Quasi-Frobenius Lie algebra
In mathematics, a quasi-Frobenius Lie algebra
- [math]\displaystyle{ (\mathfrak{g},[\,\,\,,\,\,\,],\beta ) }[/math]
over a field [math]\displaystyle{ k }[/math] is a Lie algebra
- [math]\displaystyle{ (\mathfrak{g},[\,\,\,,\,\,\,] ) }[/math]
equipped with a nondegenerate skew-symmetric bilinear form
- [math]\displaystyle{ \beta : \mathfrak{g}\times\mathfrak{g}\to k }[/math], which is a Lie algebra 2-cocycle of [math]\displaystyle{ \mathfrak{g} }[/math] with values in [math]\displaystyle{ k }[/math]. In other words,
- [math]\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 }[/math]
for all [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], [math]\displaystyle{ Z }[/math] in [math]\displaystyle{ \mathfrak{g} }[/math].
If [math]\displaystyle{ \beta }[/math] is a coboundary, which means that there exists a linear form [math]\displaystyle{ f : \mathfrak{g}\to k }[/math] such that
- [math]\displaystyle{ \beta(X,Y)=f(\left[X,Y\right]), }[/math]
then
- [math]\displaystyle{ (\mathfrak{g},[\,\,\,,\,\,\,],\beta ) }[/math]
is called a Frobenius Lie algebra.
Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form
If [math]\displaystyle{ (\mathfrak{g},[\,\,\,,\,\,\,],\beta ) }[/math] is a quasi-Frobenius Lie algebra, one can define on [math]\displaystyle{ \mathfrak{g} }[/math] another bilinear product [math]\displaystyle{ \triangleleft }[/math] by the formula
- [math]\displaystyle{ \beta \left(\left[X,Y\right],Z\right)=\beta \left(Z \triangleleft Y,X \right) }[/math].
Then one has [math]\displaystyle{ \left[X,Y\right]=X \triangleleft Y-Y \triangleleft X }[/math] and
- [math]\displaystyle{ (\mathfrak{g}, \triangleleft) }[/math]
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.
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