Lie bialgebroid

Template:Short description 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 $[\cdot ,\cdot ]$ on the sections $\mathrm{\Gamma }(A)$ of a vector bundle $A\to M$ over a smooth manifold $M$, together with a vector bundle morphism $\rho :A\to TM$ subject to the Leibniz rule

$[\varphi ,f\cdot \psi ]=\rho (\varphi )[f]\cdot \psi +f\cdot [\varphi ,\psi ],$

and Jacobi identity

$[\varphi ,[{\psi }_1,{\psi }_2]]=[[\varphi ,{\psi }_1],{\psi }_2]+[{\psi }_1,[\varphi ,{\psi }_2]]$

where $\varphi ,{\psi }_k$ are sections of $A$ and $f$ is a smooth function on $M$.

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

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

for homogeneous multivector fields $\varphi ,\psi ,X$.

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

$d_A(\alpha \wedge \beta )=(d_A\alpha )\wedge \beta +(-1{)}^{|\alpha |}\alpha \wedge d_A\beta$

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

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

and

$(d_A\alpha )[\varphi ,\psi ]=\rho (\varphi )[\alpha (\psi )]-\rho (\psi )[\alpha (\varphi )]-\alpha [\varphi ,\psi ]$

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

The definition

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

$d_{*}[\varphi ,\psi {]}_A=[d_{*}\varphi ,\psi {]}_A+[\varphi ,d_{*}\psi {]}_A$

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

Symmetry of the definition

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

Examples

1. A Lie bialgebra consists of two Lie algebras $(𝔤,[\cdot ,\cdot {]}_{𝔤})$ and $({𝔤}^{*},[\cdot ,\cdot {]}_{*})$ on dual vector spaces $𝔤$ and ${𝔤}^{*}$ such that the Chevalley–Eilenberg differential ${\delta }_{*}$ is a derivation of the $𝔤$-bracket.
2. A Poisson manifold $(M,\pi )$ gives naturally rise to a Lie bialgebroid on $TM$ (with the commutator bracket of tangent vector fields) and $T^{*}M$ (with the Lie bracket induced by the Poisson structure). The $T^{*}M$-differential is $d_{*}=[\pi ,\cdot ]$ 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 $G\rightrightarrows M$ together with a Poisson structure $\pi$ on $G$ such that the graph $m\subset G\times G\times (G,-\pi )$ of the multiplication map is coisotropic. An example of a Poisson-Lie groupoid is a Poisson-Lie group (where $M$ is a point). Another example is a symplectic groupoid (where the Poisson structure is non-degenerate on $TG$).

Differentiation of the structure

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

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

Double of a Lie bialgebroid and superlanguage of Lie bialgebroids

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

$[X+\alpha ,Y+\beta ]=[X,Y{]}_g+{\mathrm{a}\mathrm{d}}_{\alpha }Y-{\mathrm{a}\mathrm{d}}_{\beta }X+[\alpha ,\beta {]}_{*}+{\mathrm{a}\mathrm{d}}_X^{*}\beta -{\mathrm{a}\mathrm{d}}_Y^{*}\alpha$.

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