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.

A Lie algebroid consists of a bilinear skew-symmetric operation ${\displaystyle [\cdot ,\cdot ]}$ on the sections ${\displaystyle \Gamma (A)}$ of a vector bundle ${\displaystyle A\to M}$ over a smooth manifold ${\displaystyle M}$, together with a vector bundle morphism ${\displaystyle \rho :A\to TM}$ subject to the Leibniz rule

${\displaystyle [\phi ,f\cdot \psi ]=\rho (\phi )[f]\cdot \psi +f\cdot [\phi ,\psi ],}$

and Jacobi identity

${\displaystyle [\phi ,[\psi _{1},\psi _{2}]]=[[\phi ,\psi _{1}],\psi _{2}]+[\psi _{1},[\phi ,\psi _{2}]]}$

where ${\displaystyle \phi ,\psi _{k}}$ are sections of ${\displaystyle A}$ and ${\displaystyle f}$ is a smooth function on ${\displaystyle M}$.

The Lie bracket ${\displaystyle [\cdot ,\cdot ]_{A}}$ can be extended to multivector fields ${\displaystyle \Gamma (\wedge A)}$ graded symmetric via the Leibniz rule

${\displaystyle [\Phi \wedge \Psi ,\mathrm {X} ]_{A}=\Phi \wedge [\Psi ,\mathrm {X} ]_{A}+(-1)^{|\Psi |(|\mathrm {X} |-1)}[\Phi ,\mathrm {X} ]_{A}\wedge \Psi }$

for homogeneous multivector fields ${\displaystyle \phi ,\psi ,X}$.

The Lie algebroid differential is an ${\displaystyle \mathbb {R} }$-linear operator ${\displaystyle d_{A}}$ on the ${\displaystyle A}$-forms ${\displaystyle \Omega _{A}(M)=\Gamma (\wedge A^{*})}$ of degree 1 subject to the Leibniz rule

${\displaystyle d_{A}(\alpha \wedge \beta )=(d_{A}\alpha )\wedge \beta +(-1)^{|\alpha |}\alpha \wedge d_{A}\beta }$

for ${\displaystyle A}$-forms ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. It is uniquely characterized by the conditions

${\displaystyle (d_{A}f)(\phi )=\rho (\phi )[f]}$

and

${\displaystyle (d_{A}\alpha )[\phi ,\psi ]=\rho (\phi )[\alpha (\psi )]-\rho (\psi )[\alpha (\phi )]-\alpha [\phi ,\psi ]}$

for functions ${\displaystyle f}$ on ${\displaystyle M}$, ${\displaystyle A}$-1-forms ${\displaystyle \alpha \in \Gamma (A^{*})}$ and ${\displaystyle \phi ,\psi }$ sections of ${\displaystyle A}$.

A Lie bialgebroid consists of two Lie algebroids ${\displaystyle (A,\rho _{A},[\cdot ,\cdot ]_{A})}$ and ${\displaystyle (A^{*},\rho _{*},[\cdot ,\cdot ]_{*})}$ on the dual vector bundles ${\displaystyle A\to M}$ and ${\displaystyle A^{*}\to M}$, subject to the compatibility

${\displaystyle d_{*}[\phi ,\psi ]_{A}=[d_{*}\phi ,\psi ]_{A}+[\phi ,d_{*}\psi ]_{A}}$

for all sections ${\displaystyle \phi ,\psi }$ of ${\displaystyle A}$. Here ${\displaystyle d_{*}}$ denotes the Lie algebroid differential of ${\displaystyle A^{*}}$ which also operates on the multivector fields ${\displaystyle \Gamma (\wedge A)}$.

It can be shown that the definition is symmetric in ${\displaystyle A}$ and ${\displaystyle A^{*}}$, i.e. ${\displaystyle (A,A^{*})}$ is a Lie bialgebroid if and only if ${\displaystyle (A^{*},A)}$ is.

1. A Lie bialgebra consists of two Lie algebras ${\displaystyle ({\mathfrak {g}},[\cdot ,\cdot ]_{\mathfrak {g}})}$ and ${\displaystyle ({\mathfrak {g}}^{*},[\cdot ,\cdot ]_{*})}$ on dual vector spaces ${\displaystyle {\mathfrak {g}}}$ and ${\displaystyle {\mathfrak {g}}^{*}}$ such that the Chevalley–Eilenberg differential ${\displaystyle \delta _{*}}$ is a derivation of the ${\displaystyle {\mathfrak {g}}}$-bracket.
2. A Poisson manifold ${\displaystyle (M,\pi )}$ gives naturally rise to a Lie bialgebroid on ${\displaystyle TM}$ (with the commutator bracket of tangent vector fields) and ${\displaystyle T^{*}M}$ (with the Lie bracket induced by the Poisson structure). The ${\displaystyle T^{*}M}$-differential is ${\displaystyle d_{*}=[\pi ,\cdot ]}$ and the compatibility follows then from the Jacobi identity of the Schouten bracket.

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.

A Poisson groupoid is a Lie groupoid ${\displaystyle G\rightrightarrows M}$ together with a Poisson structure ${\displaystyle \pi }$ on ${\displaystyle G}$ such that the graph ${\displaystyle m\subset G\times G\times (G,-\pi )}$ of the multiplication map is coisotropic. An example of a Poisson-Lie groupoid is a Poisson-Lie group (where ${\displaystyle M}$ is a point). Another example is a symplectic groupoid (where the Poisson structure is non-degenerate on ${\displaystyle TG}$).

Remember the construction of a Lie algebroid from a Lie groupoid. We take the ${\displaystyle t}$-tangent fibers (or equivalently the ${\displaystyle s}$-tangent fibers) and consider their vector bundle pulled back to the base manifold ${\displaystyle M}$. A section of this vector bundle can be identified with a ${\displaystyle G}$-invariant ${\displaystyle t}$-vector field on ${\displaystyle G}$ which form a Lie algebra with respect to the commutator bracket on ${\displaystyle TG}$.

We thus take the Lie algebroid ${\displaystyle A\to M}$ of the Poisson groupoid. It can be shown that the Poisson structure induces a fiber-linear Poisson structure on ${\displaystyle A}$. Analogous to the construction of the cotangent Lie algebroid of a Poisson manifold there is a Lie algebroid structure on ${\displaystyle A^{*}}$ induced by this Poisson structure. Analogous to the Poisson manifold case one can show that ${\displaystyle A}$ and ${\displaystyle A^{*}}$ form a Lie bialgebroid.

For Lie bialgebras ${\displaystyle ({\mathfrak {g}},{\mathfrak {g}}^{*})}$ there is the notion of Manin triples, i.e. ${\displaystyle c={\mathfrak {g}}+{\mathfrak {g}}^{*}}$ can be endowed with the structure of a Lie algebra such that ${\displaystyle {\mathfrak {g}}}$ and ${\displaystyle {\mathfrak {g}}^{*}}$ are subalgebras and ${\displaystyle c}$ contains the representation of ${\displaystyle {\mathfrak {g}}}$ on ${\displaystyle {\mathfrak {g}}^{*}}$, vice versa. The sum structure is just

${\displaystyle [X+\alpha ,Y+\beta ]=[X,Y]_{g}+\mathrm {ad} _{\alpha }Y-\mathrm {ad} _{\beta }X+[\alpha ,\beta ]_{*}+\mathrm {ad} _{X}^{*}\beta -\mathrm {ad} _{Y}^{*}\alpha }$.

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.^[1]

The appropriate superlanguage of a Lie algebroid ${\displaystyle A}$ is ${\displaystyle \Pi A}$, the supermanifold whose space of (super)functions are the ${\displaystyle A}$-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 ${\displaystyle (A,A^{*})}$ should be ${\displaystyle \Pi A+\Pi A^{*}}$. But unfortunately ${\displaystyle d_{A}+d_{*}|\Pi A+\Pi A^{*}}$ is not a differential, basically because ${\displaystyle A+A^{*}}$ is not a Lie algebroid. Instead using the larger N-graded manifold ${\displaystyle T^{*}[2]A[1]=T^{*}[2]A^{*}[1]}$ to which we can lift ${\displaystyle d_{A}}$ and ${\displaystyle d_{*}}$ as odd Hamiltonian vector fields, then their sum squares to ${\displaystyle 0}$ iff ${\displaystyle (A,A^{*})}$ is a Lie bialgebroid.

• 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),