Quasi-Frobenius Lie algebra

In mathematics, a quasi-Frobenius Lie algebra

:(\mathfrak, ,\,\,,\,\,\,\beta )

over a field $k$ is a Lie algebra
:(\mathfrak, ,\,\,,\,\,\,)

equipped with a nondegenerate skew-symmetric bilinear form

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

:: \beta \left(\left ,Y\rightZ\right)+\beta \left(\left ,X\rightY\right)+\beta \left(\left ,Z\rightX\right)=0

for all $X$, $Y$, $Z$ in $\mathfrak$.

If $\beta$ is a coboundary, which means that there exists a linear form $f : \mathfrak\to k$ such that
:\beta(X,Y)=f(\left ,Y\right,
then
:(\mathfrak, ,\,\,,\,\,\,\beta )
is called a Frobenius Lie algebra.

Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form

If (\mathfrak, ,\,\,,\,\,\,\beta ) is a quasi-Frobenius Lie algebra, one can define on $\mathfrak$ another bilinear product $\triangleleft$ by the formula
:: \beta \left(\left ,Y\rightZ\right)=\beta \left(Z \triangleleft Y,X \right) .

Then one has
\left ,Y\rightX \triangleleft Y-Y \triangleleft X and
:$(\mathfrak, \triangleleft)$
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 {{isbn, 0-521-55884-0.

Lie algebras
Coalgebras
Symplectic topology