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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.


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