On structurable algebras
A unital involutorial algebra \((A,^-)\) over a field \(\Phi\) of characteristic \(\neq 2,3\) is called structurable if \([T_ z,V_{x,y}]=V_{T_ zx,y}-V_{x,T_{\bar z}y}\) for all x,y,z\(\in A\), where \(V_{x,y}(z)=(x\bar y)z+(z\bar y)x-(z\bar x)y\) and \(T_ x=V_{x,1}\). In particular, all unital Jordan algebras over \(\Phi\) (with the identity map as involution) are structurable. By a generalization of the Tits-Koecher construction of a Lie algebra from any Jordan algebra, every structurable algebra \((A,^-)\) yields a Lie algebra \(K(A,^-)\). Moreover, any ideal T of \(K(A,^-)\) determines a unique ideal of \((A,^-).\) Let \((A,^-)\) be a finite-dimensional structurable algebra over \(\Phi\) of characteristic 0. Taking as radical of such an algebra its maximal solvable ideal R, the author proves the following: (1) R is also the maximal Penico solvable ideal of \((A,^-)\), and it is the ideal of \((A,^-)\) determined by the radical of \(K(A,^-)\), and it is the ideal of \((A,^-)\) determined by the radical of \(K(A,^-)\). (2) \((A,^-)\) is semisimple if and only if it is a finite direct sum of simple structurable algebras. (Any simple structurable algebra is central simple over its center, and finite-dimensional central simple structurable algebras have been classified by \textit{B. N. Allison} [Math. Ann. 237, 133-156 (1978; Zbl 0368.17001)].) (3) (Wedderburn Principal Theorem) There is a subalgebra \((M,^-)\) of \((A,^-)\) such that \((A,^- )=R\oplus (M,^-)\) and \((M,^-)\cong (A/R,^-)\).
scientific article; zbMATH DE number 3879103
scientific article