Quasi-Frobenius Lie algebra

In mathematics, a quasi-Frobenius Lie algebra


 * $$(\mathfrak{g},[\,\,\,,\,\,\,],\beta )$$

over a field $$k$$ is a Lie algebra
 * $$(\mathfrak{g},[\,\,\,,\,\,\,] )$$

equipped with a nondegenerate skew-symmetric bilinear form


 * $$\beta : \mathfrak{g}\times\mathfrak{g}\to k$$, which is a Lie algebra 2-cocycle of $$\mathfrak{g}$$ with values in $$k$$. In other words,


 * $$ \beta \left(\left[X,Y\right],Z\right)+\beta \left(\left[Z,X\right],Y\right)+\beta \left(\left[Y,Z\right],X\right)=0 $$

for all $$X$$, $$Y$$, $$Z$$ in $$\mathfrak{g}$$.

If $$\beta$$ is a coboundary, which means that there exists a linear form $$f : \mathfrak{g}\to k$$ such that
 * $$\beta(X,Y)=f(\left[X,Y\right]),$$

then
 * $$(\mathfrak{g},[\,\,\,,\,\,\,],\beta )$$

is called a Frobenius Lie algebra.

Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form
If $$(\mathfrak{g},[\,\,\,,\,\,\,],\beta )$$ is a quasi-Frobenius Lie algebra, one can define on $$\mathfrak{g}$$ another bilinear product $$\triangleleft$$ by the formula
 * $$ \beta \left(\left[X,Y\right],Z\right)=\beta \left(Z \triangleleft Y,X \right) $$.

Then one has $$\left[X,Y\right]=X \triangleleft Y-Y \triangleleft X$$ and
 * $$(\mathfrak{g}, \triangleleft)$$

is a pre-Lie algebra.