Lie bialgebroid

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

:$[\backslash phi,f\backslash cdot\backslash psi]\; =\; \backslash rho(\backslash phi)[f]\backslash cdot\backslash psi\; +f\backslash cdot[\backslash phi,\backslash psi],$

and Jacobi identity

:$[\backslash phi,[\backslash psi\_1,\backslash psi\_2]]\; =\backslash phi,\backslash psi\_1],\backslash psi\_2]\; +[\backslash psi\_1,[\backslash phi,\backslash psi\_2$

where $\backslash phi,\backslash psi\_k$ are sections of $A$ and $f$ is a smooth function on $M$.

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

:$[\backslash Phi\backslash wedge\backslash Psi,\backslash Chi]\_A\; =\; \backslash Phi\backslash wedge[\backslash Psi,\backslash Chi]\_A\; +(-1)^$

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

:$(d\_Af)(\backslash phi)\; =\; \backslash rho(\backslash phi)[f]$

and

:$(d\_A\backslash alpha)[\backslash phi,\backslash psi]\; =\; \backslash rho(\backslash phi)[\backslash alpha(\backslash psi)]\; -\backslash rho(\backslash psi)[\backslash alpha(\backslash phi)]\; -\backslash alpha[\backslash phi,\backslash psi]$

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

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

:$d\_*[\backslash phi,\backslash psi]\_A\; =\; [d\_*\backslash phi,\backslash psi]\_A\; +[\backslash phi,d\_*\backslash psi]\_A$

for all sections $\backslash phi,\; \backslash psi$ of $A$. Here $d\_*$ denotes the Lie algebroid differential of $A^*$ which also operates on the multivector fields $\backslash Gamma(\backslash wedge\; A)$.

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.

1. A Lie bialgebra consists of two Lie algebras $(\backslash mathfrak\{g\},[\backslash cdot,\backslash cdot]\_\{\backslash mathfrak\{g\}\})$ and $(\backslash mathfrak\{g\}^*,[\backslash cdot,\backslash cdot]\_*)$ on dual vector spaces $\backslash mathfrak\{g\}$ and $\backslash mathfrak\{g\}^*$ such that the Chevalley–Eilenberg differential $\backslash delta\_*$ is a derivation of the $\backslash mathfrak\{g\}$-bracket.
2. A Poisson manifold $(M,\backslash pi)$ 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 $d\_*\; =\; [\backslash pi,\backslash cdot]$ 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 $G\; \backslash rightrightarrows\; M$ together with a Poisson structure $\backslash pi$ on $G$ such that the graph $m\; \backslash subset\; G\; \backslash times\; G\; \backslash times\; (G,-\backslash pi)$ 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 $TG$).

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 $A\; \backslash to\; M$ 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.

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

:$[X+\backslash alpha,Y+\backslash beta]\; =\; [X,Y]\_g\; +\backslash mathrm\{ad\}\_\backslash alpha\; Y\; -\backslash mathrm\{ad\}\_\backslash beta\; X$

+[\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.Z.-J. Liu, A. Weinstein and P. Xu: Manin triples for Lie bialgebroids, Journ. of diff. geom. vol. 45, pp. 547–574 (1997)

The appropriate superlanguage of a Lie algebroid $A$ is $\backslash Pi\; A$, 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 $\backslash Pi\; A\; +\; \backslash Pi\; A^*$. But unfortunately $d\_A\; +\; d\_*|\backslash Pi\; A\; +\; \backslash Pi\; A^*$ 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.

{{Reflist}}

• 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),

Category:Symplectic geometry

Category:Differential geometry