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In mathematics , a quasi-Frobenius Lie algebra
(
g
,
[
,
]
,
β
)
{\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )}
over a field
k
{\displaystyle k}
is a Lie algebra
(
g
,
[
,
]
)
{\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,])}
equipped with a nondegenerate skew-symmetric bilinear form
β
:
g
×
g
→
k
{\displaystyle \beta :{\mathfrak {g}}\times {\mathfrak {g}}\to k}
, which is a Lie algebra 2-cocycle of
g
{\displaystyle {\mathfrak {g}}}
with values in
k
{\displaystyle k}
. In other words,
β
(
[
X
,
Y
]
,
Z
)
+
β
(
[
Z
,
X
]
,
Y
)
+
β
(
[
Y
,
Z
]
,
X
)
=
0
{\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
X
{\displaystyle X}
,
Y
{\displaystyle Y}
,
Z
{\displaystyle Z}
in
g
{\displaystyle {\mathfrak {g}}}
.
If
β
{\displaystyle \beta }
is a coboundary, which means that there exists a linear form
f
:
g
→
k
{\displaystyle f:{\mathfrak {g}}\to k}
such that
β
(
X
,
Y
)
=
f
(
[
X
,
Y
]
)
,
{\displaystyle \beta (X,Y)=f(\left[X,Y\right]),}
then
(
g
,
[
,
]
,
β
)
{\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )}
is called a Frobenius Lie algebra .
If
(
g
,
[
,
]
,
β
)
{\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )}
is a quasi-Frobenius Lie algebra, one can define on
g
{\displaystyle {\mathfrak {g}}}
another bilinear product
◃
{\displaystyle \triangleleft }
by the formula
β
(
[
X
,
Y
]
,
Z
)
=
β
(
Z
◃
Y
,
X
)
{\displaystyle \beta \left(\left[X,Y\right],Z\right)=\beta \left(Z\triangleleft Y,X\right)}
.
Then one has
[
X
,
Y
]
=
X
◃
Y
−
Y
◃
X
{\displaystyle \left[X,Y\right]=X\triangleleft Y-Y\triangleleft X}
and
(
g
,
◃
)
{\displaystyle ({\mathfrak {g}},\triangleleft )}
is a pre-Lie algebra .
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 .