## 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

metrised algebra
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