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
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