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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,cA· 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 * be its dual space, and w:A×A * A * be a bilinear mapping. Then the author defines the “T * -extension” of A as the space AA * 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.

17A60Structure theory of general nonassociative rings and algebras
15A63Quadratic and bilinear forms, inner products
17A01General theory of nonassociative algebra