In mathematics, a quasi-Frobenius Lie algebra
over a field is a Lie algebra
equipped with a nondegenerate skew-symmetric bilinear form
, which is a Lie algebra 2-cocycle of with values in . In other words,
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 formula .
Then one has and
is a pre-Lie algebra.
See also Lie coalgebra Lie bialgebra Lie algebra cohomology Frobenius algebra Quasi-Frobenius ring
References Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge .