Trace forms for structurable algebras (Q1119994)

The authors investigate trace forms on a finite-dimensional structurable algebra \((A,^-)\) over a field of characteristic 0. A Lie algebra \(K(A,^-)\) corresponds to the structurable algebra \((A,^-)\) by a generalization of the Kantor-Tits-Koecher construction. Two trace forms are considered in the paper. The first is the symmetric bilinear form \(<x,y>=tr(L_{x\bar y+y\bar x})\) and the second is a trace form f(x,y) which is obtained by restricting the Killing form of \(K(A,^-)\) on \((A,^-).\) The authors establish that the trace form f is an invariant form. The radical of \((A,^-)\) is the radical of the form f and coincides with the radical of the form \(<, >\). Some known structure results on semisimple and radical algebras are reproved using the trace forms. Results on Hermitian idempotents are obtained, too.