Relative Algebroids: Generalizing Lie Structures

• Relative algebroids are generalizations of Lie algebroids where structures are defined relative to auxiliary data such as submersions, foliations, or fixed background frameworks.
• They provide a unified method to handle relative derivations, anchors, curvature corrections, and integrability through prolongation and tableau techniques.
• Applications span formal PDE integrability, jet-theoretic constructions, and symmetry reductions in geometric contexts, bridging classical and derived approaches.

Relative algebroids are generalizations of Lie algebroids in which the bracket, anchor, or differential is defined relative to auxiliary geometric or algebraic data rather than absolutely over a single manifold. Recent literature uses the term for several closely related constructions: algebroids relative to a submersion or foliation, designed to encode Cartan realization problems and partial differential equations (Fernandes et al., 25 Mar 2025); canonical relative algebroids attached to Pfaffian fibrations (Smilde, 1 Oct 2025); Lie algebroids rooted in a fixed background Lie algebroid $\mathcal L$ via a morphism $\phi:\mathcal E\to\mathcal L$ (Poddar et al., 2024); dg-Lie algebroids over a base cdga $A$ with anchor to $T_A=\operatorname{Der}_k(A)$ (Vezzosi, 2013); relative Lie algebroids over $X/S$ with anchor to $T_{X/S}$ (Sarkar, 2021); and curvature-controlled extensions $F\oplus E$ determined by a morphism $K:E\to F$ and a strict covariant $K$-adjustment (Fischer, 2 Jun 2025). This suggests that “relative algebroid” is presently a family of precise frameworks sharing the same organizing idea: Lie-algebroid-type structure constrained by a background map, foliation, base, or embedding.

1. Core meanings of relativity

In the submersion-based framework, relativity means that the differential no longer closes on $\Omega^\bullet(A)$, but instead takes values in forms on the pullback bundle $\phi:\mathcal E\to\mathcal L$0. In the foliation-based framework, the anchor lands in the normal bundle $\phi:\mathcal E\to\mathcal L$1 and the bracket is defined on flat sections with respect to a flat partial connection. In the rooted framework of $\phi:\mathcal E\to\mathcal L$2-Lie algebroids, relativity is encoded by a morphism to a fixed background Lie algebroid. In dg and derived settings, relativity is usually implemented by replacing the absolute tangent object with a relative tangent object such as $\phi:\mathcal E\to\mathcal L$3 or $\phi:\mathcal E\to\mathcal L$4. In extension theory, relativity is encoded by a morphism $\phi:\mathcal E\to\mathcal L$5 forcing the orbits of one algebroid to lie inside those of another, and by an extension whose anchor is the sum of the initial anchors (Fernandes et al., 25 Mar 2025, Poddar et al., 2024, Vezzosi, 2013, Sarkar, 2021, Fischer, 2 Jun 2025).

Framework	Relative datum	Characteristic structure
Submersion-relative	$\phi:\mathcal E\to\mathcal L$6	$\phi:\mathcal E\to\mathcal L$7
Foliation-relative	$\phi:\mathcal E\to\mathcal L$8	anchor $\phi:\mathcal E\to\mathcal L$9 on a flat foliated bundle
Rooted $A$0-relative	$A$1	Lie algebroid structure over a fixed background algebroid
Relative dg-Lie algebroid	base cdga $A$2	anchor $A$3
Relative algebraic/geometric	$A$4	anchor $A$5
Extension-relative	$A$6	extended algebroid $A$7 with anchor sum

A recurrent misconception is that relative algebroids already form a single universally standardized definition. Current usage is narrower and more technical: each framework has its own symbol sequence, realization theory, and functoriality. What is shared is not a single axiomatics, but a common shift from absolute anchors and differentials to constrained ones.

2. Relative derivations, anchors, and brackets

For a vector bundle $A$8 and a map $A$9, a degree-$T_A=\operatorname{Der}_k(A)$0 derivation relative to $T_A=\operatorname{Der}_k(A)$1 is a linear operator

$T_A=\operatorname{Der}_k(A)$2

satisfying

$T_A=\operatorname{Der}_k(A)$3

Dualizing gives a relative $T_A=\operatorname{Der}_k(A)$4-bracket and a relative anchor

$T_A=\operatorname{Der}_k(A)$5

with the relative Leibniz rule. The symbol sequence is

$T_A=\operatorname{Der}_k(A)$6

An almost Lie algebroid relative to a submersion $T_A=\operatorname{Der}_k(A)$7 is then a vector bundle $T_A=\operatorname{Der}_k(A)$8 together with a degree-1 derivation

$T_A=\operatorname{Der}_k(A)$9

In local submersion coordinates $X/S$0 and a local coframe $X/S$1,

$X/S$2

This is the coordinate form in which free derivatives enter: the dependence on the vertical variables $X/S$3 is not prescribed by an absolute differential, but by the relative geometry (Fernandes et al., 25 Mar 2025).

The foliation-relative variant replaces the pullback picture by a flat foliated bundle. One fixes a foliated manifold $X/S$4, a vector bundle $X/S$5, and a flat $X/S$6-connection $X/S$7. Relative forms are the flat forms $X/S$8, and a relative algebroid is a degree-1 derivation

$X/S$9

with symbol

$T_{X/S}$0

Equivalently, $T_{X/S}$1 determines an anchor to the normal bundle and a bracket on flat sections. In the Pfaffian-fibration setting this flat foliated formulation is canonical, not auxiliary (Smilde, 1 Oct 2025).

Both formulations have universal objects. For $T_{X/S}$2, the projection $T_{X/S}$3 and the tautological section define a universal relative algebroid $T_{X/S}$4, and any relative algebroid $T_{X/S}$5 is classified by a map $T_{X/S}$6 with $T_{X/S}$7. This classifying viewpoint is one reason relative algebroids interface effectively with jet-theoretic and PDE constructions (Smilde, 24 Oct 2025).

3. Prolongation, tableaux, and formal integrability

A defining feature of the submersion-relative theory is that one does not impose $T_{X/S}$8 at the outset. Instead, closure is replaced by a prolongation procedure. The basic linear object is the tableau of derivations. For a tableau $T_{X/S}$9, the Spencer differential

$F\oplus E$0

defines the first prolongation

$F\oplus E$1

higher prolongations $F\oplus E$2, and Spencer cohomology groups $F\oplus E$3. Cartan’s bound and Cartan’s test are available in this relative setting: if the Cartan characters $F\oplus E$4 are locally constant and equality holds in Cartan’s bound, then the tableau is involutive, all prolongations have locally constant rank, and $F\oplus E$5 for $F\oplus E$6, $F\oplus E$7 (Fernandes et al., 25 Mar 2025).

The obstruction theory is expressed through intrinsic torsion and curvature. For the tableau map $F\oplus E$8, the intrinsic torsion is a class

$F\oplus E$9

and, after choosing a torsionless lift $K:E\to F$0, the intrinsic curvature is a class

$K:E\to F$1

The canonical first prolongation is the locus

$K:E\to F$2

and when $K:E\to F$3 is smooth and $K:E\to F$4 is a submersion, the relative algebroid is $K:E\to F$5-integrable. The “Fundamental Theorem of Prolongation” identifies the torsion of the first prolongation with the curvature of the original relative algebroid under the natural inclusion

$K:E\to F$6

Goldschmidt’s formal integrability criterion also has a relative form: if the relative algebroid is $K:E\to F$7-integrable and $K:E\to F$8 for all $K:E\to F$9, then it is formally integrable. A formally integrable almost relative algebroid is called a relative Lie algebroid (Fernandes et al., 25 Mar 2025).

Realizations are the geometric counterpart of integrability. A realization of $K$0 is a bundle isomorphism $K$1 over $K$2 such that

$K$3

on the relevant forms. With a chosen extension $K$4, this becomes the Maurer–Cartan system

$K$5

In the analytic category, formally integrable relative Lie algebroids admit local realizations through every point and at every stage of the prolongation tower. In finite type, the tower stabilizes and one recovers an actual Lie algebroid; in infinite type, one obtains a profinite Lie algebroid whose differential squares to zero only on profinite sections. This is the mechanism by which relative algebroids bridge Cartan’s method of equivalence and the formal theory of PDEs (Fernandes et al., 25 Mar 2025).

4. PDEs, Pfaffian fibrations, and symmetry

For a PDE $K$6 with $K$7 a submersion and $K$8, the underlying relative algebroid is

$K$9

The PDE and the relative algebroid have matching prolongation theories: $\Omega^\bullet(A)$0 is $\Omega^\bullet(A)$1-integrable if and only if the associated relative algebroid is $\Omega^\bullet(A)$2-integrable; the prolongation of the relative algebroid is the relative algebroid associated to $\Omega^\bullet(A)$3; and germs of solutions of $\Omega^\bullet(A)$4 are in one-to-one correspondence with germs of realizations of the relative algebroid, modulo diffeomorphism (Fernandes et al., 25 Mar 2025).

Pfaffian fibrations provide a more intrinsic geometric source of relative algebroids. A Pfaffian fibration $\Omega^\bullet(A)$5 is a submersion $\Omega^\bullet(A)$6 with a distribution $\Omega^\bullet(A)$7 satisfying transversality $\Omega^\bullet(A)$8 and $\Omega^\bullet(A)$9-involutivity of $\phi:\mathcal E\to\mathcal L$00. Its canonical relative algebroid is built on

$\phi:\mathcal E\to\mathcal L$01

with flat $\phi:\mathcal E\to\mathcal L$02-connection $\phi:\mathcal E\to\mathcal L$03 and relative derivation

$\phi:\mathcal E\to\mathcal L$04

The explicit formulas are

$\phi:\mathcal E\to\mathcal L$05

and the bracket on flat sections is induced by Lie brackets of lifts in $\phi:\mathcal E\to\mathcal L$06 (Smilde, 1 Oct 2025).

The resulting equivalence theorem is exact. The partial and full prolongation spaces of $\phi:\mathcal E\to\mathcal L$07 coincide canonically with those of $\phi:\mathcal E\to\mathcal L$08; the tableau maps agree under the natural symbol identification; and germs of holonomic sections of the Pfaffian fibration are in bijection with germs of realizations of the associated relative algebroid. Thus Pfaffian fibrations and relative algebroids encode the same local formal geometry, but in structurally different languages (Smilde, 1 Oct 2025).

Symmetry theory also transfers. Internal symmetries are diffeomorphisms $\phi:\mathcal E\to\mathcal L$09 with $\phi:\mathcal E\to\mathcal L$10; Pfaffian symmetries additionally preserve $\phi:\mathcal E\to\mathcal L$11. Their prolongations induce actions on the prolongation spaces, and Pfaffian symmetries act by automorphisms of the canonical flat foliated bundle and preserve $\phi:\mathcal E\to\mathcal L$12: $\phi:\mathcal E\to\mathcal L$13 At the groupoid level, Pfaffian groupoid actions prolong compatibly and preserve the associated relative algebroid. This is particularly relevant for PDEs with Lie pseudogroup symmetries, where quotient procedures are more naturally expressed in the relative-algebroid language than in the raw jet-bundle language (Smilde, 1 Oct 2025).

5. Relative extensions, anchor sums, and curvature corrections

A different use of “relative” appears in the extension theory of Lie algebroids. Let $\phi:\mathcal E\to\mathcal L$14 and $\phi:\mathcal E\to\mathcal L$15 be Lie algebroids with a morphism $\phi:\mathcal E\to\mathcal L$16, so that $\phi:\mathcal E\to\mathcal L$17 and the $\phi:\mathcal E\to\mathcal L$18-orbits lie inside the $\phi:\mathcal E\to\mathcal L$19-orbits. A strict covariant $\phi:\mathcal E\to\mathcal L$20-adjustment consists of a Cartan $\phi:\mathcal E\to\mathcal L$21-connection $\phi:\mathcal E\to\mathcal L$22 on $\phi:\mathcal E\to\mathcal L$23 and a primitive $\phi:\mathcal E\to\mathcal L$24-form $\phi:\mathcal E\to\mathcal L$25 satisfying the generalized Maurer–Cartan equation

$\phi:\mathcal E\to\mathcal L$26

and the strictness condition

$\phi:\mathcal E\to\mathcal L$27

In the special case $\phi:\mathcal E\to\mathcal L$28, this reduces to

$\phi:\mathcal E\to\mathcal L$29

described as the infinitesimal version of a strict multiplicative Yang–Mills $\phi:\mathcal E\to\mathcal L$30-connection (Fischer, 2 Jun 2025).

Under these hypotheses, the original bracket on $\phi:\mathcal E\to\mathcal L$31 acquires an “action form”

$\phi:\mathcal E\to\mathcal L$32

where

$\phi:\mathcal E\to\mathcal L$33

is a fibrewise Lie bracket. The extension lives on the Whitney sum

$\phi:\mathcal E\to\mathcal L$34

with anchor

$\phi:\mathcal E\to\mathcal L$35

Its bracket is

$\phi:\mathcal E\to\mathcal L$36

equivalently

$\phi:\mathcal E\to\mathcal L$37

The kernel of

$\phi:\mathcal E\to\mathcal L$38

is $\phi:\mathcal E\to\mathcal L$39, giving a short exact sequence

$\phi:\mathcal E\to\mathcal L$40

Here the “relative” character lies in the fact that $\phi:\mathcal E\to\mathcal L$41 is inserted into $\phi:\mathcal E\to\mathcal L$42 only through the graph of $\phi:\mathcal E\to\mathcal L$43, while the anchor of the extension is the sum of the two anchors (Fischer, 2 Jun 2025).

When $\phi:\mathcal E\to\mathcal L$44 and the Cartan connection is flat, the extension bracket becomes the classical matched-pair bracket. The new feature of the curved theory is precisely the controlled failure of flatness: curvature is allowed, but only subject to the generalized Maurer–Cartan and strictness identities. The obstruction to the Jacobi identity is $\phi:\mathcal E\to\mathcal L$45, and the construction thereby points toward an obstruction theory for Cartan connections and nontrivial action algebroids (Fischer, 2 Jun 2025).

6. Algebraic, derived, rooted, and adjacent relative frameworks

In homotopical algebra, relativity is often base-relative. For a cofibrant non-positively graded cdga $\phi:\mathcal E\to\mathcal L$46, a dg-Lie algebroid over $\phi:\mathcal E\to\mathcal L$47 is an $\phi:\mathcal E\to\mathcal L$48-dg-module and a $\phi:\mathcal E\to\mathcal L$49-dg-Lie algebra equipped with an anchor

$\phi:\mathcal E\to\mathcal L$50

satisfying the graded Leibniz rule. The category $\phi:\mathcal E\to\mathcal L$51 admits a cofibrantly generated model structure in which weak equivalences are quasi-isomorphisms on the underlying $\phi:\mathcal E\to\mathcal L$52-dg-modules and fibrations are degreewise surjections. The structure is transferred along the adjunction

$\phi:\mathcal E\to\mathcal L$53

and the path object is built from

$\phi:\mathcal E\to\mathcal L$54

In the absolute case $\phi:\mathcal E\to\mathcal L$55, this recovers the Hinich model structure on dg-Lie algebras (Vezzosi, 2013).

For a morphism $\phi:\mathcal E\to\mathcal L$56, a relative Lie algebroid is a quasicoherent sheaf $\phi:\mathcal E\to\mathcal L$57 with anchor

$\phi:\mathcal E\to\mathcal L$58

Its Chevalley–Eilenberg complex computes relative Lie algebroid cohomology, and under local freeness one has

$\phi:\mathcal E\to\mathcal L$59

The universal enveloping algebroid $\phi:\mathcal E\to\mathcal L$60 and jet algebroid $\phi:\mathcal E\to\mathcal L$61 carry bialgebra structures, and the paper proves HKR-type identifications

$\phi:\mathcal E\to\mathcal L$62

and, for finite-rank $\phi:\mathcal E\to\mathcal L$63,

$\phi:\mathcal E\to\mathcal L$64

For $\phi:\mathcal E\to\mathcal L$65, these become the relative tangent and relative jet cases (Sarkar, 2021).

A sheaf-theoretic rooted version is the $\phi:\mathcal E\to\mathcal L$66-Lie algebroid. Here $\phi:\mathcal E\to\mathcal L$67 is itself a Lie algebroid, together with a morphism

$\phi:\mathcal E\to\mathcal L$68

This is equivalent to giving $\phi:\mathcal E\to\mathcal L$69 the structure of a $\phi:\mathcal E\to\mathcal L$70-Gerstenhaber algebra; if $\phi:\mathcal E\to\mathcal L$71 is locally free of finite rank, it is also equivalent to an $\phi:\mathcal E\to\mathcal L$72-dga structure on $\phi:\mathcal E\to\mathcal L$73. The same rooted formalism extends to relative Lie bialgebroids, BV generators, PBW compatibility for universal enveloping algebroids, and factorization of homology–cohomology duality through $\phi:\mathcal E\to\mathcal L$74. In this framework, relativity is not to a base map but to a fixed background Lie algebroid carrying the ambient Gerstenhaber and BV structures (Poddar et al., 2024).

In derived geometry, the relative tangent target is $\phi:\mathcal E\to\mathcal L$75 or, affinely, $\phi:\mathcal E\to\mathcal L$76. Free dg-Lie algebroids over anchored modules $\phi:\mathcal E\to\mathcal L$77 admit explicit universal enveloping and jet algebras, and derived twisted connections are controlled by the HKR class

$\phi:\mathcal E\to\mathcal L$78

Vanishing of $\phi:\mathcal E\to\mathcal L$79 is equivalent to the existence of a derived $\phi:\mathcal E\to\mathcal L$80-connection on $\phi:\mathcal E\to\mathcal L$81, and the canonical twisted square-zero extension $\phi:\mathcal E\to\mathcal L$82 realizes the same lifting problem. Under the hypotheses of the paper,

$\phi:\mathcal E\to\mathcal L$83

This places relative dg-Lie algebroids at the center of derived intersection theory and formal neighborhood calculations (2002.01285).

A nearby but distinct use of “relative” occurs for relative Rota–Baxter operators on Lie algebroids. Given a LieRep pair $\phi:\mathcal E\to\mathcal L$84 and a bundle map $\phi:\mathcal E\to\mathcal L$85, the Maurer–Cartan condition $\phi:\mathcal E\to\mathcal L$86 defines a Lie algebroid structure on $\phi:\mathcal E\to\mathcal L$87, and the cohomology $\phi:\mathcal E\to\mathcal L$88 governs infinitesimal deformations and obstruction classes. This is not itself a definition of relative algebroid, but it is a relative construction internal to Lie algebroid theory (Liu et al., 2021). In a different direction again, the Banach–Lie groupoid of partially invertible elements of a $\phi:\mathcal E\to\mathcal L$89-algebra yields orbitwise Atiyah algebroids $\phi:\mathcal E\to\mathcal L$90, described as relative to the projection lattice and written in explicit operator coordinates (Odzijewicz et al., 2014).

Taken together, these frameworks show that relative algebroids are best understood not as a single replacement for Lie algebroids, but as a cluster of Lie-algebroid-based formalisms adapted to relative tangent data, foliations, base cdgas, fixed ambient algebroids, and curvature-controlled extensions. Their common role is to transport Lie-theoretic methods—Maurer–Cartan equations, tableaux, Spencer cohomology, PBW theory, HKR maps, enveloping constructions, and symmetry reduction—into settings where the underlying geometry is constrained by PDEs, embeddings, or auxiliary background structures.