Quasi-Frobenius Lie algebra

In mathematics, a quasi-Frobenius Lie algebra

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

over a field ${\displaystyle k}$ is a Lie algebra

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

equipped with a nondegenerate skew-symmetric bilinear form

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

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

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

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

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

then

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

is called a Frobenius Lie algebra.

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

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

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

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

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

is a pre-Lie algebra.

See also

• Lie coalgebra
• Lie bialgebra
• Lie algebra cohomology
• Frobenius algebra
• Quasi-Frobenius ring

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.