- Source: Lie bialgebroid
In differential geometry, a field in mathematics, a Lie bialgebroid consists of two compatible Lie algebroids defined on dual vector bundles. Lie bialgebroids are the vector bundle version of Lie bialgebras.
Definition
= Preliminary notions
=A Lie algebroid consists of a bilinear skew-symmetric operation
[
⋅
,
⋅
]
{\displaystyle [\cdot ,\cdot ]}
on the sections
Γ
(
A
)
{\displaystyle \Gamma (A)}
of a vector bundle
A
→
M
{\displaystyle A\to M}
over a smooth manifold
M
{\displaystyle M}
, together with a vector bundle morphism
ρ
:
A
→
T
M
{\displaystyle \rho :A\to TM}
subject to the Leibniz rule
[
ϕ
,
f
⋅
ψ
]
=
ρ
(
ϕ
)
[
f
]
⋅
ψ
+
f
⋅
[
ϕ
,
ψ
]
,
{\displaystyle [\phi ,f\cdot \psi ]=\rho (\phi )[f]\cdot \psi +f\cdot [\phi ,\psi ],}
and Jacobi identity
[
ϕ
,
[
ψ
1
,
ψ
2
]
]
=
[
[
ϕ
,
ψ
1
]
,
ψ
2
]
+
[
ψ
1
,
[
ϕ
,
ψ
2
]
]
{\displaystyle [\phi ,[\psi _{1},\psi _{2}]]=[[\phi ,\psi _{1}],\psi _{2}]+[\psi _{1},[\phi ,\psi _{2}]]}
where
ϕ
,
ψ
k
{\displaystyle \phi ,\psi _{k}}
are sections of
A
{\displaystyle A}
and
f
{\displaystyle f}
is a smooth function on
M
{\displaystyle M}
.
The Lie bracket
[
⋅
,
⋅
]
A
{\displaystyle [\cdot ,\cdot ]_{A}}
can be extended to multivector fields
Γ
(
∧
A
)
{\displaystyle \Gamma (\wedge A)}
graded symmetric via the Leibniz rule
[
Φ
∧
Ψ
,
X
]
A
=
Φ
∧
[
Ψ
,
X
]
A
+
(
−
1
)
|
Ψ
|
(
|
X
|
−
1
)
[
Φ
,
X
]
A
∧
Ψ
{\displaystyle [\Phi \wedge \Psi ,\mathrm {X} ]_{A}=\Phi \wedge [\Psi ,\mathrm {X} ]_{A}+(-1)^{|\Psi |(|\mathrm {X} |-1)}[\Phi ,\mathrm {X} ]_{A}\wedge \Psi }
for homogeneous multivector fields
ϕ
,
ψ
,
X
{\displaystyle \phi ,\psi ,X}
.
The Lie algebroid differential is an
R
{\displaystyle \mathbb {R} }
-linear operator
d
A
{\displaystyle d_{A}}
on the
A
{\displaystyle A}
-forms
Ω
A
(
M
)
=
Γ
(
∧
A
∗
)
{\displaystyle \Omega _{A}(M)=\Gamma (\wedge A^{*})}
of degree 1 subject to the Leibniz rule
d
A
(
α
∧
β
)
=
(
d
A
α
)
∧
β
+
(
−
1
)
|
α
|
α
∧
d
A
β
{\displaystyle d_{A}(\alpha \wedge \beta )=(d_{A}\alpha )\wedge \beta +(-1)^{|\alpha |}\alpha \wedge d_{A}\beta }
for
A
{\displaystyle A}
-forms
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
. It is uniquely characterized by the conditions
(
d
A
f
)
(
ϕ
)
=
ρ
(
ϕ
)
[
f
]
{\displaystyle (d_{A}f)(\phi )=\rho (\phi )[f]}
and
(
d
A
α
)
[
ϕ
,
ψ
]
=
ρ
(
ϕ
)
[
α
(
ψ
)
]
−
ρ
(
ψ
)
[
α
(
ϕ
)
]
−
α
[
ϕ
,
ψ
]
{\displaystyle (d_{A}\alpha )[\phi ,\psi ]=\rho (\phi )[\alpha (\psi )]-\rho (\psi )[\alpha (\phi )]-\alpha [\phi ,\psi ]}
for functions
f
{\displaystyle f}
on
M
{\displaystyle M}
,
A
{\displaystyle A}
-1-forms
α
∈
Γ
(
A
∗
)
{\displaystyle \alpha \in \Gamma (A^{*})}
and
ϕ
,
ψ
{\displaystyle \phi ,\psi }
sections of
A
{\displaystyle A}
.
= The definition
=A Lie bialgebroid consists of two Lie algebroids
(
A
,
ρ
A
,
[
⋅
,
⋅
]
A
)
{\displaystyle (A,\rho _{A},[\cdot ,\cdot ]_{A})}
and
(
A
∗
,
ρ
∗
,
[
⋅
,
⋅
]
∗
)
{\displaystyle (A^{*},\rho _{*},[\cdot ,\cdot ]_{*})}
on the dual vector bundles
A
→
M
{\displaystyle A\to M}
and
A
∗
→
M
{\displaystyle A^{*}\to M}
, subject to the compatibility
d
∗
[
ϕ
,
ψ
]
A
=
[
d
∗
ϕ
,
ψ
]
A
+
[
ϕ
,
d
∗
ψ
]
A
{\displaystyle d_{*}[\phi ,\psi ]_{A}=[d_{*}\phi ,\psi ]_{A}+[\phi ,d_{*}\psi ]_{A}}
for all sections
ϕ
,
ψ
{\displaystyle \phi ,\psi }
of
A
{\displaystyle A}
. Here
d
∗
{\displaystyle d_{*}}
denotes the Lie algebroid differential of
A
∗
{\displaystyle A^{*}}
which also operates on the multivector fields
Γ
(
∧
A
)
{\displaystyle \Gamma (\wedge A)}
.
= Symmetry of the definition
=It can be shown that the definition is symmetric in
A
{\displaystyle A}
and
A
∗
{\displaystyle A^{*}}
, i.e.
(
A
,
A
∗
)
{\displaystyle (A,A^{*})}
is a Lie bialgebroid if and only if
(
A
∗
,
A
)
{\displaystyle (A^{*},A)}
is.
Examples
A Lie bialgebra consists of two Lie algebras
(
g
,
[
⋅
,
⋅
]
g
)
{\displaystyle ({\mathfrak {g}},[\cdot ,\cdot ]_{\mathfrak {g}})}
and
(
g
∗
,
[
⋅
,
⋅
]
∗
)
{\displaystyle ({\mathfrak {g}}^{*},[\cdot ,\cdot ]_{*})}
on dual vector spaces
g
{\displaystyle {\mathfrak {g}}}
and
g
∗
{\displaystyle {\mathfrak {g}}^{*}}
such that the Chevalley–Eilenberg differential
δ
∗
{\displaystyle \delta _{*}}
is a derivation of the
g
{\displaystyle {\mathfrak {g}}}
-bracket.
A Poisson manifold
(
M
,
π
)
{\displaystyle (M,\pi )}
gives naturally rise to a Lie bialgebroid on
T
M
{\displaystyle TM}
(with the commutator bracket of tangent vector fields) and
T
∗
M
{\displaystyle T^{*}M}
(with the Lie bracket induced by the Poisson structure). The
T
∗
M
{\displaystyle T^{*}M}
-differential is
d
∗
=
[
π
,
⋅
]
{\displaystyle d_{*}=[\pi ,\cdot ]}
and the compatibility follows then from the Jacobi identity of the Schouten bracket.
Infinitesimal version of a Poisson groupoid
It is well known that the infinitesimal version of a Lie groupoid is a Lie algebroid (as a special case, the infinitesimal version of a Lie group is a Lie algebra). Therefore, one can ask which structures need to be differentiated in order to obtain a Lie bialgebroid.
= Definition of Poisson groupoid
=A Poisson groupoid is a Lie groupoid
G
⇉
M
{\displaystyle G\rightrightarrows M}
together with a Poisson structure
π
{\displaystyle \pi }
on
G
{\displaystyle G}
such that the graph
m
⊂
G
×
G
×
(
G
,
−
π
)
{\displaystyle m\subset G\times G\times (G,-\pi )}
of the multiplication map is coisotropic. An example of a Poisson-Lie groupoid is a Poisson-Lie group (where
M
{\displaystyle M}
is a point). Another example is a symplectic groupoid (where the Poisson structure is non-degenerate on
T
G
{\displaystyle TG}
).
= Differentiation of the structure
=Remember the construction of a Lie algebroid from a Lie groupoid. We take the
t
{\displaystyle t}
-tangent fibers (or equivalently the
s
{\displaystyle s}
-tangent fibers) and consider their vector bundle pulled back to the base manifold
M
{\displaystyle M}
. A section of this vector bundle can be identified with a
G
{\displaystyle G}
-invariant
t
{\displaystyle t}
-vector field on
G
{\displaystyle G}
which form a Lie algebra with respect to the commutator bracket on
T
G
{\displaystyle TG}
.
We thus take the Lie algebroid
A
→
M
{\displaystyle A\to M}
of the Poisson groupoid. It can be shown that the Poisson structure induces a fiber-linear Poisson structure on
A
{\displaystyle A}
. Analogous to the construction of the cotangent Lie algebroid of a Poisson manifold there is a Lie algebroid structure on
A
∗
{\displaystyle A^{*}}
induced by this Poisson structure. Analogous to the Poisson manifold case one can show that
A
{\displaystyle A}
and
A
∗
{\displaystyle A^{*}}
form a Lie bialgebroid.
Double of a Lie bialgebroid and superlanguage of Lie bialgebroids
For Lie bialgebras
(
g
,
g
∗
)
{\displaystyle ({\mathfrak {g}},{\mathfrak {g}}^{*})}
there is the notion of Manin triples, i.e.
c
=
g
+
g
∗
{\displaystyle c={\mathfrak {g}}+{\mathfrak {g}}^{*}}
can be endowed with the structure of a Lie algebra such that
g
{\displaystyle {\mathfrak {g}}}
and
g
∗
{\displaystyle {\mathfrak {g}}^{*}}
are subalgebras and
c
{\displaystyle c}
contains the representation of
g
{\displaystyle {\mathfrak {g}}}
on
g
∗
{\displaystyle {\mathfrak {g}}^{*}}
, vice versa. The sum structure is just
[
X
+
α
,
Y
+
β
]
=
[
X
,
Y
]
g
+
a
d
α
Y
−
a
d
β
X
+
[
α
,
β
]
∗
+
a
d
X
∗
β
−
a
d
Y
∗
α
{\displaystyle [X+\alpha ,Y+\beta ]=[X,Y]_{g}+\mathrm {ad} _{\alpha }Y-\mathrm {ad} _{\beta }X+[\alpha ,\beta ]_{*}+\mathrm {ad} _{X}^{*}\beta -\mathrm {ad} _{Y}^{*}\alpha }
.
= Courant algebroids
=It turns out that the naive generalization to Lie algebroids does not give a Lie algebroid any more. Instead one has to modify either the Jacobi identity or violate the skew-symmetry and is thus lead to Courant algebroids.
= Superlanguage
=The appropriate superlanguage of a Lie algebroid
A
{\displaystyle A}
is
Π
A
{\displaystyle \Pi A}
, the supermanifold whose space of (super)functions are the
A
{\displaystyle A}
-forms. On this space the Lie algebroid can be encoded via its Lie algebroid differential, which is just an odd vector field.
As a first guess the super-realization of a Lie bialgebroid
(
A
,
A
∗
)
{\displaystyle (A,A^{*})}
should be
Π
A
+
Π
A
∗
{\displaystyle \Pi A+\Pi A^{*}}
. But unfortunately
d
A
+
d
∗
|
Π
A
+
Π
A
∗
{\displaystyle d_{A}+d_{*}|\Pi A+\Pi A^{*}}
is not a differential, basically because
A
+
A
∗
{\displaystyle A+A^{*}}
is not a Lie algebroid. Instead using the larger N-graded manifold
T
∗
[
2
]
A
[
1
]
=
T
∗
[
2
]
A
∗
[
1
]
{\displaystyle T^{*}[2]A[1]=T^{*}[2]A^{*}[1]}
to which we can lift
d
A
{\displaystyle d_{A}}
and
d
∗
{\displaystyle d_{*}}
as odd Hamiltonian vector fields, then their sum squares to
0
{\displaystyle 0}
iff
(
A
,
A
∗
)
{\displaystyle (A,A^{*})}
is a Lie bialgebroid.
References
C. Albert and P. Dazord: Théorie des groupoïdes symplectiques: Chapitre II, Groupoïdes symplectiques. (in Publications du Département de Mathématiques de l’Université Claude Bernard, Lyon I, nouvelle série, pp. 27–99, 1990)
Y. Kosmann-Schwarzbach: The Lie bialgebroid of a Poisson–Nijenhuis manifold. (Lett. Math. Phys., 38:421–428, 1996)
K. Mackenzie, P. Xu: Integration of Lie bialgebroids (1997),
K. Mackenzie, P. Xu: Lie bialgebroids and Poisson groupoids (Duke J. Math, 1994)
A. Weinstein: Symplectic groupoids and Poisson manifolds (AMS Bull, 1987),
Kata Kunci Pencarian:
- Lie bialgebroid
- Bialgebroid
- Lie algebroid
- Associative bialgebroid
- Algebroid
- Poisson manifold
- Courant algebroid
