- Source: Lie bialgebra
In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible.
It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary.
They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson–Lie group.
Lie bialgebras occur naturally in the study of the Yang–Baxter equations.
Definition
A vector space
g
{\displaystyle {\mathfrak {g}}}
is a Lie bialgebra if it is a Lie algebra,
and there is the structure of Lie algebra also on the dual vector space
g
∗
{\displaystyle {\mathfrak {g}}^{*}}
which is compatible.
More precisely the Lie algebra structure on
g
{\displaystyle {\mathfrak {g}}}
is given
by a Lie bracket
[
,
]
:
g
⊗
g
→
g
{\displaystyle [\ ,\ ]:{\mathfrak {g}}\otimes {\mathfrak {g}}\to {\mathfrak {g}}}
and the Lie algebra structure on
g
∗
{\displaystyle {\mathfrak {g}}^{*}}
is given by a Lie
bracket
δ
∗
:
g
∗
⊗
g
∗
→
g
∗
{\displaystyle \delta ^{*}:{\mathfrak {g}}^{*}\otimes {\mathfrak {g}}^{*}\to {\mathfrak {g}}^{*}}
.
Then the map dual to
δ
∗
{\displaystyle \delta ^{*}}
is called the cocommutator,
δ
:
g
→
g
⊗
g
{\displaystyle \delta :{\mathfrak {g}}\to {\mathfrak {g}}\otimes {\mathfrak {g}}}
and the compatibility condition is the following cocycle relation:
δ
(
[
X
,
Y
]
)
=
(
ad
X
⊗
1
+
1
⊗
ad
X
)
δ
(
Y
)
−
(
ad
Y
⊗
1
+
1
⊗
ad
Y
)
δ
(
X
)
{\displaystyle \delta ([X,Y])=\left(\operatorname {ad} _{X}\otimes 1+1\otimes \operatorname {ad} _{X}\right)\delta (Y)-\left(\operatorname {ad} _{Y}\otimes 1+1\otimes \operatorname {ad} _{Y}\right)\delta (X)}
where
ad
X
Y
=
[
X
,
Y
]
{\displaystyle \operatorname {ad} _{X}Y=[X,Y]}
is the adjoint.
Note that this definition is symmetric and
g
∗
{\displaystyle {\mathfrak {g}}^{*}}
is also a Lie bialgebra, the dual Lie bialgebra.
Example
Let
g
{\displaystyle {\mathfrak {g}}}
be any semisimple Lie algebra.
To specify a Lie bialgebra structure we thus need to specify a compatible Lie algebra structure on the dual vector space.
Choose a Cartan subalgebra
t
⊂
g
{\displaystyle {\mathfrak {t}}\subset {\mathfrak {g}}}
and a choice of positive roots.
Let
b
±
⊂
g
{\displaystyle {\mathfrak {b}}_{\pm }\subset {\mathfrak {g}}}
be the corresponding opposite Borel subalgebras, so that
t
=
b
−
∩
b
+
{\displaystyle {\mathfrak {t}}={\mathfrak {b}}_{-}\cap {\mathfrak {b}}_{+}}
and there is a natural projection
π
:
b
±
→
t
{\displaystyle \pi :{\mathfrak {b}}_{\pm }\to {\mathfrak {t}}}
.
Then define a Lie algebra
g
′
:=
{
(
X
−
,
X
+
)
∈
b
−
×
b
+
|
π
(
X
−
)
+
π
(
X
+
)
=
0
}
{\displaystyle {\mathfrak {g'}}:=\{(X_{-},X_{+})\in {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}\ {\bigl \vert }\ \pi (X_{-})+\pi (X_{+})=0\}}
which is a subalgebra of the product
b
−
×
b
+
{\displaystyle {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}}
, and has the same dimension as
g
{\displaystyle {\mathfrak {g}}}
.
Now identify
g
′
{\displaystyle {\mathfrak {g'}}}
with dual of
g
{\displaystyle {\mathfrak {g}}}
via the pairing
⟨
(
X
−
,
X
+
)
,
Y
⟩
:=
K
(
X
+
−
X
−
,
Y
)
{\displaystyle \langle (X_{-},X_{+}),Y\rangle :=K(X_{+}-X_{-},Y)}
where
Y
∈
g
{\displaystyle Y\in {\mathfrak {g}}}
and
K
{\displaystyle K}
is the Killing form.
This defines a Lie bialgebra structure on
g
{\displaystyle {\mathfrak {g}}}
, and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group.
Note that
g
′
{\displaystyle {\mathfrak {g'}}}
is solvable, whereas
g
{\displaystyle {\mathfrak {g}}}
is semisimple.
Relation to Poisson–Lie groups
The Lie algebra
g
{\displaystyle {\mathfrak {g}}}
of a Poisson–Lie group G has a natural structure of Lie bialgebra.
In brief the Lie group structure gives the Lie bracket on
g
{\displaystyle {\mathfrak {g}}}
as usual, and the linearisation of the Poisson structure on G
gives the Lie bracket on
g
∗
{\displaystyle {\mathfrak {g^{*}}}}
(recalling that a linear Poisson structure on a vector space is the same thing as a Lie bracket on the dual vector space).
In more detail, let G be a Poisson–Lie group, with
f
1
,
f
2
∈
C
∞
(
G
)
{\displaystyle f_{1},f_{2}\in C^{\infty }(G)}
being two smooth functions on the group manifold. Let
ξ
=
(
d
f
)
e
{\displaystyle \xi =(df)_{e}}
be the differential at the identity element. Clearly,
ξ
∈
g
∗
{\displaystyle \xi \in {\mathfrak {g}}^{*}}
. The Poisson structure on the group then induces a bracket on
g
∗
{\displaystyle {\mathfrak {g}}^{*}}
, as
[
ξ
1
,
ξ
2
]
=
(
d
{
f
1
,
f
2
}
)
e
{\displaystyle [\xi _{1},\xi _{2}]=(d\{f_{1},f_{2}\})_{e}\,}
where
{
,
}
{\displaystyle \{,\}}
is the Poisson bracket. Given
η
{\displaystyle \eta }
be the Poisson bivector on the manifold, define
η
R
{\displaystyle \eta ^{R}}
to be the right-translate of the bivector to the identity element in G. Then one has that
η
R
:
G
→
g
⊗
g
{\displaystyle \eta ^{R}:G\to {\mathfrak {g}}\otimes {\mathfrak {g}}}
The cocommutator is then the tangent map:
δ
=
T
e
η
R
{\displaystyle \delta =T_{e}\eta ^{R}\,}
so that
[
ξ
1
,
ξ
2
]
=
δ
∗
(
ξ
1
⊗
ξ
2
)
{\displaystyle [\xi _{1},\xi _{2}]=\delta ^{*}(\xi _{1}\otimes \xi _{2})}
is the dual of the cocommutator.
See also
Lie coalgebra
Manin triple
References
H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG, 1989, Springer-Verlag Berlin, ISBN 3-540-53503-9.
Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.
Beisert, N.; Spill, F. (2009). "The classical r-matrix of AdS/CFT and its Lie bialgebra structure". Communications in Mathematical Physics. 285 (2): 537–565. arXiv:0708.1762. Bibcode:2009CMaPh.285..537B. doi:10.1007/s00220-008-0578-2. S2CID 8946457.