nLab foliation of a Lie algebroid



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Higher Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids



Related topics


\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



A generalization of the notion of foliation of a smooth manifold from manifolds to Lie algebroids.


One of several equivalent definitions of a (regular) foliation of a smooth manifold is


A regular foliation of a smooth manifold XX is a wide sub-Lie algebroid of its tangent Lie algebroid, hence a Lie algebroid 𝒫\mathcal{P} over XX with injective anchor map

𝒫 TX X = X. \array{ \mathcal{P} &\hookrightarrow & T X \\ \downarrow && \downarrow \\ X &=& X } \,.

In this spirit there is an evident generalization of the notion to a notion of foliations of Lie algebroids.


A (regular) foliation of a Lie algebroid AA is a sub-double Lie algebroid of the tangent double Lie algebroid which is wide over AA

𝒫 TA dp A p TA 𝒫 0 TX A p A X. \array{ \mathcal{P} &\hookrightarrow& T A \\ \downarrow && {}^{\mathllap{d p_A}}\downarrow & \searrow^{\mathrlap{p_{T A}}} \\ \mathcal{P}_0 &\hookrightarrow& T X && A \\ && & \searrow & \downarrow^{\mathrlap{p_A}} \\ && && X } \,.

Foliations of a Lie algbroid AXA \to X according to def. are in natural bijection to the following data:

  1. an ordinary foliation 𝒫 0TX\mathcal{P}_0 \hookrightarrow T X

  2. 𝒫 1A\mathcal{P}_1 \hookrightarrow A a sub-vector bundle

    (this is the joint kernel 𝒫 1=ker(p TA)ker(dp A)\mathcal{P}_1 = ker(p_{T A}) \cap ker(d p_{A}) naturally identified as a subspace of AA)

  3. \nabla a flat connection on the quotient bundle A/𝒫 1A/\mathcal{P}_1 partially defined over vector fields in 𝒫 0\mathcal{P}_0

    (this is the induced linear foliation of the total space AA regarded as a horizontal-subspace distribution)

such that over every open subset UXU \hookrightarrow X

  1. The sections of AA that become \nabla-constant in A/𝒫 1A/\mathcal{P}_1 form a sub-Lie algebra of Γ U(A| U)\Gamma_U(A|_U) (an integrable distribution of subspaces);

  2. the sections of 𝒫 1\mathcal{P}_1 are a Lie ideal inside this sub-Lie algebra;

  3. the image of 𝒫 1\mathcal{P}_1 under the anchor map is in 𝒫 0\mathcal{P}_0;

  4. the quotiented anchor map A/𝒫 1TX/𝒫 0A/\mathcal{P}_1 \to T X / \mathcal{P}_0 intertwines \nabla with the 𝒫 0\mathcal{P}_0-Bott connection.

This is (EH, theorem 7.2).


According to prop. the notion of foliation of a Lie algebroid generalizes the notion of ideal system of a Lie algebroid (Higgins-Mackenzie, Mackenzie): A foliation as in def. comes from an ideal system in this sense precisely of 𝒫 0\mathcal{P}_0 is a simple foliation (the quotient map exists in smooth manifolds and is a surjective submersion) and the holonomy of \nabla is trivial.



Let 𝒢 \mathcal{G}_\bullet be a Lie groupoid and let

𝒫 T𝒢 1 ds dt 𝒫 0 T𝒢 0 \array{ \mathcal{P} &\hookrightarrow& T \mathcal{G}_1 \\ \downarrow\downarrow && {}^{\mathllap{d s}}\downarrow \downarrow^{\mathrlap{d t}} \\ \mathcal{P}_0 &\hookrightarrow& T \mathcal{G}_0 }

be a foliation of a Lie groupoid, regarded as an internal groupoid in Lie algebroids. Then applying Lie differentiation yields a foliation of the Lie algebroid Lie(𝒢 )Lie(\mathcal{G}_\bullet).


Maybe the first discussion of foliations of Lie algebroids appears in

Ideal systems of Lie algebroids have been introduced and studied in

  • Philip Higgins, Kirill Mackenzie, Algebraic constructions in the category of Lie algebroids . J. Algebra

    129 (1990), no. 1, 194–230.MR1037400

  • Kirill Mackenzie, General theory of Lie groupoids and Lie algebroids Cambridge Univ. Press, Cambridge (2005)

Related discussion is in

Last revised on August 14, 2017 at 06:31:13. See the history of this page for a list of all contributions to it.