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Nondegenerate invariant bilinear forms on nonassociative algebras. (English) Zbl 1014.17003

Let \(A\) be a (not necessarily associative) algebra over a field \(K\) with a nondegenerate bilinear form \(f\) which is invariant in the sense that \(f(ab,c)=f(a,bc)\) for all \(a,b,c\in A.\) For example, the form \(f(a,b):=trace(ab)\) on a matrix algebra, and the Killing form on a Lie algebra are invariant. If, in addition, \(f\) is symmetric, then the author calls the pair \((A,f)\) a “metrised algebra”. Let \(A\) be any algebra, \(A^{\ast}\) be its dual space, and \(w:A\times A^{\ast }\rightarrow A^{\ast}\) be a bilinear mapping. Then the author defines the “\(T^{\ast}\)-extension” of \(A\) as the space \(A\oplus A^{\ast}\) with a twisted multiplication which depends on \(w.\) Under suitable conditions on \(w\) this new algebra \(T_{w}^{\ast}A\) is a metrised algebra, and properties such as solvability and nilpotence are preserved in going from \(A\) to \(T_{w}^{\ast}A.\)
The main part of the paper studies various properties of these \(T^{\ast} \)-extensions and gives criteria for when a metrised algebra can be embedded isometrically in a suitable \(T^{\ast}\)-extension. For example, the author shows that every finite-dimensional nilpotent metrised algebra over an algebraically closed field of characteristic not 2 is isometric to a nondegenerate ideal of codimension 1 in a \(T^{\ast}\)-extension of a nilpotent algebra of nilindex roughly half of the nilindex of \(A.\) The author explains how known theorems for associative and Lie algebras generalize in this context.
Reviewer: J.D.Dixon (Ottawa)

MSC:

17A60 Structure theory for nonassociative algebras
15A63 Quadratic and bilinear forms, inner products
17A01 General theory of nonassociative rings and algebras
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