Quasi-Frobenius Lie algebra

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Quasi-Frobenius Lie algebra

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In mathematics, a quasi-Frobenius Lie algebra

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

over a field $k$ is a Lie algebra

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

equipped with a nondegenerate skew-symmetric bilinear form

$\beta : \mathfrak{g}\times\mathfrak{g}\to k$ , 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 $\mathfrak{g}$ .

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

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

then

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

is called a Frobenius Lie algebra.

Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form

If $(\mathfrak{g},[\,\,\,,\,\,\,],\beta )$ is a quasi-Frobenius Lie algebra, one can define on $\mathfrak{g}$ another bilinear product $\triangleleft$ by the formula

$\beta \left(\left[X,Y\right],Z\right)=\beta \left(Z \triangleleft Y,X \right)$ .

Then one has $\left[X,Y\right]=X \triangleleft Y-Y \triangleleft X$ and

$(\mathfrak{g}, \triangleleft)$

is a pre-Lie algebra.

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.

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