Lie bialgebroid

A Lie bialgebroid is a mathematical structure in the area of non-Riemannian differential geometry. In brief a Lie bialgebroid are two compatible Lie algebroids defined on dual vector bundles. They form the vector bundle version of a Lie bialgebra.

Definition

Preliminary notions

Remember that a Lie algebroid is defined as a skew-symmetric operation [.,.] on the sections Γ(A) of a vector bundle A→M over a smooth manifold M together with a vector bundle morphism ρ: A→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 Φ, ψ_k are sections of A and f is a smooth function on M.

The Lie bracket [.,.]_A can be extended to multivector fields Γ(⋀A) graded symmetric via the Leibniz rule

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

for homogeneous multivector fields Φ, Ψ, Χ.

The Lie algebroid differential is an R-linear operator d_A on the A-forms Ω_A(M) = Γ(⋀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 α and β. It is uniquely characterized by the conditions

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

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 α∈Γ(A^*) and Φ, ψ sections of A.

The definition

A Lie bialgebroid are two Lie algebroids (A,ρ_A,[.,.]_A) and (A^*,ρ_*,[.,.]_*) on dual vector bundles A→M and A^*→M subject to the compatibility

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

for all sections Φ, ψ of A. Here d_* denotes the Lie algebroid differential of A^* which also operates on the multivector fields Γ(∧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 iff (A^*,A) is.

Examples

1. A Lie bialgebra are two Lie algebras (g,[.,.]_g) and (g^*,[.,.]_*) on dual vector spaces g and g^* such that the Chevalley–Eilenberg differential δ_* is a derivation of the g-bracket.

2. A Poisson manifold (M,π) 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_*= [π, .] 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⇉M) together with a Poisson structure π on G such that the multiplication graph m ⊂ G×G×(G,−π) is coisotropic. An example of a Poisson Lie groupoid is a Poisson Lie group (where M=pt, just 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→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

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

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 ΠA, 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 ΠA+ΠA^*. But unfortunately d_A +d_*|ΠA+ΠA^* 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),