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.

See also

References

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