Lie algebras, 2-groups and cotriangular spaces
Let \((P, L)\) be a partial linear space with the point set \(P\) and the set \(L\) of lines carrying \(3\) points. If one provides each line \( l = \{x, y, z\}\) with a cyclic orientation \(\sigma ( l)\) putting \(\sigma ( l) = \{x, y, z\}\) or \(\sigma ( l) = \{y, x, z\}\), then one can construct for a field \(\mathbb F\) of characteristic different from \(2\) an algebra with bilinear multiplication taking the points of \(P\) as basis of an \(\mathbb F\)-vector space defining the multiplication on this basis by \([x, y] = 0\) if \(x = y\) or \(x, y\) noncollinear and either \([x, y] = z\) or \([x, y] = -z\) depending on whether \( l =\{x, y, z\}\in L\) and \(\sigma ( l) = \{x, y, z\}\) or \( l =\{x, y, z\}\in L\) and \(\sigma ( l) = \{y, x, z\}\). The authors call these algebras Kaplansky algebras and classify the partial linear spaces \((P, L)\), the corresponding Kaplanky algebras of which are Lie algebras. The authors determine all such Lie algebras if they are simple.
scientific article; zbMATH DE number 6045764
scientific article