On structurable algebras
type: Item
instance of: scholarly article
title: On structurable algebras
zbMATH Open document ID: 0552.17001
DOI: 10.1016/0021-8693(85)90131-0
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: 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 \((M,^-)\cong (A/R,^-)\).
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 17A60
zbMATH DE Number: 3879103
zbMATH Keywords: characteristic zero
zbMATH Keywords: unital involutorial algebra
zbMATH Keywords: unital Jordan algebras
zbMATH Keywords: structurable algebra
zbMATH Keywords: maximal Penico solvable ideal
author: Schafer, R. D.
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90131-0
OpenAlex ID: W2017101385
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description: scientific article; zbMATH DE number 3879103