Quasi-Frobenius Lie algebra

In mathematics, a quasi-Frobenius Lie algebra

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

over a field $k$ is a Lie algebra

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

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

k

. In other words,

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

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

[edit] See also

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.