In mathematics, a quasi-Frobenius Lie algebra
![(\mathfrak{g},[\,\,\,,\,\,\,],\beta )](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8997357d58354e97fdc54f19ed28ca05198eef8)
over a field
is a Lie algebra
![(\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- of
with values in
. 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](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
![\beta(X,Y)=f(\left[X,Y\right]),](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a38800adb937135bd582aaccc0abe763739be7f)
then
![(\mathfrak{g},[\,\,\,,\,\,\,],\beta )](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8997357d58354e97fdc54f19ed28ca05198eef8)
is called a Frobenius Lie algebra.
Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form[]
If
is a quasi-Frobenius Lie algebra, one can define on
another bilinear product
by the formula
.
Then one has
and
![(\mathfrak{g}, \triangleleft)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b3f56973dac4773ab849c978b9512096a9b996)
is a pre-Lie algebra.
See also[]
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.