¹

Introduction↩︎

This article reconciles two modern approaches to PDEs and geometric structures with their symmetries, namely Pfaffian fibrations and groupoids, and relative algebroids.

The Pfaffian approach, developed by Accornero, Cattafi, Crainic, Yudilevich and Salazar in [1]–[8], studies the essential data of a PDE, encoding it in a Pfaffian fibration, as well as the essential data of a Lie pseudogroup: a Pfaffian groupoid. The focus of their work has been on geometric structures. A comprehensive treatment of the theory will be published in the upcoming book [3]. The structural object of a Pfaffian fibration is a distribution.

The study of relative algebroids, initiated by Yudelivich, Fernandes and the author of this article in [9] (see also [10]), draws from aspects of Lie theory to study existence and classification problems for geometric structures [11], and unifies the theory of Lie algebroids with that of formal PDEs. It explains the appearance of Lie algebroids behind several classification problems for geometric structures [12]–[15], and codifies what is left of a PDE after taking the quotient by symmetries. The structural object of a relative algebroid is a relative derivation.

Both approaches are seemingly different but build on the formal theory of PDEs, and many analogies can be observed. The present article establishes a robust relationship between the frameworks in two fundamental ways:

1. Every Pfaffian fibration gives rise to a relative algebroid in a natural way. The underlying relative algebroid shares the same formal theory as the Pfaffian fibration it is induced from.

2. We introduce a framework for both pseudogroup and groupoid symmetries of Pfaffian fibrations, and show that they preserve the underlying relative algebroid in an appropriate sense.

We believe that these results provide a clear picture of their interactions.

The framework for symmetries of Pfaffian fibrations presented here is a generalization of PDEs with symmetries, but with some adaptations to make the interaction with relative algebroids more clear. More specifically:

1. We introduce two different types of symmetries: internal and Pfaffian symmetries. Internal symmetries are a direct generalization of internal symmetries for PDEs [16], while Pfaffian symmetries of a Pfaffian fibration interact well with the underlying relative algebroid.

2. We show that internal symmetries prolong to Pfaffian symmetries, and that conversely every Pfaffian symmetry of the prolongation is the prolongation of an internal symmetry when the Pfaffian fibration is involutive (Theorem 10).

3. We develop a theory of groupoid actions of Pfaffian groupoids on Pfaffian fibrations by internal as well as by Pfaffian symmetries. Our notions more general than the Pfaffian actions developed in [1], [3], [4], [7], which allows for natural examples of PDEs with symmetries that are not Pfaffian actions (see Section 4.4).

4. Pfaffian and internal actions are related as follows. The prolongation of an action by internal symmetries is a partial action by Pfaffian symmetries. Partial groupoid actions have appeared in the literature before [17], [18], and seem to appear naturally in our context.

The various notions of in this article are related as depicted by the following diagram. \[\begin{tikzcd}[column sep=6em] {\substack{Pfaffian fibrations and\\Pfaffian groupoids} } \arrow[r] & \substack{Relative\arrow[dl, shift left, dashed, "\text{Applications}"] \\algebroids} \\ {\substack{PDEs and\\Lie pseudogroups} } \arrow[u, hook] \arrow[ru, shift left] & \end{tikzcd}\] The dashed arrow—indicating applications of the results of this article to PDEs with symmetries—will be a topic of further study. We will briefly indicate to the possible directions for applications at the end of this introduction. Furthermore, we believe that the connection between symmetries of Pfaffian fibrations and the current literature on symmetries of PDEs could be further developed.

Overview of the article↩︎

In this paper, we work with algebroids relative to a foliation. Such an object \((B, {\overline{\nabla}}, {\mathrm{D}})\) consists of:

• a manifold with foliation \((M, \mathcal{F})\), where \(\mathcal{F}\subset TM\) is an involutive subbundle,

• a vector bundle \((B, {\overline{\nabla}})\to (M, \mathcal{F})\) with flat \(\mathcal{F}\)-connection \({\overline{\nabla}}\),

• a derivation of degree 1 relative to \(\mathcal{F}\) : \[{\mathrm{D}}\colon \Omega^\bullet_{(B, {\overline{\nabla}})} \to \Omega^{\bullet+1}_B.\]

Here, \(\Omega^\bullet_{B}\) stands for the sheaf of forms on \(B\) (i.e. sections of \(\wedge^\bullet B^*\)), and \(\Omega^\bullet_{(B, {\overline{\nabla}})}\) denotes the sheaf of flat forms (w.r.t. \(\mathrel{{\overline{\nabla}}}\)). We refer to Appendix 5, or the articles [9], [10] for more details on this notion.

From Pfaffian fibrations to relative algebroids↩︎

In Section 1, we recall the basic definitions of Pfaffian fibrations and explain how every Pfaffian fibration induces a relative algebroid in a natural way. In this article, we mostly work from the perspective of distributions, so, to us, a Pfaffian fibration \((P, \mathcal{C}, \pi)\) is a submersion \(\pi\colon P\to X\) with a distribution \(\mathcal{C}\subseteq TP\) that is transverse to \(\ker T\pi\) and for which \(\mathcal{C}^\pi:= \mathcal{C}\cap \ker T\pi\) is involutive. A (local) solution to a Pfaffian fibration is a (local) holonomic section of \(\pi\), i.e. a (local) section \(\sigma\colon X\dasharrow P\) with image tangent to \(\mathcal{C}\).

A particularly important example is the bundle of first jets \(\pi\colon J^1q\to X\) of a submersion \(q\colon Q\to X\). The Cartan distribution \(\mathcal{C}^1\) makes \((J^1q, \mathcal{C}^1, \pi)\) into a Pfaffian fibration. Locally, this is the universal example [5]. In this case, the holonomic sections are precisely those of the form \(j^1\sigma\) for some section \(\sigma\) of \(q\).

A partial differential equation (PDE) is a subspace \(\mathcal{E}\subset J^1q\) for some submersion \(q: Q\to X\). By restricting the Cartan distribution \(\mathcal{C}^1\) on \(J^1q\) to \(\mathcal{E}\) we obtain a Pfaffian fibration \((\mathcal{E}, \mathcal{C}^1, \pi)\) whose holonomic sections correspond precisely to the solutions of \(\mathcal{E}\).

A central result in [9] is that every PDE canonically induces a relative algebroid. If \(\mathcal{E}\subset J^1q\) is a PDE on sections of some submersion \(q\colon Q\to X\), then the vector bundle of the relative algebroid underlying the PDE is \(\pi^*TX\to J^1q\), and the derivation corresponds to the horizontal derivation in the corner of the variational bicomplex.

We extend this result to Pfaffian fibrations in Section 1.2. In particular, we show that to any Pfaffian fibration \((P, \mathcal{C}, \pi)\) one can naturally associate a relative algebroid \((\pi^*TX, {\overline{\nabla}}, {\mathrm{D}}^\mathcal{C})\), where the derivation is entirely determined by the distribution \(\mathcal{C}\subset TP\) and the de Rham differential \({\mathrm{d}}\) on \(TX\). This algebroid is not relative to the submersion \(\pi\) but rather relative to the foliation \(\mathcal{C}^\pi= \mathcal{C}\cap \ker T\pi\). The following theorem establishes a tight relationship between the formal theory of a Pfaffian fibration and of its underlying relative algebroid.

Pfaffian groupoids↩︎

In Section 2, we recall some general notions about Pfaffian groupoids. In this article, a Pfaffian groupoid \((\mathcal{G}, \mathcal{H})\rightrightarrows M\) is a Lie groupoid \(\mathcal{G}\rightrightarrows M\) with a multiplicative distribution \(\mathcal{H}\subset T\mathcal{G}\) that makes the source map \(s\colon \mathcal{G}\to M\) into a Pfaffian fibration. Because a Pfaffian groupoid is in particular a Pfaffian fibration, we can talk about prolongations and \(k\)-integrability.

Symmetries of Pfaffian fibrations↩︎

In Section 3, we turn to our second theme of symmetries of Pfaffian fibrations. The essential distributions of a Pfaffian fibration \((P, \mathcal{C}, \pi)\) are \(\mathcal{C}\) and \(\mathcal{C}^\pi = \mathcal{C}\cap \ker T\pi\). The pseudgroups of internal symmetries and Pfaffian symmetries are, respectively, \[\begin{align} \mathop{\mathrm{Diff}}_{\mathrm{loc}}(P,\mathcal{C}) &= \{ \varphi\in \mathop{\mathrm{Diff}}_{\mathrm{loc}}(P) \;| \;T\varphi(\mathcal{C})\subseteq \mathcal{C}\},\\ \mathop{\mathrm{Diff}}_{\mathrm{loc}}(P, \mathcal{C}, \pi) &= \{ \varphi\in \mathop{\mathrm{Diff}}_{\mathrm{loc}}(P, \mathcal{C}) \;| \;T \varphi(\mathcal{C}^\pi) \subseteq \mathcal{C}^\pi \}. \end{align}\] We show that:

1. Pfaffian symmetries induce local isomorphisms of the underlying relative algebroid (Proposition 12).

2. Internal symmetries of \((P, \mathcal{C}, \pi)\) induce Pfaffian symmetries of the prolongation \((P^{(1)}, \mathcal{C}^1, \pi^{(1)})\). Moreover, when \((P, \mathcal{C}, \pi)\) is involutive, Pfaffian symmetries of the prolongation are (locally) prolongations of internal symmetries (Theorem 10).

Perhaps more interesting are the Pfaffian groupoids of internal and Pfaffian symmetries, defined as subgroupoids of the groupoid first jets of diffeomorphisms: \[\begin{align} \mathcal{G}_\mathcal{C}&= \{ j^1_x\varphi \in J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(P) \;| \;T_x\varphi(\mathcal{C}_x) \subseteq \mathcal{C}_{\varphi(x)}\},\\ \mathcal{G}_{\mathcal{C}, \pi} &= \{ j^1_x\varphi\in J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(P) \;| \; T_x\varphi(\mathcal{C}_x) \subseteq \mathcal{C}_{\varphi(x)} , \;T_x\varphi(\mathcal{C}^\pi_x) \subseteq \mathcal{C}^\pi_{\varphi(x)} \}, \end{align}\] and are equipped with the Cartan distribution \(\mathcal{H}^1\) inherited from \(J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(P)\). The central result of this part is the following:

Groupoid actions by Pfaffian symmetries↩︎

Often, a PDE comes with a specific pseudogroup symmetry given by the context (e.g. diffeomorphism-invariance). In the final part, Section 4, we capture this by introducing a notion of an action of a Pfaffian groupoid on a Pfaffian fibration by internal symmetries and by Pfaffian symmetries. More precisely, a groupoid action \[\xymatrix{ (\mathcal{G}, \mathcal{H}) \ar@<0.5ex>[d] \ar@<-0.5ex>[d] & {(P, \mathcal{C}, \pi)} \ar@`{[l]+/l-0.2pc/+/d+1.8pc/,[l]+/l-0.2pc/+/u+1.8pc/}[0, 0]_{m} \ar[ld]^{\mu} \\ M & }\] of a Pfaffian groupoid \((\mathcal{G}, \mathcal{H})\rightrightarrows M\) on a Pfaffian fibration \((P, \mathcal{C}, \pi)\) is an action by internal symmetries when it satisfies \[Tm\left( \mathcal{H}\leftindex_{Ts}{\times}_{T\mu} \mathcal{C}\right) \subseteq \mathcal{C},\] and an action by Pfaffian symmetries when in addition \[Tm \left( \mathcal{H}\leftindex_{Ts}{\times}_{T\mu} \mathcal{C}^\pi \right) \subseteq \mathcal{C}^\pi\] is satisfied. These imply that the holonomic bisections of \((\mathcal{G}, \mathcal{H})\) act by internal (resp. Pfaffian) symmetries of \((P, \mathcal{C}, \pi)\).

As for the pseudogroups defined above, actions by internal symmetries and by Pfaffian symmetries are related through prolongation. However, if \((\mathcal{G}, \mathcal{H})\curvearrowright (P, \mathcal{C}, \pi)\) is an action by internal symmetires, then it only induces a partial action (sometimes called a local action²) on the prolongation. In Section 2.4, we briefly recall what a partial action of a groupoid is (see also [17], [18]).

Finally, if \((\mathcal{G}, \mathcal{H})\rightrightarrows M\) is sufficiently regular, we show that a Pfaffian action it leaves the derivation of the Pfaffian fibration invariant.

Often, the action of \(\mathcal{G}^{(1)}\) on \(\pi^*TX\) descends to an action of \(\mathcal{G}\) itself on \(\pi^*TX\). This happens for instance when the action is the prolongation of an action by internal symmetries. In that case, the corollary implies that the derivation preserves \(\mathcal{G}\)-invariant sections, so it descends to the quotient!

PDEs with symmetries and Pfaffian actions↩︎

In the last part, Section 4.4, we demonstrate that the notion of a Pfaffian action, developed in [2]–[4], is not sufficient to capture PDEs with symmetries. Our example comes from the point symmetries of the first jet space \(J^1q\) of a submersion \(q\colon Q\to X\). It gives rise to the following action \[\xymatrix{ (J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q), \omega_Q) \ar@<0.5ex>[d] \ar@<-0.5ex>[d] & {(J^1q, \theta_Q, \pi)} \ar@`{[l]+/l-2.2pc/+/d+1.8pc/,[l]+/l-2.2pc/+/u+1.8pc/}[0, 0]_{m} \ar[ld]^{p_1} \\ Q,& }\] where \(\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q)\) is the pseudogroup of diffeomorphisms preserving \(\ker Tq\), and \(\omega_Q\) and \(\theta_Q\) are the Cartan forms on \(J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q)\) and \(J^1q\), respectively.

There is a groupoid map \(\overline{q}\colon J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q)\to J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(X)\), and the action satisfies the following “twisted Pfaffian" equation: \[\mathop{\mathrm{pr}}^*_\mathcal{G}\omega_Q - (g\cdot h) \circ \left(\overline{q}^*\omega_X\right) = m^*\theta_Q - g \cdot \left(\mathop{\mathrm{pr}}_{J^1q}^*\theta_Q\right),\] which shows that the action is not a Pfaffian action in the sense of [3], [4]. Instead, it is an action by Pfaffian symmetries.

Appendices↩︎

Appendix 5 gives a summary of the main notions around relative derivations and relative algebroids, and states specific results used in the main part of the article. Some parts are new, such as the notions of a groupoid representation on flat foliated bundles and on relative algebroids (Sections 5.9 and 5.10).

Appendix 6 discusses involutivity, Cartan characters, and regular flags for tableau maps (introduced as generalized tableau in [7]), providing a technical result necessary for the proof of Theorem 10.

Applications↩︎

A fruitful direction of applications will come from the study of the quotient relative algebroid of a PDE by symmetries. Let us first briefly mention how PDEs with pseudogroup symmetries fit into the framework of this paper.

For any pseudogroup \(\Gamma\) (of diffeomorphisms) on a manifold \(M\), the \(k\)-jets form a groupoid \(J^k\Gamma\) whose multiplication is induced from composition of diffeomorphisms. If \(J^k\Gamma\rightrightarrows M\) is smooth, it becomes a Pfaffian groupoid \((J^k\Gamma, \mathcal{H}^k)\) with the Cartan distribution \(\mathcal{H}^k\) inherited from \(J^k\mathop{\mathrm{Diff}}_{\mathrm{loc}}(M)\).

A PDE with symmetries \((E, \Gamma)\) is a PDE \(E\subset J^k q\) on section of a submersion \(q\colon Q\to X\) together with a pseudogroup \(\Gamma\) on \(E\) that preserves the Cartan distribution. Interpreting \(E\) as a Pfaffian fibration, \(\Gamma\) is a pseudogroup of internal symmetries. In this case there are partial actions \(J^k \Gamma \hat{\curvearrowright} E^{(k)}\) on the prolongation spaces of \(E\). Such actions are instances of actions by Pfaffian symmetries. PDEs with symmetries have been extensively studied in the literature [16], [19]–[24].

A huge advantage of the framework over relative algebroids is that it is stable under quotients. The current article sets it up by showing that the derivation of the relative algebroid underlying a Pfaffian fibration is invariant under symmetries. When quotient spaces exist, it descends to the structure of a relative algebroid on the quotient.

One instance of this phenomenon was already presented in [10] to construct a relative algebroid on the space of \(k\)-jets of \(n\)-dimensional submanifolds \(J^k(M, n)\) of a manifold \(M\), and to construct the relative algebroid associated to a relative distribution.

In a subsequent paper, we will study this quotient construction in detail and apply it to PDEs with symmetries. For instance, through a quotient construction, there should be a direct relationships between diffeomorphsim-invariant PDEs on Riemannian metrics and the relative algebroids found through Bryant’s equations in [11].

The quotient of a PDE by symmetries lives over the space of invariants, and we expect there to be a close relationship between the quotient relative algebroid and the Lie-Tresse theorem [21].

A different direction is to establish the relationship between relative algebroids the moving-frame method for Lie pseudogroups developed by Olver and Pohjanpelto [23].

Acknowledgments↩︎

This article was written to properly answer a question raised by Luca Vitagliano following a talk by the author during the Mini-Workshop on Groupoids, Algebroids & Differentiable Stacks, which took place at the University of Napoli Federico II, June 11–14, 2024.

The author especially thanks Luca Accornero for pointing out that actions by Pfaffian symmetries should be treated from the distributional sense (Remark 25), a foundational remark that inspired much of the content of this article.

Additionally, the author thanks Luca Accornero, Francesco Cattafi, Marius Crainic and Maria Amelia Salazar for their answers to many questions about the framework of Pfaffian fibrations and Pfaffian actions, which will appear in full detail in their upcoming book [3]. Access to an early draft has been particularly helpful.

1 Pfaffian fibrations↩︎

In this section, we establish the direct connection between Pfaffian fibrations and relative algebroids. A rapid introduction to relative algebroids has been included in Appendix 5, but for more details and examples we refer to [9], [10].

1.1 Pfaffian fibrations↩︎

We start by summarizing the main aspects of the theory on Pfaffian fibrations—for a more detailed introduction that includes motivation and examples, see [4], [5], [7].

Let \((P, \mathcal{C}, \pi)\) be a Pfaffian fibration. The curvature map of a Pfaffian fibration \((P, \mathcal{C}, \pi)\) is given by \[\kappa_\mathcal{C}\colon \mathcal{C}\times \mathcal{C}\to TP/\mathcal{C}, \quad \kappa_\mathcal{C}(X, Y) = [\tilde{X}, \tilde{Y}]\mod \mathcal{C}.\] where \(\tilde{X}, \tilde{Y}\in \Gamma(\mathcal{C})\) are any sections through \(X, Y\in \mathcal{C}\).

The tableau map or symbol map of a Pfaffian fibration \((P, \mathcal{C}, \pi)\) is \[\tau_\mathcal{C}\colon \mathcal{C}^\pi \to \operatorname{Hom}(\pi^*TX, TP/\mathcal{C}), \quad \tau_\mathcal{C}(v)(X) = \kappa_\mathcal{C}(v, \tilde{X})\] for \(v\in \mathcal{C}^\pi\), \(X\in TX\) and \(\tilde{X}\in \mathcal{C}\) such that \(T\pi(\tilde{X}) = X\).

Recall that \(J^1\pi\) can be identified with the bundle of (pointwise) splittings of \(T\pi\): \[J^1\pi = \{ h\colon T_{\pi(x)} X\to T_xP \; | \; T_x\pi \circ h = \operatorname{id}\}\] Any holonomic section \(\sigma\colon X\to P\) satisfies \(\sigma^*\theta_\mathcal{C}= 0\) and consequently \(\sigma^*\kappa_\mathcal{C}= 0\). The prolongation of a Pfaffian fibration takes into account these two differential consequences.

The partial prolongation is always smooth and defines an affine bundle \(p_1\colon J^1_\mathcal{C}P\to P\) modeled on \(\operatorname{Hom}(\pi^*TX, \mathcal{C}^\pi)\). The prolongation, however, may not be smooth. When it does, and \(p_1\colon P^{(1)}\to P\) is a submersion, we call \((P, \mathcal{C}, \pi)\) 1-integrable. In that case, \(p_1\colon P^{(1)}\to P\) is an affine bundle modeled on \[\tau_\mathcal{C}^{(1)} = \{ \xi\in \operatorname{Hom}(\pi^*TX, \mathcal{C}^\pi) \;| \;\delta_\tau(\xi) = 0\}\] where \(\delta_\tau(\xi) (X, Y) = \tau(\xi(X))(Y) - \tau(\xi(Y))(X)\) for \(X, Y\in \pi^*TX\).

Note that both the partial prolongation and the prolongations are PDEs, as they are subspaces of \(J^1\pi\). This way, we can regard the (partial) prolongation as a new Pfaffian fibration \((P^{(1)}, \mathcal{C}^{1}, \pi^{(1)})\) (or \((J^1_\mathcal{C}P, \mathcal{C}^{1}, \pi^{(1)})\)) with the Cartan distribution of a PDE (Example 1).

The higher prolongations of \((P, \mathcal{C}, \pi)\) are defined iteratively: \[(P, \mathcal{C}, \pi)^{(k)} := (P^{(k)}, \mathcal{C}^k, \pi^{(k)}) = (P^{(k-1)}, \mathcal{C}^{k-1}, \pi^{(k-1)})^{(1)}.\] When the first \(k\) prolongations exist, then the Pfaffian fibration is called \(k\)-integrable. If it is \(k\)-integrable for all \(k\), then it is formally integrable.

1.2 Pfaffian fibrations as relative algebroids↩︎

Next we explain how a Pfaffian fibrations gives rise to a relative algebroid. We start by explaining how the various components arise from the Pfaffian fibration. The two essential distributions of a Pfaffian fibration \((P, \mathcal{C}, \pi)\) are the Cartan distribution \(\mathcal{C}\) and the foliation \(\mathcal{C}^\pi = \mathcal{C}\cap \ker T\pi\). The transversality condition implies that there is a canonical splitting \[\nu(\mathcal{C}^\pi) \cong \mathcal{C}/\mathcal{C}^\pi \oplus \ker T\pi /\mathcal{C}^\pi\] for which the projection \(\overline{T\pi}\colon \nu(\mathcal{C}^\pi)\to \pi^*TX\) induces an isomorphism \(\mathcal{C}/\mathcal{C}^\pi \cong \pi^*TX\). We denote the projections and inclusions associated to this splitting by the following: \[\begin{align} \Pi:= \overline{T\pi}&\colon \nu(\mathcal{C}^\pi) \to \pi^*TX, & I&\colon \pi^*T^*X \hookrightarrow \nu^*(\mathcal{C}^\pi),\\ I&\colon\pi^*TX\to \nu(\mathcal{C}^\pi), & \quad \Pi&\colon \nu^*(\mathcal{C}^\pi) \to \pi^*T^*X. \end{align}\]

The flat foliated bundle↩︎

The bundle of the relative algebroid is \(B\colon= \pi^*TX\to P\). It has a canonical flat \(\ker T\pi\)-connection \({\overline{\nabla}}\) that can be restricted to \(\mathcal{C}^\pi \subset \ker T\pi\) to obtain a canonical flat \(\mathcal{C}^\pi\)-bundle \((B, {\overline{\nabla}})\). In other words, the connection of the bundle \((B, {\overline{\nabla}})\) is characterized by \[{\overline{\nabla}}_v \pi^*X = 0, \quad for v\in \Gamma(\mathcal{C}^\pi) and X\in \mathfrak{X}(X),\] and extended through the Leibniz rule.

There is a more concrete description of \({\overline{\nabla}}\) in terms of the canonical flat \(\mathcal{C}^\pi\)-bundle \((\nu(\mathcal{C}^\pi), {\overline{\nabla}}^{\mathrm{Bott}})\). While \({\overline{\nabla}}^{\mathrm{Bott}}\) does not preserve \(\Gamma(\mathcal{C})\), we claim that \[\label{eq:FlatConnectionPfaffian} {\overline{\nabla}}_v b = \Pi \left({\overline{\nabla}}^{\mathrm{Bott}}_v I(b) \right)\tag{2}\] for \(b\in \Gamma(B)\). Indeed, both sides vanish when \(b = \pi^*X\) for some \(X\in \mathfrak{X}(X)\), and satisfy the same Leibniz rule.

For future reference we will also derive an explicit formula for \({\overline{\nabla}}\) on \(B^*\). First, we check that the Bott connection \({\overline{\nabla}}^{\mathrm{Bott}}\) on \(\nu^*(\mathcal{C}^\pi)\) is given by \[{\overline{\nabla}}^{\mathrm{Bott}}_v \alpha = -\iota_v {\mathrm{d}}\alpha\] for \(v\in \Gamma(\mathcal{C}^\pi)\) and \(\alpha\in \Gamma(\nu^*(\mathcal{C}^\pi))\). By regarding \(\nu^*(\mathcal{C}^\pi) \subseteq T^*P\) as the annihilator of \(\mathcal{C}^\pi\), the expression \({\mathrm{d}}\alpha\) makes sense. Dualizing Equation (2 ) yields: \[\label{eq:DualFlatConnectionPfaffian} {\overline{\nabla}}_v \alpha = - \Pi(\iota_v {\mathrm{d}}I(\alpha))\tag{3}\] for \(v\in \Gamma(\mathcal{C}^\pi)\), \(\alpha \in \Gamma(B^*)\)

The canonical relative derivation↩︎

The relative derivation \({\mathrm{D}}^\mathcal{C}\) induced by the Pfaffian fibration on \((B, {\overline{\nabla}})\) is characterized by the symbol \(I\colon B\to \nu(\mathcal{C}^\pi)\) and the equation \({\mathrm{D}}^\mathcal{C}\pi^*\alpha = \pi^*{\mathrm{d}}\alpha\) for \(\alpha\in\Omega^\bullet_{TX}\). It can be seen as an extension of the de Rham differential on \(TX\) to \((\pi^*TX, {\overline{\nabla}})\), a statement that will be made precise in Proposition 4. Alternatively, the derivation \({\mathrm{D}}^\mathcal{C}\) can be defined through the equation \[\label{eq:PfaffianDerivationExplicit} {\mathrm{D}}^\mathcal{C}\alpha = \Pi ({\mathrm{d}}I(\alpha)),\tag{4}\] for \(\alpha\in \Omega^\bullet_{(\pi^*TX, {\overline{\nabla}})}\). Here, \(I(\alpha)\) is regarded as a section of \(\nu^*(\mathcal{C}^\pi)\subset T^*P\) so that the exterior derivative makes sense. In Equation (4 ), it is crucial that \(\alpha\) is a flat section of \((\wedge^\bullet B^*, {\overline{\nabla}})\), otherwise \({\mathrm{d}}I(\alpha)\) would not be a section of \(\wedge^{\bullet+1}\nu^*(\mathcal{C}^\pi)\).

The bracket \([\cdot, \cdot]^\mathcal{C}\) dual to the derivation \({\mathrm{D}}^\mathcal{C}\) has a similar description. It is determined by \[[\pi^*X, \pi^*Y]^\mathcal{C}= \pi^*[X, Y], \quad X, Y\in \mathfrak{X}(X)\] with anchor \(I\colon B\to \nu(\mathcal{C}^\pi)\). Alternatively, the bracket is given by \[[b_1, b_2]^\mathcal{C}= \Pi(\overline{[X_1, X_2]}),\] where \(X_1, X_2\in \Gamma_{\mathcal{C}}\) are such that \(\Pi(X_i) = b_i\) for two flat section \(b_1, b_2\in \Gamma_{(B, {\overline{\nabla}})}\). The bar above the Lie bracket indicates the projection to \(\nu(\mathcal{C}^\pi)\). We have proven the following statement.

Next, we make precise how the Pfaffian fibration and the underlying relative algebroid relate. First, we need some preliminary results that characterize Pfaffian structures as derivations.

Let \(\pi\colon P\to X\) be a submersion and \(\mathcal{F}\subset \ker T\pi\) a foliation. By restricting the canonical \(\ker T\pi\)-connection on \(\pi^*TX\) to \(\mathcal{F}\), we obtain a foliated flat bundle \((\pi^* TX, {\overline{\nabla}})\to (P, \mathcal{F})\). Because \(\mathcal{F}\) is tangent to the fibers of \(\pi\), there is a natural map \(\pi_*\colon\mathcal{D}^1_{(TX, {\overline{\nabla}})}\to \pi^*\mathcal{D}^1_{TX}\) obtained by restricting a derivation \({\mathrm{D}}\colon \Omega^\bullet_{(\pi^*TX, {\overline{\nabla}})} \to \Omega^{\bullet + 1}_{\pi^*TX}\) to \(\pi^*\Omega^\bullet_{TX}\subset\Omega^\bullet_{(\pi^*TX, {\overline{\nabla}})}\). Also, recall that we can interpret the de Rham differential \({\mathrm{d}}\) as a section of \(\mathcal{D}^1_{TX}\).

1.3 Morphisms of Pfaffian fibrations↩︎

Next, we discuss the naturality of the construction of the relative algebroid underlying a Pfaffian fibration.

The next proposition shows that the algebroid underlying the Pfaffian fibration is natural.

The next lemma is a strengthening of parts of [5].

2 Pfaffian groupoids↩︎

Next, we recall the basic notions of Pfaffian groupoids, introduced in [7]. In Section 2.4, we explain how (partial) action groupoids of a Pfaffian groupoid again a Pfaffian groupoid.

2.1 Pfaffian groupoids↩︎

A Pfaffian groupoid is a Lie groupoid that is also a Pfaffian fibration which is multiplicative in the following sense.

A local bisection \(\sigma\in \mathop{\mathrm{Bis}}_{\mathrm{loc}}(\mathcal{G})\) of a Pfaffian groupoid \((\mathcal{G}, \mathcal{H}) \rightrightarrows M\) is holonomic when \(\operatorname{im}T\sigma \subset \mathcal{H}\). The collection of local holonomic bisections \(\mathop{\mathrm{Bis}}_{\mathrm{loc}}(\mathcal{G}, \mathcal{H})\) form what’s called a generalized pseudogroup, meaning that it is closed under restriction, inversion and composition, and satisfies a gluing axiom.

2.2 Prolongations of Pfaffian groupoids↩︎

As for Pfaffian fibrations, there is a one-form (with coefficients, now multiplicative) on \(\mathcal{G}\) whose kernel is exactly \(\mathcal{H}\), given by the projection \[\omega_\mathcal{H}\colon T\mathcal{G}\to T\mathcal{G}/\mathcal{H}.\] The partial prolongation of a Pfaffian groupoid \((\mathcal{G}, \mathcal{H})\rightrightarrows M\) is \[J^1_\mathcal{H}\mathcal{G}= \{ h\in J^1\mathcal{G}\;| \;h^*\omega_\mathcal{H}= 0\}.\] Equipped with the Cartan distribution \(\mathcal{H}^1\) inherited from \(J^1\mathcal{G}\), we obtain a new Pfaffian groupoid \((J^1_\mathcal{H}\mathcal{G}, \mathcal{H}^1)\rightrightarrows M\).

The prolongation of \((\mathcal{G}, \mathcal{H})\rightrightarrows M\) consists of those elements with vanishing curvature: \[\mathcal{G}^{(1)} = \{ h \in J^1\mathcal{G}\;| \;h^*\omega_\mathcal{H}=0, \;h^*\kappa_\mathcal{H}=0\}\] If smooth, it becomes a Pfaffian groupoid \((\mathcal{G}^{(1)}, \mathcal{H}^1)\rightrightarrows M\) when equipped with the Cartan distribution \(\mathcal{H}^1\) restricted from \(J^1\mathcal{G}\).

Like for Pfaffian fibrations, we can talk about \(k\)-integrability and formal integrability for Pfaffian groupoids.

The reduced Pfaffian groupoid of \((\mathcal{G}, \mathcal{H})\) is the locus where the intrinsic torsion vanishes (cf. Remark 2). It can also be defined as the image of the prolongation space under the projection \[\overline{\mathcal{G}} = p_1(\mathcal{G}^{(1)}).\] If smooth, we obtain a new Pfaffian groupoid \((\overline{\mathcal{G}}, \mathcal{H})\rightrightarrows M\). If \(\mathcal{G}^{(1)}\) is in addition smooth, then \((\overline{\mathcal{G}}, \mathcal{H})\rightrightarrows M\) is 1-integrable, even though \((\mathcal{G}, \mathcal{H})\) may not be.

2.3 Morphisms of Pfaffian groupoids↩︎

While it is possible to consider more general morphisms of Pfaffian groupoids over different bases, the following notion is sufficient for the purposes of this paper.

2.4 Partial actions and Pfaffian groupoids↩︎

In Section 4, we will encounter natural groupoid actions that are partial, as opposed to being complete. Partial actions are different from local actions–partial refers to the space the groupoid acts on, not to the groupoid itself. Every partial action still has a well-defined action groupoid. We spend a couple of words on partial actions of groupoids, as discussed in [17], [18].

The domain of the multiplication \(m\) is assumed to be open (hidden in the notion of a local map). The conditions imply that \(\mathcal{G}\hat{\ltimes}P := \operatorname{dom}m\rightrightarrows P\) is a Lie groupoid, which we call the (partial) action groupoid.

Perhaps surprisingly, the partial action groupoid \(\mathcal{G}\hat{\ltimes}P \rightrightarrows P\) is a global Lie groupoid, and not just a local one. A partial action is distinct from a local action of a Lie groupoid, where the locality is concentrated around the unit section of the groupoid (which would give rise to a local action groupoid).

If \(\Phi\colon \mathcal{G}_1 \to \mathcal{G}_2\) is a Lie group morphism and \(\mathcal{G}_2\mathbin{\hat{\curvearrowright}}P\) is a partial action, then it induces naturally a partial action \(\mathcal{G}_1 \mathbin{\hat{\curvearrowright}}P\).

In Section 4 we will encounter partial actions of Pfaffian groupoids. If \((\mathcal{G}, \mathcal{H})\rightrightarrows M\) is a Pfaffian groupoid and \(\mathcal{G}\mathbin{\hat{\curvearrowright}}P\) a partial action along \(\mu\colon P\to M\), then we can pull back the distributions \(\mathcal{H}\subset T\mathcal{G}\) to the action groupoid \(\mathcal{G}\hat{\ltimes}P\) along the projection \(\mathop{\mathrm{pr}}_\mathcal{G}\colon \mathcal{G}\hat{\ltimes}P \to \mathcal{G}\). Concretely, \[\mathop{\mathrm{pr}}_\mathcal{G}^*\mathcal{H}= \left\{ (X, Y)\in T\mathcal{G}\leftindex_{Ts}{\times}_{T\mu} TP\big\vert_{\mathcal{G}\hat{\ltimes}P} \;\big\vert \;X\in \mathcal{H}\right\}\subset T(\mathcal{G}\hat{\ltimes}P).\] If \(\omega_\mathcal{H}\) is the form corresponding to \(\mathcal{H}\), note that \(\mathop{\mathrm{pr}}_\mathcal{G}^*\mathcal{H}= \ker \mathop{\mathrm{pr}}_\mathcal{G}^*\omega_\mathcal{H}\). After (canonically) identifying \(T(\mathcal{G}\hat{\ltimes}\mathcal{H})/ \mathop{\mathrm{pr}}_\mathcal{G}^*\mathcal{H}\cong \mathop{\mathrm{pr}}^*_\mathcal{G}\left(T\mathcal{G}/\mathcal{H}\right)\), it actually holds that \[\mathop{\mathrm{pr}}^*_\mathcal{G}\omega_\mathcal{H}= \omega_{\mathop{\mathrm{pr}}^*_\mathcal{G}\mathcal{H}}.\]

3 Symmetries of Pfaffian fibrations↩︎

We have now arrived at the second part of the paper, where we investigate the symmetries of a Pfaffian fibration, and explain how those symmetries give rise to symmetries of the underlying relative algebroid.

3.1 Internal symmetries of Pfaffian fibrations↩︎

The most general notion of symmetry of a Pfaffian fibration is one that preserves the Cartan distribution. More precisely, the pseudogroup of internal symmetries of a Pfaffian fibration \((P, \mathcal{C}, \pi)\) is given by \[\mathop{\mathrm{Diff}}_{\mathrm{loc}}(P, \mathcal{C}) := \left\{ \varphi\in \mathop{\mathrm{Diff}}_{\mathrm{loc}}(P) \;| \;T\varphi (\mathcal{C}) \subseteq \mathcal{C}\right\}.\] Here, \(\mathop{\mathrm{Diff}}_{\mathrm{loc}}(P)\) is the pseudogroup of local diffeomorphisms on \(P\). If \((P, \mathcal{C}, \pi)\) is comes from a PDE (Example 1), this notion corresponds to the notion of internal symmetries for PDEs [16].

An internal symmetry \(\varphi\) of \((P, \mathcal{C}, \pi)\) sends solutions to solutions, but only when it well-defined. This is already visible at the level of first jets. The prolongation of an internal symmetry is the local diffeomorphism \[\varphi^{(1)}\colon J^1_\mathcal{C}P\dasharrow J^1_\mathcal{C}P, \quad \varphi^{(1)}\colon P^{(1)}\to P^{(1)}\] given by \(\varphi^{(1)} (H_x) = T_x\varphi(H_x)\) for \(H_x\subset \mathcal{C}_x\) with \(H_x\oplus \mathcal{C}^\pi_x = \mathcal{C}_x\). This is only well-defined when \(T_x\varphi(H_x) + \mathcal{C}^\pi_{\varphi(x)} = \mathcal{C}_{\varphi(x)}\). If instead we identity elements of \(J^1_\mathcal{C}P\) with splittings \(h_x\colon T_{\pi(x)}\to \mathcal{C}_x\), then \[\varphi^{(1)}(h_x)= T_x\varphi \circ h_x \circ \left( T_{\varphi(x)}\pi\circ T_x\varphi \circ h_x \right)^{-1}.\] The domain of \(\varphi^{(1)}\), given by the subset where \(T\pi\circ T\varphi \circ h\) is invertible, is open and dense over the domain of \(\varphi\) (if \((P, \mathcal{C}, \pi)\) is involutive, this is also true for \(\varphi^{(1)}\) restricted to \(P^{(1)}\)).

3.2 Pfaffian symmetries of Pfaffian fibrations↩︎

A Pfaffian symmetry of a Pfaffian fibration \((P, \mathcal{C}, \pi)\) is an internal symmetry that also preserves \(\mathcal{C}^\pi\). They form the pseudogroup of Pfaffian symmetries, given by \[\mathop{\mathrm{Diff}}_{\mathrm{loc}}(P, \mathcal{C}, \pi) := \left\{ \varphi\in \mathop{\mathrm{Diff}}_{\mathrm{loc}}(P) \;\big\vert \; T\varphi(\mathcal{C}) \subseteq \mathcal{C}, \;T\varphi(\mathcal{C}^\pi)\subseteq \mathcal{C}^\pi \right\}.\]

Since a local symmetry \(\varphi\in \mathop{\mathrm{Diff}}_{\mathrm{loc}}(P, \mathcal{C}, \pi)\) is in particular an internal symmetry, it prolongs to a local diffeomorphism \[\varphi^{(1)}\colon J^1_\mathcal{C}P \dasharrow J^1_\mathcal{C}P, \quad \varphi^{(1)}\colon P^{(1)}\dasharrow P^{(1)},\] but this time, since the \(\varphi\) also preserves \(\mathcal{C}^\pi\), the transversality condition \(T_x\varphi(H_x) + \mathcal{C}^\pi_{\varphi(x)}= \mathcal{C}_{\varphi(x)}\) is automatically satisfied, so the domain of \(\varphi^{(1)}\) is exactly \(J^1_\mathcal{C}P\) restricted to the domain of \(\varphi\).

The following result relates internal symmetries to Pfaffian symmetries of the prolongation.

3.3 Pseudogroup symmetries of the underlying relative algebroid↩︎

The symmetries of a Pfaffian fibration are defined in such a way that it interacts well with the underlying relative algebroid.

3.4 Groupoids of symmetries of Pfaffian fibrations↩︎

The pseudogroups of internal and Pfaffian symmetries are determined by their first jet. So, for a Pfaffian fibration \((P, \mathcal{C}, \pi)\) it is perhaps more interesting to instead consider the Pfaffian groupoids \[\begin{align} \mathcal{G}_\mathcal{C}&:= \left\{ j^1_x\varphi\in J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(P)\;\big\vert \; T_x\varphi(\mathcal{C}_x) \subseteq \mathcal{C}_{\varphi(x)} \right\}, \\ \mathcal{G}_{\mathcal{C}, \pi} &:= \{ j^1_x\varphi\in \mathcal{G}_\mathcal{C}\; \big\vert \; T_x\varphi(\mathcal{C}^\pi_x) \subseteq \mathcal{C}^\pi_{\varphi(x)} \}, \end{align}\] equipped with the Cartan distribution \(\mathcal{H}^1\) on \(J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(P)\) (Example 3). Their holonomic bisections have the property that \[\mathop{\mathrm{Bis}}_{\mathrm{loc}}(\mathcal{G}_\mathcal{C}, \mathcal{H}^1)\cong \mathop{\mathrm{Diff}}_{\mathrm{loc}}(P, \mathcal{C}), \quad \mathop{\mathrm{Bis}}_{\mathrm{loc}}(\mathcal{G}_{\mathcal{C}, \pi}, \mathcal{H}^1) \cong \mathop{\mathrm{Diff}}_{\mathrm{loc}}(P, \mathcal{C}, \pi).\] Note that, in general, \(J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(P, \mathcal{C})\) (resp. \(J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(P, \mathcal{C}, \pi)\)) is strictly contained in \(\mathcal{G}_\mathcal{C}\) (resp. \(\mathcal{G}_{\mathcal{C}, \pi}\)).

3.5 Groupoids of symmetries and the underlying relative algebroid↩︎

It is the groupoid of Pfaffian symmetries \((\mathcal{G}_{\mathcal{C}, \pi}, \mathcal{H}^1)\rightrightarrows P\) of a Pfaffian fibration \((P, \mathcal{C}, \pi)\) that interacts well with the underlying relative algebroid structure, as we will discuss next.

The canonical representation \(\mathcal{G}_{\mathcal{C}, \pi}\curvearrowright TP\) preserves both \(\mathcal{C}\) and \(\mathcal{C}^\pi\), and thus descends to a representation \(\mathcal{G}_{\mathcal{C}, \pi}\curvearrowright \mathcal{C}/\mathcal{C}^\pi\cong \pi^*TX\).

4 Actions by Pfaffian groupoids on Pfaffian fibrations↩︎

In practice, rather than considering the full groupoid of symmetries of a Pfaffian fibration, one is often interested specific symmetries of the PDE, e.g. diffeomorphisms or gauge symmetries. Such symmetries are often described by a certain action of a Pfaffian groupoid on a Pfaffian fibration.

Analogous to the two types of symmetries of Pfaffian fibrations, there will be two types of actions by Pfaffian groupoids: actions by internal symmetries and actions by Pfaffian symmetries. The groupoids acting by internal symmetries, however, only prolong to partial groupoid actions on the prolongation.

4.1 Actions by internal symmetries↩︎

Let \((\mathcal{G}, \mathcal{H})\rightrightarrows M\) be a Pfaffian groupoid and \((P, \mathcal{C}, \pi)\) a Pfaffian fibration. We say that a partial action \(m\colon \mathcal{G}\mathbin{\hat{\curvearrowright}}P\) along a moment map \(\mu\colon P\to M\) is by internal symmetries when the action map satisfying \[\label{eq:DefinitionInternalAction} Tm\left( \mathcal{H}\leftindex_{Ts}{\times}_{T\mu} \mathcal{C}\big\vert_{\operatorname{dom}m} \right)\subseteq \mathcal{C}.\tag{11}\]

4.2 Actions by Pfaffian symmetries↩︎

Let \((\mathcal{G}, \mathcal{H})\rightrightarrows M\) be a Pfaffian groupoid and \((P, \mathcal{C}, \pi)\) a Pfaffian fibration. We say that \((\mathcal{G}, \mathcal{H})\) acts on \(P\) along \(\mu\colon P\to M\) by Pfaffian symmetries when the (partial) action map \(m\colon \mathcal{G}\leftindex_{s}{\times}_\mu P\dasharrow P\) satisfies \[\label{eq:DefinitionGroupoidActionByPfaffianSymmetries} Tm\left(\mathcal{H}\leftindex_{Ts}{\times}_{T\mu} \mathcal{C}\vert_{\operatorname{dom}m} \right)\subseteq \mathcal{C}, \quad Tm\left( \mathcal{H}\leftindex_{Ts}{\times}_{T\mu} \mathcal{C}^\pi \vert_{\operatorname{dom}m} \right) \subseteq \mathcal{C}^\pi.\tag{12}\]

4.3 Actions by symmetries and the relative algebroid↩︎

Finally, we are in a position where we can state how groupoid actions by symmetries of Pfaffian fibrations interact with the relative algebroid.

4.4 Remarks on Pfaffian actions and PDEs with symmetries↩︎

In this section we show that not all PDEs with symmetries are captured by a Pfaffian action in the sense of [3], [4]. Pfaffian actions arise naturally in the context of geometric structures. However, the example below illustrates that the study of PDEs with symmetries from a Pfaffian perspective demands a broader notion of symmetry.

4.4.1 Pfaffian groupoids, fibrations and actions from the perspective of forms↩︎

The notion of a Pfaffian action relies on the Pfaffian form of a Pfaffian fibration, in contrast to this article, where only the Pfaffian distribution played a role. Let us recall the definitions using Pfaffian forms.

Let \(\mathcal{G}\rightrightarrows M\) be a Lie groupoid and \(E\to M\) a representation of \(\mathcal{G}\). A one-form \(\omega\in \Omega^1(\mathcal{G}; t^*E)\) is multiplicative when [25]: \[(m^*\omega)_{(g, h)} = \mathop{\mathrm{pr}}_1^*\omega + L_g\circ \mathop{\mathrm{pr}}_2^*\omega,\] for any composable pair \((g, h)\in \mathcal{G}\leftindex_{s}{\times}_t \mathcal{G}\). Here \(L_g\colon E_{s(g)}\to E_{t(g)}\) is translation, \(\mathop{\mathrm{pr}}_1\) (resp. \(\mathop{\mathrm{pr}}_2\)) is the projection onto the left (resp. right) factor, and \(m\) is the groupoid multiplication.

If \((\mathcal{G}, \mathcal{H})\rightrightarrows M\) is a Pfaffian groupoid in the sense of Definition 4, then, after identifying \(T\mathcal{G}/\mathcal{H}\cong t^*(T\mathcal{G}/\mathcal{H}\vert_M)\), the one-form \(\omega_\mathcal{H}\in \Omega^1(\mathcal{G};t^*E)\), with \(E=T\mathcal{G}/\mathcal{H}\vert_M\), is multiplicative.

In this section, a Pfaffian groupoid \((\mathcal{G}, \omega)\rightrightarrows M\) is a Lie groupoid with multiplicative one-form \(\omega\in \Omega^1(\mathcal{G};t^*E)\) for which \((\mathcal{G}, \ker \omega)\rightrightarrows M\) is a Pfaffian groupoid in the sense of Definition 4.

Similarly, by a Pfaffian fibration \((P, \theta, \pi)\) we mean a submersion \(\pi\colon P\to M\) and a one-form \(\theta\in \Omega^1(P;E)\) for which \((P, \ker \theta, \pi)\) is a Pfaffian fibration in the sense of Definition 1.

4.4.2 A PDE with symmetry that is not a Pfaffian action↩︎

Let \(q\colon Q\to X\) be a submersion. The point symmetries of \(J^1q\) come from the pseudogroup \(\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q)\), generated by local diffeomorphism \(\varphi\in \mathop{\mathrm{Diff}}_{\mathrm{loc}}(Q)\) lifting some diffeomorphism \(\overline{\varphi}\in \mathop{\mathrm{Diff}}_{\mathrm{loc}}(X)\). The pseudogroup \(\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q)\) is the same as the pseudogroup of diffeomorphisms preserving \(\ker Tq\). There is also a natural groupoid morphism \[\overline{q}\colon J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q)\to J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(X),\] sending \(j^1_x\varphi\) to \(j^1_{q(x)}\overline{\varphi}\), where \(\varphi\) lifts \(\overline{\varphi}\) in a neighborhood of \(x\in Q\). Note that \(\overline{q}\colon J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q)\to J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(X)\) is a groupoid morphism covering \(q\colon Q\to X\) and the Pfaffian forms satisfy \(\overline{q}^*\omega_X = Tq\circ \omega_Q\). Here \(\omega_Q\in \Omega^1(J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q); t^*TQ)\) and \(\omega_X\in \Omega^1(J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(X);t^*TX)\) are the Cartan forms (Example 6).

Clearly, \(\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q)\) acts on germs of sections of \(q\): \[\varphi\cdot \sigma = \varphi \circ \sigma \circ \overline{\varphi}^{-1}\] where \(\varphi\in \mathop{\mathrm{Diff}}_{\mathrm{loc}}(q)\) lifts \(\overline{\varphi}\in \mathop{\mathrm{Diff}}_{\mathrm{loc}}(X)\). This action descends to the level of jets and gives rise to a groupoid action \[\xymatrix{ (J^1\mathop{\mathrm{Diff}}_{\mathrm{loc}}(q), \omega_Q) \ar@<0.5ex>[d] \ar@<-0.5ex>[d] & {(J^1q, \theta_Q, \pi)} \ar@`{[l]+/l-2.5pc/+/d+1.5pc/,[l]+/l-2.5pc/+/u+1.5pc/}[0, 0]_{m} \ar[ld]^{p_1} \\ Q & }\] that is neither principal nor Pfaffian. Even though the coefficients of \(\omega_Q\) and \(\theta_Q\) can be matched through the inclusion \(\ker Tq\subset TQ\), Equation (13 ) can not hold for these forms because the RHS would be surjective onto \(TQ\) while the LHS only has image contained in \(\ker Tq\). Nevertheless, there is still an elegant identity relating all the forms, stated in Proposition 23 below, which almost looks like a “twisted Pfaffian action".

We regard \(J^1q\) as the space of pointwise splittings of \(Tq\): \[J^1q=\{ h\colon T_{q(x)}X\to T_xQ \;| \;T_xq \circ h = \operatorname{id}\}.\]

5 Relative algebroids↩︎

In this section, we give a rapid introduction to relative derivations and relative algebroids necessary to state our results. For more details, we refer to [9], [10]. Only Sections 5.5, 5.9 and 5.10 can not be found in the main sources.

5.1 Derivations of a vector bundle↩︎

Let \(V\to N\) be a vector bundle, and let \(\Omega^\bullet(V) := \Gamma(\wedge^\bullet V^*)\). Recall that a \(k\)-derivation \({\mathrm{D}}\in \operatorname{Der}^k(V)\) on \(V\) is a linear map \[{\mathrm{D}}\colon \Omega^\bullet(V)\to \Omega^{\bullet + k}(V)\] satisfying a Leibniz rule: \[{\mathrm{D}}(\alpha\wedge\beta) = ({\mathrm{D}}\alpha)\wedge \beta + (-1)^{|\alpha| k }\alpha\wedge({\mathrm{D}}\beta).\] Every derivation \({\mathrm{D}}\in \operatorname{Der}(V)\) has a symbol \(\sigma({\mathrm{D}})\colon \wedge^kV\to TN\) determined by \[\left \langle{\mathrm{D}}f, v_1\wedge \dots\wedge v_k\right\rangle = \left\langle{\mathrm{d}}f, \sigma({\mathrm{D}})(v_1, \dots, v_k)\right\rangle\] for \(f\in C^\infty(N)\) and \(v_1, \dots, v_k\in \Gamma(V)\).

The space of derivations \(\operatorname{Der}^k(V)\) is a \(C^\infty(N)\)-module and can be identified with the space of sections of a vector bundle \(\mathcal{D}^k_V\to N\). The symbol map induces the following exact sequence of vector bundles over \(N\): \[0 \to \operatorname{Hom}(\wedge^{k+1}V, V)\to \mathcal{D}^k_V \to \operatorname{Hom}(\wedge^kV, TN) \to 0.\] Any derivation \({\mathrm{D}}\in \operatorname{Der}^k(V)\) determines and is determined by a pair of a \(k\)-bracket \([\cdot, \dots, \cdot]\colon \wedge^{k+1}\Gamma(V)\to \Gamma(V)\) and anchor \(\rho\colon \wedge^kV\to TN\) satisfying a Leibniz rule: \[[fv_0, \dots, v_k] = f[v_0, \dots, v_k] + (-1)^k \mathcal{L}_{\rho(v_1, \dots, v_k)}(f)v_0\] for \(f\in C^\infty(N)\) and \(v_0, \dots, v_k\in \Gamma(V)\).

5.2 Derivations relative to a submersion↩︎

Let \(V\to N\) be a vector bundle and \(p\colon M\to N\) a map (often a submersion). Denote by \(W= p^*V\) and \(p_*\colon W\to V\) the canonical map. A \(k\)-derivation on \(V\) relative to \(p\), written \({\mathrm{D}}\in \operatorname{Der}^k(p_*)\), is a linear map \[{\mathrm{D}}\colon \Omega^\bullet(V) \to \Omega^{\bullet + k}(W)\] satisfying the Leibniz rule with respect to \(p\): \[{\mathrm{D}}(\alpha\wedge\beta) = ({\mathrm{D}}\alpha)\wedge p^*\beta+ (-1)^{|\alpha|k} (p^*\alpha)\wedge({\mathrm{D}}\beta).\] The symbol of a derivation \({\mathrm{D}}\in \operatorname{Der}^k(p)\) relative to \(p\) is the bundle map \(\sigma({\mathrm{D}}) \colon \wedge^k W\to p^*TN\) determined by \[\left\langle {\mathrm{D}}f, w_1\wedge \dots\wedge w_k\right\rangle = \left\langle {\mathrm{d}}f, \sigma({\mathrm{D}})(w_1, \dots, w_k)\right\rangle,\] for \(f\in C^\infty(N)\) and \(w_1, \dots, w_k\in W\).

This time, the space of relative derivation \(\operatorname{Der}^k(p_*)\) is a \(C^\infty(N)\)-module and can be identified with the space of sections of the vector bundle \(p^*\mathcal{D}^k_V\). The symbol induces a short exact sequence \[0 \to \operatorname{Hom}(\wedge^{k+1}W, W) \to p^*\mathcal{D}^k_V \xrightarrow{\sigma} \operatorname{Hom}(\wedge^k W, p^*TN)\to 0.\]

As for ordinary derivations, a relative derivation \({\mathrm{D}}\in \operatorname{Der}^k(p_*)\) determines and is determined by a \(k\)-bracket relative to \(p\), which consists of a bracket and anchor \[[\cdot, \dots, \cdot]\colon \wedge^{k+1}\Gamma(V) \to \Gamma(W), \quad \rho\colon \wedge^k W \to p^*TN\] satisfying the Leibniz rule: \[[fv_0, \dots, v_k] = p^*f [v_0, \dots, v_k] + (-1)^k \mathcal{L}_{\rho(p^*v_1, \dots, p^*v_k)}(f)v_0,\] where \(f\in C^\infty(N)\) and \(v_0, \dots, v_k\in \Gamma(V)\). Note that the derivative makes sense because \(\rho(v_1, \dots, v_k)\) takes values in \(p^*TN\), so it’s possible to differentiate a function on \(N\) along \(\rho(v_1, \dots, v_k)\), but the result will be a function on \(M\).

Given an ordinary derivation \({\mathrm{D}}\in \operatorname{Der}^k(W)\), it is possible to restrict it only to the forms \(p^*\alpha\) for \(\alpha\in \Omega^\bullet(V)\). This determines a derivation relative to \(p\). This process is \(C^\infty(M)\)-linear so there is an induced bundle map \(p_*\colon \mathcal{D}^k_W\to p^*\mathcal{D}^k_V\). This bundle map fits into a short exact sequence \[0\to \operatorname{Hom}(W, \ker Tp) \to \mathcal{D}^k_W\xrightarrow{p_*} p^*\mathcal{D}^k_V\to 0.\]

5.3 Flat foliated vector bundles↩︎

Let \(W\to M\) be a vector bundle and \(\mathcal{F}\subset TM\) a foliation on \(M\). An \(\mathcal{F}\)-connection on \(W\) is a partial connection \[\overline{\nabla}\colon \Gamma(\mathcal{F})\times \Gamma(W)\to \Gamma(W)\] satisfying the usual properties of a connection. The curvature of an \(\mathcal{F}\)-connection \({\overline{\nabla}}\) is given by \[R_{{\overline{\nabla}}}\in \Omega^2(\mathcal{F};\operatorname{End}(W)), \quad R_{{\overline{\nabla}}}(X, Y)w = {\overline{\nabla}}_X{\overline{\nabla}}_Yw-{\overline{\nabla}}_X{\overline{\nabla}}_Yw-{\overline{\nabla}}_{[X, Y]}w.\] The \(\mathcal{F}\)-connection is called flat when \(R_{\overline{\nabla}}=0\). In this case we call \((W, {\overline{\nabla}})\to (M, \mathcal{F})\) a flat foliated vector bundle.

A local section \(w\in \Gamma_W\) is called flat when \({\overline{\nabla}}_X w=0\) for all \(X\in \Gamma_\mathcal{F}\). When \({\overline{\nabla}}\) is flat, then locally there always exist a flat section through every \(w\in W\), but nonzero global flat sections may fail to exist. We denote by \(\Gamma_{(W, {\overline{\nabla}})}\) the sheaf of flat sections of \((W, {\overline{\nabla}})\).

A map of foliations between two foliated manifold \((M_1, \mathcal{F}_1)\) and \((M_2, \mathcal{F}_2)\) is a smooth map \(\varphi\colon M_1\to M_2\) such that \(T\varphi(\mathcal{F}_1)\subseteq \mathcal{F}_2\). Let \((W_1, {\overline{\nabla}}^1)\) and \((W_2, {\overline{\nabla}}^2)\) are two foliated vector bundles over \((M_1, \mathcal{F}_1\) and \((M_2, \mathcal{F}_2)\), respectively, a map of foliated vector bundles is a bundle map \((\Phi, \varphi)\colon W_1\to W_2\) covering a map of foliations \(\varphi\) such that \[\label{eq:MapOfFoliatedVectorBundles} \Phi^*\left( {\overline{\nabla}}^2_{\varphi_*(X)} \alpha\right) = {\overline{\nabla}}^1_{X} \Phi^*\alpha,\tag{14}\] for all \(\alpha\in \Omega^1(W_2)\) and \(X\in \mathcal{F}_1\). If both are foliated flat vector bundles, then Equation (14 ) implies that \(\Phi^*\) preserves flat forms. If \(\Phi\) is in addition fiberwise an isomorphism, then we get a map between flat sections \[\Phi^*\colon \Gamma_{(W_2, {\overline{\nabla}}^2)} \to \Gamma_{(W_1, {\overline{\nabla}}^1)}, \quad (\Phi^*w_2)(x) = \Phi_x^{-1}(w_2(\varphi(x)))\] The converse is also true.

5.4 Derivations relative to foliations↩︎

The derivations associated to a Pfaffian fibration are only locally derivations relative to a submersion and their global geometry can best be described by the notion of derivation relative to a foliation.

The space of relative derivations \(\operatorname{Der}^k(W, {\overline{\nabla}})\) can be realized as the sections of a vector bundle \(\mathcal{D}^k_{(W, {\overline{\nabla}})}\to M\) which itself is actually also a foliated flat bundle: the flat sections of \(\mathcal{D}^k_{(W, {\overline{\nabla}})}\) correspond to those derivations that preserve \(\Omega^\bullet_{(W, {\overline{\nabla}})}\). The symbol map induces a short exact sequence of flat foliated bundles \[0 \to (\operatorname{Hom}(\wedge^{k+1}W, W), {\overline{\nabla}}) \to (\mathcal{D}^k_{(W, {\overline{\nabla}})}, {\overline{\nabla}}) \xrightarrow{\sigma} (\operatorname{Hom}(\wedge^kW, \nu(\mathcal{F})), {\overline{\nabla}}) \to 0.\]

As before, a derivation relative to a foliation determines and is determined by a \(k\)-bracket relative to \(\mathcal{F}\), which consists of a partial bracket and anchor \[[\cdot, \dots, \cdot]\colon \wedge^{k+1} \Gamma_{(W, {\overline{\nabla}})}\to \Gamma_W, \quad \rho\colon \wedge^kW\to \nu(\mathcal{F}),\] satisfying the Leibniz rule: \[[fw_0, \dots, w_k] = f[w_0, \dots, w_k] + (-1)^k \mathcal{L}_{\rho(w_1, \dots, w_k)}(f)v_0.\] Note that differentiating \(f\in C^\infty_\mathcal{F}\) in the direction of \(\nu(\mathcal{F})\) makes sense because \(f\) is basic for the foliation \(\mathcal{F}\).

An ordinary derivation \({\mathrm{D}}\in \operatorname{Der}^k(W)\) can be restricted to \(\Omega^\bullet_{(W, {\overline{\nabla}})}\) to yield a derivation relative to the foliation. This process induces a bundle map \(\Pi\colon \mathcal{D}^k_W\to \mathcal{D}^k_{(W, {\overline{\nabla}})}\) that fits into the following short exact sequence \[0 \to \operatorname{Hom}(\wedge^k W, \mathcal{F})\to \mathcal{D}^k_W \xrightarrow{\Pi} \mathcal{D}^k_{(W, {\overline{\nabla}})} \to 0.\]

5.5 Derivations as maps on first jets↩︎

If \(W\to M\) is a vector bundle, a derivation \({\mathrm{D}}\in \operatorname{Der}^k(W)\) factors (pointwise) through the first jet, and is therefore determined by a bundle map: \[{\mathrm{D}}\colon J^1W^*\to \wedge^{1+k} W^*.\] The investigation of its properties with respect to the structure of \(J^1W^*\) is interesting but omitted from this paper. Likewise, a pointwise derivation \({\mathrm{D}}_x\in \mathcal{D}^k_W\) is now a linear map \[{\mathrm{D}}_x\colon (J^1W^*)_x\to \wedge^{\bullet+k}W_x^*.\]

If \((W, {\overline{\nabla}})\to (M, \mathcal{F})\) is a flat foliated vector bundle, then the first order PDE defining flat sections can be regarded as a subspace \[J^1_{\overline{\nabla}}W^*\subset J^1W^*.\] A derivation relative to \(\mathcal{F}\) induces a bundle map \[{\mathrm{D}}\colon J^1_{{\overline{\nabla}}} W^*\to \wedge^{k+1}W^*.\] and a pointwise derivation \({\mathrm{D}}_x\in (\mathcal{D}^k_{(W, {\overline{\nabla}})})_x\) induces a linear map \[{\mathrm{D}}_x\colon (J^1_{{\overline{\nabla}}}W^*)_x\to \wedge^{k+1}W_x^*.\] This perspective is useful when dealing with actions of jets of bundle maps on bundles of derivations.

If \((\Phi, \varphi)\colon W\to W\) is a bundle map (which is a local isomorphism), then its first jet acts on a pointwise derivation \({\mathrm{D}}_x\in \mathcal{D}^1_W\) as \[j^1_x\Phi\cdot {\mathrm{D}}_x = \Phi_*\circ {\mathrm{D}}_x \circ j^1_{\varphi(x)}\Phi^*\colon (J^1W^*)_{\varphi(x)}\to \wedge^{k+1} W_{\varphi(x)}^*.\] where \(j^1_{\varphi(x)}\Phi^*\colon (J^1W^*)_{\varphi(x)}\to (J^1W^*)_x\) is defined as \[j^1_{\varphi(x)}\Phi^*\cdot j^1_{\varphi(x)}\alpha = j^1_x(\Phi^*\alpha).\] If, at \(x\in M\), the action of \(j^1_{\varphi(x)}\Phi^*\) preserves \(J^1_{\overline{\nabla}}W^*\), then the jet also acts on a pointwise derivation \({\mathrm{D}}_x\in \mathcal{D}^1_{(W, {\overline{\nabla}})}\) by the same formula, but \(\Phi\) itself does not have to be a map of flat foliated bundles away from \(x\). This is the true advantage of working with jets over local sections.

Recall that derivations \({\mathrm{D}}\colon\Omega^\bullet_{(W, {\overline{\nabla}})}\to \Omega^{\bullet+k}\) correspond to relative \(k\)-brackets \([\cdot, \dots, \cdot]\colon \wedge^{k+1}\Gamma_{(W, {\overline{\nabla}})} \to \Gamma_W\). The bracket also depends only on first jets, so a pointwise derivation \({\mathrm{D}}_x\in (\mathcal{D}^1_{(W, {\overline{\nabla}})})_x\) corresponds to a pointwise bracket \[[\cdot, \dots, \cdot]_x\colon \wedge^{k+1}(J^1_{{\overline{\nabla}}}W)_x \to W_x\] If \(j^1_{x}\Phi\) preserves \(J^1_{{\overline{\nabla}}}W\), then it acts on the partial bracket as \[j^1_{x}\Phi\cdot[\cdot, \dots, \cdot]_x =\Phi_x\left( \left[(j^1_x\Phi)^{-1}(\cdot),\dots, (j^1_x\Phi)^{-1}(\cdot)\right]_x \right)\colon \wedge^{k+1}(J^1_{{\overline{\nabla}}}W)_{\varphi(x)} \to W_{\varphi(x)}.\]

5.6 Relative algebroids↩︎

A relative algebroid is nothing more than a flat foliated vector bundle together with a relative 1-derivation, and we now have the language to understand all of its ingredients.

Next, we introduce the concept of “solutions" in the context of relative algebroids.

A realization is a particular example of a morphism of relative algebroids (Definition 12).

The tableau map of a relative algebroid is a tableau of derivations [9]. The symbol of the tableau map defines a classical tableau map. In terms of the anchor \(\rho\colon B\to \nu(\mathcal{F})\), it is given by \[\label{eq:SymbolTableauMap} \sigma(\tau)(X)(b) = ({\overline{\nabla}}_X \rho)(b) = {\overline{\nabla}}_X (\rho(b)) - \rho({\overline{\nabla}}_X b).\tag{15}\]

Next, we discuss two fundamental constructions of relative algebroids.

5.7 Prolongation and formal integrability↩︎

For this section, let us fix a relative algebroid \((B, {\overline{\nabla}}, {\mathrm{D}})\) over \((M, \mathcal{F})\). A prolongation of \(B\) is an attempt to complete the derivation \({\mathrm{D}}\) in such a way that “\({\mathrm{D}}^2=0\)", but, instead of considering one specific completion, we consider all of them at once.

Recall from Section 5.4 that there is a map \(\Pi\colon \mathcal{D}^1_B\to \mathcal{D}^1_{(B, {\overline{\nabla}})}\) obtained by restricting derivations to flat forms. The space of pointwise extensions (or the partial prolongation space) of \((B, {\overline{\nabla}}, {\mathrm{D}})\) is the space \[J^1_{\mathrm{D}}M := \Pi^{-1}(\operatorname{im}{\mathrm{D}}) = \left\{ \tilde{{\mathrm{D}}}_x\in \mathcal{D}^1_{B}\;| \;\Pi(\tilde{{\mathrm{D}}}_x) = {\mathrm{D}}_x\in \mathcal{D}^1_{(B, {\overline{\nabla}})} \right\}.\] It is an affine bundle over \(M\) modeled on \(\operatorname{Hom}(B, \mathcal{F})\).

Since \(p_1\colon J^1_{\mathrm{D}}M\to M\) is a surjective submersion, the inclusion \(i\colon J^1_{\mathrm{D}}M\hookrightarrow \mathcal{D}^1_{B}\) satisfies the conditions of Proposition 29 and thus the universal relative algebroid \((B, p_1, \Breve{\mathrm{D}}))\) of \(B\to M\) from Example 10 can be restricted to \(J^1_{\mathrm{D}}M\) to obtain a relative algebroid that we call the partial prolongation of \((B, {\overline{\nabla}}, {\mathrm{D}})\).

However, to complete \((B, {\overline{\nabla}}, {\mathrm{D}})\) to an object that is closer to satisfying “\({\mathrm{D}}^2 =0\)" we have to take that condition into account. The space of pointwise completions (or the prolongation space) of \((B, {\overline{\nabla}}, {\mathrm{D}})\) is \[M^{(1)} = \{ \tilde{{\mathrm{D}}}_x\in \mathcal{D}^1_B \; | \;\Pi(\tilde{{\mathrm{D}}}_x) = {\mathrm{D}}_x, \;\tilde{{\mathrm{D}}}_x\circ {\mathrm{D}}=0 \}\] The space \(M^{(1)}\) is not guaranteed to be smooth, nor does \(p_1\colon M^{(1)}\to M\) have to be surjective. In fact, the image \(p_1(M^{(1)})\subseteq M\) is precisely the locus where the intrinsic torsion of \((B, {\overline{\nabla}}, {\mathrm{D}})\) vanishes. We don’t discuss torsion and curvature in this paper and refer to [9] for more details.

When \(M^{(1)}\) is smooth and \(p_1\colon M^{(1)}\to M\) is surjective, we call \((B, {\overline{\nabla}}, {\mathrm{D}})\) 1-integrable. In that case, the inclusion \(M^{(1)}\hookrightarrow \mathcal{D}^1_B\) satisfies the conditions of Proposition 29 and thus the universal relative algebroid restricts to \(M^{(1)}\) (Proposition 29). The resulting relative algebroid, denoted \((B, p_1, {\mathrm{D}}^{(1)})\) is called the first prolongation of \((B, {\overline{\nabla}}, {\mathrm{D}})\).

Higher prolongations are defined iteratively, with \[(B, {\overline{\nabla}}, {\mathrm{D}})^{(k)} :=(B^{(k-1)}, p_k, {\mathrm{D}}^{(k)}) = (B^{(k-2)}, p_{k-1}, {\mathrm{D}}^{(k-1)} )^{(1)}.\] If the first \(k\) prolongations exist, we call \((B, {\overline{\nabla}}, {\mathrm{D}})\) \(k\)-integrable, and \((B, {\overline{\nabla}}, {\mathrm{D}})\) is formally integrable if it is \(k\)-integrable for all \(k\). In that case, it gives rise to what should be a profinite Lie algebroid (of finite type): \[\xymatrix{ {\left(B^{(\infty)}, {\mathrm{D}}^{(\infty)}\right)\colon\ldots} \ar[r] \ar@<-12pt>[d] & B^{(k)} \ar[d] \ar[r] & B^{(k-1)} \ar[r] \ar[d] \ar@{-->}@/^1pc/[l]^{{\mathrm{D}}^{(k)}} & \ldots \ar[r] & B^{(1)} \ar[r] \ar[d] & B \ar[d] \ar@{-->}@/^1pc/[l]^{{\mathrm{D}}^{(1)}}\\ M^{(\infty)}\colon\ldots \ar[r] & M^{(k)} \ar[r]_{p_k} & M^{(k-1)} \ar[r] & \ldots \ar[r] & M^{(1)} \ar[r]_{p_1} & M}\] If \((B, {\overline{\nabla}}, {\mathrm{D}})\) is formally integrable, we say that it is a relative Lie algebroid, placing the “Lie" back into the terminology.

5.8 Morphisms of relative algebroids↩︎

Recall from Section 5.3 that a morphism of flat foliated vector bundles \((\Phi, \varphi)\colon (B_1, {\overline{\nabla}}^1)\to (B_2, {\overline{\nabla}}^2)\) induces a map \[\Phi^*\colon \Omega^\bullet_{(B_2, {\overline{\nabla}}^2)} \to \Omega^\bullet_{(B_1, {\overline{\nabla}}^1)}.\]

If the morphism \(\Phi\) is fiberwise an isomorphism, then Equation (16 ) can be expressed in terms of brackets as \[\Phi^*[b_1, b_2]_{{\mathrm{D}}_2} = [\Phi^*b_1, \Phi^*b_2]_{{\mathrm{D}}_1},\] for all \(b_1, b_2\in \Gamma_{(B_2, {\overline{\nabla}}^2)}\).

This notion of morphism pairs well with the constructions built out of a relative algebroid.

5.9 Representations on flat foliated vector bundles↩︎

The flat sections of a foliated bundle \((W, {\overline{\nabla}})\) are the solutions to the PDE \({\overline{\nabla}}w=0\). Since \({\overline{\nabla}}w\) only depends on the first jet of \(w\), the PDE is determined by a subspace of \(J^1W\) denoted by \[J^1_{\overline{\nabla}}W = \{ j^1_xw\in J^1W \;| \;({\overline{\nabla}}w)_x = 0\}\subseteq J^1W.\] The torsion of this PDE can be identified with the curvature of \({\overline{\nabla}}\), and when it vanishes, it is PDE-integrable, meaning that there exists flat sections trough every point \(w\in W\).

If \(\mathcal{G}\rightrightarrows M\) is a groupoid with a representation on a vector bundle \(W\to M\), then \(J^1\mathcal{G}\rightrightarrows M\) has a representation on \(J^1W\to M\). In general, it is too much to ask that \(J^1\mathcal{G}\) preserves the PDE for parallel section \(J^1_{{\overline{\nabla}}}W\). Therefore, we propose the following definition.

Restricting the Cartan distribution \(\mathcal{H}^1\) on \(J^1\mathcal{G}\) to \(\mathcal{G}_1\) (Example 2), we obtain a Pfaffian groupoid \((\mathcal{G}_1, \mathcal{H}^1)\to M\) that acts on \(J^1_{\overline{\nabla}}W\) by Pfaffian symmetries. Thus, essentially, a prolongation pair is a first order PDE on the bisections of a Lie groupoid, but our notation remembers the underlying groupoid and its representation on \(W\).

5.10 Representations on relative algebroids↩︎

Let \((\mathcal{G}_1, \mathcal{G})\rightrightarrows M\) be a prolongation pair and \((B, {\overline{\nabla}}, {\mathrm{D}})\) a relative algebroid over \((M, \mathcal{F})\). A representation of \((\mathcal{G}_1, \mathcal{G})\) on \((B, {\overline{\nabla}}, {\mathrm{D}})\) by symmetries of relative algebroids is a representation on a flat foliated bundle \((\mathcal{G}_1, \mathcal{G})\curvearrowright (B, {\overline{\nabla}})\) for which \({\mathrm{D}}\) is a \(\mathcal{G}_1\)-invariant section of \(\mathcal{D}^1_{(B, {\overline{\nabla}})}\).

6 Tableau maps and involutivity↩︎

In the classical theory of partial differential equations, a tableau is simply a subspace \(\mathfrak{g} \subset \operatorname{Hom}(V, W)\) for two (finite-dimensional) vector spaces \(V\) and \(W\). In the context of Pfaffian fibrations or relative algebroids, one encounters tableau maps \[\tau\colon \mathfrak{g}\to \operatorname{Hom}(V, W)\] that are not necessarily injective. These were called generalized tableaux in [7], but we stick to the term tableau maps to avoid confusion with other generalizations of tableaux (such as tableaux of derivations in [9]). In this section, we show that several natural extensions of definitions of involutivity for tableau maps coincide.

A tableau map corresponds intuitively to a simple system of first order linear differential equation on pairs \((f, g)\) of functions \(f\colon V\to W\) and \(g\colon V\to \mathfrak{g}\) such that \[{\mathrm{d}}_v f = \tau(g(v))\] for all \(v\in \mathfrak{g}\). In case \(\mathfrak{g} = \ker \tau \oplus \operatorname{im}\tau\) is split, then this is equivalent to a pair of functions \((f, h)\) with \(f\colon V\to W\) and \(h\colon V\to \ker \tau\) such that \({\mathrm{d}}f \mod \operatorname{im}\tau = 0\), with no further equations on \(h\).

When \(\tau\colon \mathfrak{g}\to \operatorname{Hom}(V, W)\) is a tableau map, the Spencer differential w.r.t. \(\tau\) is \[\delta_\tau\colon \operatorname{Hom}(V, \mathfrak{g})\to \operatorname{Hom}(\wedge^2V, W), \quad \delta_\tau(\xi)(v, w) = \tau(\xi(v))(w) - \tau(\xi(w))(v),\] for \(v, w\in V\).

A flag for an \(n\)-dimensional vector space \(V\) is a nested sequence of subspaces \(\{V_k\}\) such that \(\dim V_k = k\). A basis adapted to the flag \(\{V_k\}\) is a basis \(\{e_k\}_k\) with the property that \(\{e_1, \dots e_k\}\) is a basis of \(V_k\).

For a tableau map \(\tau\colon \mathfrak{g}\to \operatorname{Hom}(V, W)\), we set \[\mathfrak{g}_k = \{ \xi \in \mathfrak{g} \;| \;\tau(\xi)\vert_{V_k} = 0\}.\] Note that \(\mathfrak{g}_0 = \mathfrak{g}\) and \(\mathfrak{g}_n = 0\).

The Cartan characters of \(\tau\) (w.r.t. the flag \((V_k)_k\)) are defined to be \[s^{\mathfrak{g}}_k = \dim \mathfrak{g}_{k-1} - \dim \mathfrak{g}_k.\] In the proposition below, we set \(\mathfrak{h} = \operatorname{im}\tau\subset \operatorname{Hom}(V, W)\). Note that \(\mathfrak{h}\) is a tableau in the classical sense.

In other words, the tableau map \(\tau\) is involutive if and only if \(\operatorname{im}\tau\) is involutive as a classical tableau.

References↩︎

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1. This work was partially supported by NSF grant DMS-2303586 and UIUC Campus Research Board Award RB25014.↩︎

2. For this article, we stick to the term partial action and prefer to use local actions for actions of local Lie groupoids, which do not play a role in this article.↩︎