transpositions. They also prove that Kaplansky's list is complete.
scientific article; zbMATH DE number 701275