ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 66,   2   (1997)
pp.   151-201

NONDEGENERATE INVARIANT BILINEAR FORMS ON NONASSOCIATIVE ALGEBRAS
M. BORDEMANN

Abstract.  A bilinear form $f$ on a nonassociative algebra $A$ is said to be invariant iff $f(ab,c) = f(a,bc)$ for all $a,b,c \in A$. Finite-dimensional complex semisimple Lie algebras (with their Killing form) and certain associative algebras (with a trace) carry such a structure. We discuss the ideal structure of $A$ if $f$ is nondegenerate and introduce the notion of $T$-extension of an arbitrary algebra $B$ (i.e. by its dual space $B$) where the natural pairing gives rise to a nondegenerate invariant symmetric bilinear form on $A:= B\oplus B$. The $T$-extension involves the third scalar cohomology $H^3(B,\field)$ if $B$ is Lie and the second cyclic cohomology $HC^2(B)$ if $B$ is associative in a natural way. Moreover, we show that every nilpotent finite-dimensional algebra $A$ over an algebraically closed field carrying a nondegenerate invariant symmetric bilinear form is a suitable $T$-extension. As a Corollary, we prove that every complex Lie algebra carrying a nondegenerate invariant symmetric bilinear form is always a special type of Manin pair in the sense of Drinfel'd but not always isomorphic to a Manin triple. Examples involving the Heisenberg and filiform Lie algebras (whose third scalar cohomology is computed) are discussed.

AMS subject classification.  15A63, 17B30, 17B56, 18G60; Secondary 16D25, 17A01, 17A30
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