Lie bialgebroid

Preliminary notions A Lie algebroid consists of a bilinear skew-symmetric operation [\cdot,\cdot] on the sections \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 : [\phi,f\cdot\psi] = \rho(\phi)[f]\cdot\psi +f\cdot[\phi,\psi], and Jacobi identity : [\phi,[\psi_1,\psi_2 = \phi,\psi_1],\psi_2] +[\psi_1,[\phi,\psi_2 where \phi,\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 \Gamma(\wedge A) graded symmetric via the Leibniz rule : [\Phi\wedge\Psi,\Chi]_A = \Phi\wedge[\Psi,\Chi]_A +(-1)^\alpha\wedge d_A\beta for A-forms \alpha and \beta. It is uniquely characterized by the conditions : (d_Af)(\phi) = \rho(\phi)[f] and : (d_A\alpha)[\phi,\psi] = \rho(\phi)[\alpha(\psi)] -\rho(\psi)[\alpha(\phi)] -\alpha[\phi,\psi] for functions f on M, A-1-forms \alpha \in \Gamma(A^*) and \phi, \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_*[\phi,\psi]_A = [d_*\phi,\psi]_A +[\phi,d_*\psi]_A for all sections \phi, \psi of A. Here d_* denotes the Lie algebroid differential of A^* which also operates on the multivector fields \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 ==

• A Lie bialgebra consists of two Lie algebras (\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}}) and (\mathfrak{g}^*,[\cdot,\cdot]_*) on dual vector spaces \mathfrak{g} and \mathfrak{g}^* such that the Chevalley–Eilenberg differential \delta_* is a derivation of the \mathfrak{g}-bracket. • 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 (\mathfrak{g},\mathfrak{g}^*) there is the notion of Manin triples, i.e. c = \mathfrak{g} + \mathfrak{g}^* can be endowed with the structure of a Lie algebra such that \mathfrak{g} and \mathfrak{g}^* are subalgebras and c contains the representation of \mathfrak{g} on \mathfrak{g}^*, vice versa. The sum structure is just : [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 is \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 \Pi A + \Pi A^*. But unfortunately d_A + d_*|\Pi A + \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 ==