Let be a (not necessarily associative) algebra over a field with a nondegenerate bilinear form which is invariant in the sense that for all For example, the form on a matrix algebra, and the Killing form on a Lie algebra are invariant. If, in addition, is symmetric, then the author calls the pair a “metrised algebra”. Let be any algebra, be its dual space, and be a bilinear mapping. Then the author defines the “-extension” of as the space with a twisted multiplication which depends on Under suitable conditions on this new algebra is a metrised algebra, and properties such as solvability and nilpotence are preserved in going from to
The main part of the paper studies various properties of these -extensions and gives criteria for when a metrised algebra can be embedded isometrically in a suitable -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 -extension of a nilpotent algebra of nilindex roughly half of the nilindex of The author explains how known theorems for associative and Lie algebras generalize in this context.