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
- , 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
References
- Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4
- Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.