In mathematics, a quasi-Frobenius Lie algebra
![{\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8997357d58354e97fdc54f19ed28ca05198eef8)
over a field
is a Lie algebra
![{\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/007a3516c1d50e9723cc7ebb671c165cc9458ade)
equipped with a nondegenerate skew-symmetric bilinear form
, which is a Lie algebra 2-cocycle of
with values in
. 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b23be708e4579afbecba44d2d4d9db61ac80d1db)
for all
,
,
in
.
If
is a coboundary, which means that there exists a linear form
such that
![{\displaystyle \beta (X,Y)=f(\left[X,Y\right]),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a38800adb937135bd582aaccc0abe763739be7f)
then
![{\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8997357d58354e97fdc54f19ed28ca05198eef8)
is called a Frobenius Lie algebra.