Infinitesimal Multiplicative Weightings

• Infinitesimally multiplicative weightings are defined as Lie algebroid-level filtrations that ensure compatibility between weighted structures and Lie algebroid operations.
• They utilize graded subbundles of higher tangent bundles to encode anisotropic vanishing orders and maintain structure-preserving properties.
• This framework connects Lie groupoid multiplicative weightings to their infinitesimal counterparts via deformation spaces, Poisson geometry, and graded algebra techniques.

Infinitesimally multiplicative weightings are the Lie-algebroid-level counterparts of multiplicative weightings on Lie groupoids. They arise within the theory of weightings along submanifolds, where a single vanishing order along $N\subset M$ is replaced by a filtration of smooth functions that records anisotropic or nonisotropic order. In the Lie-theoretic setting, the groupoid-level requirement that the weighting respect source, target, units, multiplication, and inversion differentiates to a compatibility condition on the Lie algebroid. The resulting infinitesimal object can be described as a linear weighting on a Lie algebroid whose anchor and bracket preserve weighted degree, and equivalently through higher tangent geometry, degree-$0$ linear Poisson structures, or filtration-preserving homological vector fields (Meinrenken, 15 Jan 2026, Hudson, 14 Aug 2025).

1. Weightings along submanifolds

A weighting along a submanifold $N\subset M$ is a coordinate-free refinement of ordinary vanishing order. Instead of a single ideal filtration by powers of $I_N$, one considers a decreasing filtration

$$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$

such that

$C^\infty(M)_{(1)}=I_N.$

Locally, in suitable weighted coordinates $x_a$, a function belongs to $C^\infty(M)_{(i)}$ precisely when all Taylor monomials have total weight at least $i$. Intrinsically, the filtration is required to satisfy three conditions: the first layer is $I_N$; the differentials of functions of degree $0$0 span a subbundle $0$1; and the filtration is compatible with multiplication through

$0$2

This formalism is designed to encode “orders of vanishing” richer than the usual one, especially in Lie filtrations, Carnot-type structures, weighted blow-ups, and tangent groupoids (Meinrenken, 15 Jan 2026).

The associated constructions are the weighted normal bundle and the weighted deformation space, obtained from the associated graded algebra and the Rees algebra. In one formulation,

$0$3

with fibers

$0$4

The weighted normal bundle is generally a graded bundle rather than canonically a vector bundle, and the weighted blow-up is obtained from the deformation space by removing the zero section and quotienting by positive scaling. These constructions generalize the ordinary normal bundle, deformation to the normal cone, and blow-up to a weighted setting (Loizides et al., 2020).

2. Higher tangent encoding

A decisive structural insight is that a weighting can be encoded by a graded subbundle of a higher tangent bundle. One formulation uses

$0$5

and associates to a weighting a graded subbundle $0$6 defined by

$0$7

The weighting is recovered from $0$8 by

$0$9

The survey formulation uses the equivalent idea that the weighted structure is encoded by a submanifold in a higher tangent bundle, with the filtration reconstructed from vanishing of jet lifts (Loizides et al., 2020, Meinrenken, 15 Jan 2026).

This higher tangent viewpoint is what makes multiplicativity and infinitesimal multiplicativity natural. Once the basic weighted geometry has been translated into a graded subbundle $N\subset M$0, compatibility with Lie groupoid or Lie algebroid structure becomes the requirement that $N\subset M$1 be a subgroupoid or subalgebroid of the appropriate higher tangent object. A weighting is therefore not merely a filtration of functions; it is a geometric structure functorial enough to interact with tangent prolongation, deformation spaces, and weighted normal constructions (Loizides et al., 2020).

3. Multiplicative weightings on Lie groupoids

Let $N\subset M$2 be a Lie groupoid and $N\subset M$3 a Lie subgroupoid. A weighting of $N\subset M$4 along $N\subset M$5 is called multiplicative when it is compatible with the groupoid structure. One formulation requires that $N\subset M$6 be a weighted submanifold, that $N\subset M$7 be weighted submersions, that multiplication $N\subset M$8 be a weighted morphism or weighted submersion, and that inversion be a weighted morphism. Hudson’s theorem gives a particularly direct criterion: multiplicativity is equivalent to weightedness of $N\subset M$9, $I_N$0, the unit manifold, and the multiplication map. An equivalent tangent-bundle formulation requires that each annihilator piece $I_N$1 be a Lie subgroupoid of the tangent groupoid $I_N$2, together with weightedness of units and the multiplication graph (Meinrenken, 15 Jan 2026, Hudson, 14 Aug 2025).

The graph formulation is often more usable. The thesis proves that a weighting on $I_N$3 along $I_N$4 is multiplicative if and only if $I_N$5 is a weighted submanifold of $I_N$6, the graph $I_N$7 is a weighted submanifold, and the filtration of $I_N$8 is by subgroupoids

$I_N$9

The deformation-space formulation is equally central: multiplicativity is equivalent to the unique extension of the product groupoid $$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$0 to a Lie groupoid

$$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$1

In the survey notation, the weighted normal and weighted deformation spaces inherit Lie groupoid structures,

$$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$2

The infinitesimal counterpart of this global compatibility is the notion of an infinitesimally multiplicative weighting on the Lie algebroid (Hudson, 14 Aug 2025).

4. Lie algebroid definition of infinitesimally multiplicative weightings

For a Lie algebroid $$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$3, a linear weighting is a $$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$4-graded filtration of the sheaf of sections, compatible with the weighted structure on the base. An infinitesimally multiplicative weighting, or IM weighting, is obtained by imposing compatibility with the Lie algebroid structure: $$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$5

$$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$6

A Lie algebroid equipped with such a linear weighting is called a weighted Lie algebroid. This is the explicit algebroid-level realization of the infinitesimal counterpart announced in the survey literature (Hudson, 14 Aug 2025, Meinrenken, 15 Jan 2026).

The thesis gives two equivalent characterizations. First, under the standard correspondence between Lie algebroid structures on $$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$7 and linear Poisson structures on $$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$8, an IM weighting is equivalent to the canonical linear Poisson bivector $$C^\infty(M)=C^\infty(M)_{(0)} \supset C^\infty(M)_{(1)} \supset C^\infty(M)_{(2)} \supset \cdots$$9 having filtration degree $C^\infty(M)_{(1)}=I_N.$0, meaning

$C^\infty(M)_{(1)}=I_N.$1

Second, in supergeometric language, it is equivalent to the Lie algebroid differential

$C^\infty(M)_{(1)}=I_N.$2

being filtration preserving, or equivalently to the homological vector field on $C^\infty(M)_{(1)}=I_N.$3 being filtration preserving. In a weighted frame $C^\infty(M)_{(1)}=I_N.$4 of degrees $C^\infty(M)_{(1)}=I_N.$5, the local structure functions satisfy

$C^\infty(M)_{(1)}=I_N.$6

The higher tangent description remains valid on the algebroid side: a weighting on $C^\infty(M)_{(1)}=I_N.$7 along a Lie subalgebroid $C^\infty(M)_{(1)}=I_N.$8 is infinitesimally multiplicative exactly when the associated graded subbundle $C^\infty(M)_{(1)}=I_N.$9 is a Lie subalgebroid of $x_a$0 (Hudson, 14 Aug 2025, Loizides et al., 2020).

5. Differentiation, integration, and special cases

A central theorem states that multiplicative weightings differentiate to IM weightings. If $x_a$1 is a weighted Lie groupoid with Lie algebroid $x_a$2, the induced filtration on $x_a$3 is defined by

$x_a$4

This filtration is an IM weighting, and the weighted normal and weighted deformation constructions differentiate compatibly: $x_a$5 for both the weighted normal bundle and the weighted deformation bundle. The proof passes through the cotangent groupoid $x_a$6: the canonical Poisson structure on $x_a$7 has filtration degree $x_a$8, and the induced linear Poisson structure on $x_a$9 therefore has degree $C^\infty(M)_{(i)}$0, yielding the IM weighting on $C^\infty(M)_{(i)}$1 (Hudson, 14 Aug 2025).

The integration problem is subtler. For wide Lie subalgebroids, the thesis gives a complete criterion. If

$C^\infty(M)_{(i)}$2

is a Lie filtration of $C^\infty(M)_{(i)}$3, $C^\infty(M)_{(i)}$4 is an $C^\infty(M)_{(i)}$5-connected wide subgroupoid with $C^\infty(M)_{(i)}$6, and

$C^\infty(M)_{(i)}$7

together with constancy of

$C^\infty(M)_{(i)}$8

then the filtration integrates to a multiplicative weighting of $C^\infty(M)_{(i)}$9 along $i$0. In this wide setting, multiplicative weightings along $i$1-connected wide subgroupoids are in bijection with infinitesimally multiplicative weightings along wide Lie subalgebroids. For general subalgebroids, a sufficient criterion is that the graded subbundle $i$2 integrate to an $i$3-connected Lie subgroupoid $i$4; the resulting $i$5 then defines a multiplicative weighting of $i$6 along $i$7 (Hudson, 14 Aug 2025).

Several special cases organize the theory. Weight $i$8 recovers the ordinary vanishing filtration. Order $i$9 weightings are equivalent to a submanifold $I_N$0 together with a subbundle $I_N$1. Graded bundles produce weightings via a scaling action $I_N$2, with $I_N$3 of degree $I_N$4 exactly when $I_N$5. Lie filtrations and Carnot-type geometry yield weightings whose weighted normal and deformation spaces reproduce osculating and tangent-groupoid constructions. Along units, the theory becomes հատկապես rigid: for $I_N$6-connected groupoids, multiplicative weightings along the units are classified by Lie filtrations of the Lie algebroid,

$I_N$7

In particular, a filtered manifold $I_N$8 corresponds to a multiplicative weighting of the pair groupoid $I_N$9 along its units, and for a Lie group $0$00, a multiplicative weighting is a filtration of its Lie algebra by Lie subalgebras compatible with the bracket (Meinrenken, 15 Jan 2026, Hudson, 14 Aug 2025).

6. Position within the broader theory of infinitesimally multiplicative structures

Infinitesimally multiplicative weightings belong to a broader Lie-theoretic pattern in which a multiplicative object on a Lie groupoid is encoded by explicit compatibility data on the Lie algebroid. For multiplicative differential forms, the infinitesimal data are IM forms given by a pair

$0$01

satisfying the identities

$0$02

$0$03

$0$04

and on source-simply-connected groupoids these integrate bijectively to multiplicative forms (Bursztyn et al., 2010). For multiplicative connections on $0$05, the infinitesimal counterpart is an IM connection encoded by $0$06, and source-simply-connected groupoids again yield a bijection between multiplicative connections and IM connections (Pugliese et al., 2020). For multiplicative Dirac structures on Lie groups, the infinitesimal classification is by an ideal $0$07 together with a Lie bialgebra structure on $0$08 (Ortiz, 2009).

IM weightings follow the same architecture, but the infinitesimal data are filtrations and weighted degrees rather than forms, tensors, or Dirac structures. A recurrent misconception is to treat a weighting as only a filtration of functions or sections. The weighted-geometric literature instead presents it as a mechanism that simultaneously produces weighted normal bundles, weighted deformation spaces, and weighted blow-ups, and whose compatibility with Lie groupoids or Lie algebroids is most naturally expressed in higher tangent, Poisson, and homological terms (Meinrenken, 15 Jan 2026, Loizides et al., 2020). This suggests that IM weightings are best understood not as an isolated Lie-algebraic gadget but as the infinitesimal layer of a deformation-theoretic geometry adapted to anisotropic order.