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) cdot\psi +f\cdot phi,\psi

and Jacobi identity

: phi,[\psi_1,\psi_2 = \phi,\psi_1">psi_1,\psi_2.html" ;"title="phi,[\psi_1,\psi_2">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 ,.sub>''A'' can be extended to polyvector field, multivector fields Γ(⋀''A'') graded symmetric via the Leibniz rule
: Phi\wedge\Psi,\ChiA = \Phi\wedge Psi,\ChiA +(-1)^ Phi,\ChiA\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
:$d_A(\alpha\wedge\beta) = (d_A\alpha)\wedge\beta +(-1)^\alpha\wedge d_A\beta$
for ''A''-forms ''α'' and ''β''. It is uniquely characterized by the conditions
: (d_Af)(\phi) = \rho(\phi) /math>
and
: (d_A\alpha) phi,\psi= \rho(\phi) alpha(\psi)-\rho(\psi) alpha(\phi)-\alpha phi,\psi/math>
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'', ,.sub>''A'') and (''A''^*,ρ_*, ,.sub>*) on dual vector bundles ''A→M'' and ''A''^*→''M'' subject to the compatibility
: d_* phi,\psiA = _*\phi,\psiA + phi,d_*\psiA
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, ,.sub>g) and (g^*, ,.sub>*) 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_*= �, .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'',−) 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
: +\alpha,Y+\beta= ,Yg +\mathrm_\alpha Y -\mathrm_\beta X
+ alpha,\beta* +\mathrm^*_X\beta -\mathrm^*_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.Z.-J. Liu, A. Weinstein and P. Xu: Manin triples for Lie bialgebroids, Journ. of diff. geom. vol. 45, pp. 547–574 (1997)

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^* ' = ''T^{* *} ' 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

{{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),

Symplectic geometry