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,

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

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

$\beta (X,Y)=f([X,Y]),$

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 $◃$ by the formula

$\beta ([X,Y],Z)=\beta (Z◃Y,X)$.

Then one has $[X,Y]=X◃Y-Y◃X$ and

$(\mathfrak{𝔤},◃)$

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.