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

• 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. 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.