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foliation of a Lie algebroid

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential 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}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

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)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

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

Definition

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

Definition

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.

Definition

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 } \,.
Proposition

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).

Remark

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.

Examples

Example

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).

Referemces

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 02:31:13. See the history of this page for a list of all contributions to it.