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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 $f\left(ab,c\right)=f\left(a,bc\right)$ for all $a,b,c\in A·$ For example, the form $f\left(a,b\right):=trace\left(ab\right)$ 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 $\left(A,f\right)$ a “metrised algebra”. Let $A$ be any algebra, ${A}^{*}$ be its dual space, and $w:A×{A}^{*}\to {A}^{*}$ be a bilinear mapping. Then the author defines the “${T}^{*}$-extension” of $A$ as the space $A\oplus {A}^{*}$ with a twisted multiplication which depends on $w·$ Under suitable conditions on $w$ this new algebra ${T}_{w}^{*}A$ is a metrised algebra, and properties such as solvability and nilpotence are preserved in going from $A$ to ${T}_{w}^{*}A·$

The main part of the paper studies various properties of these ${T}^{*}$-extensions and gives criteria for when a metrised algebra can be embedded isometrically in a suitable ${T}^{*}$-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 ${T}^{*}$-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.

##### MSC:
 17A60 Structure theory of general nonassociative rings and algebras 15A63 Quadratic and bilinear forms, inner products 17A01 General theory of nonassociative algebra
metrised algebra