Quasi-Frobenius Lie algebra
In mathematics, a quasi-Frobenius Lie algebra
over a field is a Lie algebra
equipped with a nondegenerate skew-symmetric bilinear form
for all,, in.
If is a coboundary, which means that there exists a linear form such that
then
is called a Frobenius Lie algebra.
Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form
If is a quasi-Frobenius Lie algebra, one can define on another bilinear product by the formulaThen one has
and
is a pre-Lie algebra.