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 [ ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ]} on the sections Γ ( A ) {\displaystyle \Gamma (A)} of a vector bundle A → M {\displaystyle A\to M} over a smooth manifold M {\displaystyle M}, together with a vector bundle morphism ρ : A → T M {\displaystyle \rho :A\to TM} subject to the Leibniz rule

[ ϕ , f ⋅ ψ ] = ρ ( ϕ ) [ f ] ⋅ ψ + f ⋅ [ ϕ , ψ ] , {\displaystyle [\phi ,f\cdot \psi ]=\rho (\phi )[f]\cdot \psi +f\cdot [\phi ,\psi ],}

and Jacobi identity

[ ϕ , [ ψ 1 , ψ 2 ] ] = [ [ ϕ , ψ 1 ] , ψ 2 ] + [ ψ 1 , [ ϕ , ψ 2 ] ] {\displaystyle [\phi ,[\psi _{1},\psi _{2}]]=[[\phi ,\psi _{1}],\psi _{2}]+[\psi _{1},[\phi ,\psi _{2}]]}

where ϕ , ψ k {\displaystyle \phi ,\psi _{k}} are sections of A {\displaystyle A} and f {\displaystyle f} is a smooth function on M {\displaystyle M}.

The Lie bracket [ ⋅ , ⋅ ] A {\displaystyle [\cdot ,\cdot ]_{A}} can be extended to multivector fields Γ ( ∧ A ) {\displaystyle \Gamma (\wedge A)} graded symmetric via the Leibniz rule

[ Φ ∧ Ψ , X ] A = Φ ∧ [ Ψ , X ] A + ( − 1 ) | Ψ | ( | X | − 1 ) [ Φ , X ] A ∧ Ψ {\displaystyle [\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 , X {\displaystyle \phi ,\psi ,X}.

The Lie algebroid differential is an R {\displaystyle \mathbb {R} }-linear operator d A {\displaystyle d_{A}} on the A {\displaystyle A}-forms Ω A ( M ) = Γ ( ∧ A ∗ ) {\displaystyle \Omega _{A}(M)=\Gamma (\wedge A^{*})} of degree 1 subject to the Leibniz rule

d A ( α ∧ β ) = ( d A α ) ∧ β + ( − 1 ) | α | α ∧ d A β {\displaystyle d_{A}(\alpha \wedge \beta )=(d_{A}\alpha )\wedge \beta +(-1)^{|\alpha |}\alpha \wedge d_{A}\beta }

for A {\displaystyle A}-forms α {\displaystyle \alpha } and β {\displaystyle \beta }. It is uniquely characterized by the conditions

( d A f ) ( ϕ ) = ρ ( ϕ ) [ f ] {\displaystyle (d_{A}f)(\phi )=\rho (\phi )[f]}

and

( d A α ) [ ϕ , ψ ] = ρ ( ϕ ) [ α ( ψ ) ] − ρ ( ψ ) [ α ( ϕ ) ] − α [ ϕ , ψ ] {\displaystyle (d_{A}\alpha )[\phi ,\psi ]=\rho (\phi )[\alpha (\psi )]-\rho (\psi )[\alpha (\phi )]-\alpha [\phi ,\psi ]}

for functions f {\displaystyle f} on M {\displaystyle M}, A {\displaystyle A}-1-forms α ∈ Γ ( A ∗ ) {\displaystyle \alpha \in \Gamma (A^{*})} and ϕ , ψ {\displaystyle \phi ,\psi } sections of A {\displaystyle A}.

The definition

A Lie bialgebroid consists of two Lie algebroids ( A , ρ A , [ ⋅ , ⋅ ] A ) {\displaystyle (A,\rho _{A},[\cdot ,\cdot ]_{A})} and ( A ∗ , ρ ∗ , [ ⋅ , ⋅ ] ∗ ) {\displaystyle (A^{*},\rho _{*},[\cdot ,\cdot ]_{*})} on the dual vector bundles A → M {\displaystyle A\to M} and A ∗ → M {\displaystyle A^{*}\to M}, subject to the compatibility

d ∗ [ ϕ , ψ ] A = [ d ∗ ϕ , ψ ] A + [ ϕ , d ∗ ψ ] A {\displaystyle d_{*}[\phi ,\psi ]_{A}=[d_{*}\phi ,\psi ]_{A}+[\phi ,d_{*}\psi ]_{A}}

for all sections ϕ , ψ {\displaystyle \phi ,\psi } of A {\displaystyle A}. Here d ∗ {\displaystyle d_{*}} denotes the Lie algebroid differential of A ∗ {\displaystyle A^{*}} which also operates on the multivector fields Γ ( ∧ A ) {\displaystyle \Gamma (\wedge A)}.

Symmetry of the definition

It can be shown that the definition is symmetric in A {\displaystyle A} and A ∗ {\displaystyle A^{*}}, i.e. ( A , A ∗ ) {\displaystyle (A,A^{*})} is a Lie bialgebroid if and only if ( A ∗ , A ) {\displaystyle (A^{*},A)} is.

Examples

  1. A Lie bialgebra consists of two Lie algebras ( g , [ ⋅ , ⋅ ] g ) {\displaystyle ({\mathfrak {g}},[\cdot ,\cdot ]_{\mathfrak {g}})} and ( g ∗ , [ ⋅ , ⋅ ] ∗ ) {\displaystyle ({\mathfrak {g}}^{*},[\cdot ,\cdot ]_{*})} on dual vector spaces g {\displaystyle {\mathfrak {g}}} and g ∗ {\displaystyle {\mathfrak {g}}^{*}} such that the Chevalley–Eilenberg differential δ ∗ {\displaystyle \delta _{*}} is a derivation of the g {\displaystyle {\mathfrak {g}}}-bracket.
  2. A Poisson manifold ( M , π ) {\displaystyle (M,\pi )} gives naturally rise to a Lie bialgebroid on T M {\displaystyle TM} (with the commutator bracket of tangent vector fields) and T ∗ M {\displaystyle T^{*}M} (with the Lie bracket induced by the Poisson structure). The T ∗ M {\displaystyle T^{*}M}-differential is d ∗ = [ π , ⋅ ] {\displaystyle 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 ⇉ M {\displaystyle G\rightrightarrows M} together with a Poisson structure π {\displaystyle \pi } on G {\displaystyle G} such that the graph m ⊂ G × G × ( G , − π ) {\displaystyle 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 {\displaystyle M} is a point). Another example is a symplectic groupoid (where the Poisson structure is non-degenerate on T G {\displaystyle TG}).

Differentiation of the structure

Remember the construction of a Lie algebroid from a Lie groupoid. We take the t {\displaystyle t}-tangent fibers (or equivalently the s {\displaystyle s}-tangent fibers) and consider their vector bundle pulled back to the base manifold M {\displaystyle M}. A section of this vector bundle can be identified with a G {\displaystyle G}-invariant t {\displaystyle t}-vector field on G {\displaystyle G} which form a Lie algebra with respect to the commutator bracket on T G {\displaystyle TG}.

We thus take the Lie algebroid A → M {\displaystyle A\to M} of the Poisson groupoid. It can be shown that the Poisson structure induces a fiber-linear Poisson structure on A {\displaystyle A}. Analogous to the construction of the cotangent Lie algebroid of a Poisson manifold there is a Lie algebroid structure on A ∗ {\displaystyle A^{*}} induced by this Poisson structure. Analogous to the Poisson manifold case one can show that A {\displaystyle A} and A ∗ {\displaystyle A^{*}} form a Lie bialgebroid.

Double of a Lie bialgebroid and superlanguage of Lie bialgebroids

For Lie bialgebras ( g , g ∗ ) {\displaystyle ({\mathfrak {g}},{\mathfrak {g}}^{*})} there is the notion of Manin triples, i.e. c = g + g ∗ {\displaystyle c={\mathfrak {g}}+{\mathfrak {g}}^{*}} can be endowed with the structure of a Lie algebra such that g {\displaystyle {\mathfrak {g}}} and g ∗ {\displaystyle {\mathfrak {g}}^{*}} are subalgebras and c {\displaystyle c} contains the representation of g {\displaystyle {\mathfrak {g}}} on g ∗ {\displaystyle {\mathfrak {g}}^{*}}, vice versa. The sum structure is just

[ X + α , Y + β ] = [ X , Y ] g + a d α Y − a d β X + [ α , β ] ∗ + a d X ∗ β − a d Y ∗ α {\displaystyle [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 {\displaystyle A} is Π A {\displaystyle \Pi A}, the supermanifold whose space of (super)functions are the A {\displaystyle 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 ∗ ) {\displaystyle (A,A^{*})} should be Π A + Π A ∗ {\displaystyle \Pi A+\Pi A^{*}}. But unfortunately d A + d ∗ | Π A + Π A ∗ {\displaystyle d_{A}+d_{*}|\Pi A+\Pi A^{*}} is not a differential, basically because A + A ∗ {\displaystyle A+A^{*}} is not a Lie algebroid. Instead using the larger N-graded manifold T ∗ [ 2 ] A [ 1 ] = T ∗ [ 2 ] A ∗ [ 1 ] {\displaystyle T^{*}[2]A[1]=T^{*}[2]A^{*}[1]} to which we can lift d A {\displaystyle d_{A}} and d ∗ {\displaystyle d_{*}} as odd Hamiltonian vector fields, then their sum squares to 0 {\displaystyle 0} iff ( A , A ∗ ) {\displaystyle (A,A^{*})} is a Lie bialgebroid.

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