Lie bialgebroid explained

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

[ ⋅ , ⋅ ] 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 ⋅ \psi] = \rho(\phi)[f] ⋅ \psi + f ⋅ [\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

[ ⋅ , ⋅ ] _{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 ) ^{|\Psi|(|\Chi|-1)} [\Phi,\Chi] _{A\wedge\Psi} for homogeneous multivector fields \phi, \psi, X .

The Lie algebroid differential is an

R -linear operator d _{A} on the A -forms \Omega _{A} (M) = \Gamma (\wedge A ^{*)} of degree 1 subject to the Leibniz ruled _{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 _{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,[ ⋅ , ⋅ ]} _{A)} and (A ^{*,\rho} _{*,[ ⋅ , ⋅ ]} _{*)} on the dual vector bundles A \to M and A ^{*} \to M , subject to the compatibilityd _{*[\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

1. A Lie bialgebra consists of two Lie algebras

( ak{g},[ ⋅, ⋅ ] _{ak{g} }) and ( ak{g} ^{*,[ ⋅ , ⋅ ]} _{*)} on dual vector spaces ak{g} and ak{g} ^{*} such that the Chevalley–Eilenberg differential \delta _{*} is a derivation of the ak{g} -bracket.

1. 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, ⋅ ] 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 x G x (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

( ak{g}, ak{g} ^{*)} there is the notion of Manin triples, i.e. c = ak{g} + ak{g} ^{*} can be endowed with the structure of a Lie algebra such that ak{g} and ak{g} ^{*} are subalgebras and c contains the representation of ak{g} on ak{g} ^{*} , vice versa. The sum structure is just[X + \alpha,Y + \beta] = [X,Y] _{g} + ad _{\alpha} Y - ad _{\beta} X + [\alpha,\beta] _{*} + ad ^{*} _{X\beta} - 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.^[1]

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

• 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),

Notes and References

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