Quasi-Frobenius Lie algebra

In mathematics, a quasi-Frobenius Lie algebra

:$(\backslash mathfrak\{g\},[\backslash ,\backslash ,\backslash ,,\backslash ,\backslash ,\backslash ,],\backslash beta\; )$

over a field $k$ is a Lie algebra

:$(\backslash mathfrak\{g\},[\backslash ,\backslash ,\backslash ,,\backslash ,\backslash ,\backslash ,]\; )$

equipped with a nondegenerate skew-symmetric bilinear form

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

::$\backslash beta\; \backslash left(\backslash left[X,Y\backslash right],Z\backslash right)+\backslash beta\; \backslash left(\backslash left[Z,X\backslash right],Y\backslash right)+\backslash beta\; \backslash left(\backslash left[Y,Z\backslash right],X\backslash right)=0$

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

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

:$\backslash beta(X,Y)=f(\backslash left[X,Y\backslash right]),$

then

:$(\backslash mathfrak\{g\},[\backslash ,\backslash ,\backslash ,,\backslash ,\backslash ,\backslash ,],\backslash beta\; )$

is called a Frobenius Lie algebra.