Quasi-Frobenius Lie algebra

In mathematics, a quasi-Frobenius Lie algebra

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

over a field ${\displaystyle k}$ is a Lie algebra

${\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,])}$

equipped with a nondegenerate skew-symmetric bilinear form

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

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

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

If ${\displaystyle \beta }$ is a coboundary, which means that there exists a linear form ${\displaystyle f:{\mathfrak {g}}\to k}$ such that

${\displaystyle \beta (X,Y)=f(\left[X,Y\right]),}$

then

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

is called a Frobenius Lie algebra.