Lie bialgebroid

• Comment: I have requested help from WikiProject Mathematics to review this Roger (Dodger67) (talk) 14:50, 31 October 2014 (UTC)

• Comment: This topic gets 329 hits in GScolar including many papers in which Lie bialgebroids are the main subject of the paper. There seems to be plenty of well-cited material to satisfy notability guidelines per WP:GNG. From what little I have read of this field, the information presented here seems correct and with due weight. Sources are provided with few citations, but this is fairly common practice in advanced math articles. I'd say as a start-class article, this draft is ready for mainspace. Nice work, MelchiorG! --Mark viking (talk) 21:24, 31 October 2014 (UTC)

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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 bialgebroids (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 dA 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),

Category:Differential geometry Category:Symplectic geometry