Holonomy Chen Lie Algebra

• The holonomy Chen Lie algebra is a metabelian quotient of the holonomy Lie algebra, defined using the cup product on H1 and H2 and serving as a quadratic approximation for Chen invariants.
• For finitely presented groups, explicit computations via Magnus expansions and Fox derivatives yield cup-product relations that determine the algebra’s structure.
• The framework extends to matroids and CDGAs, enabling resonance analysis and comparisons with lower-central series to detect formality and non-formality properties.

The holonomy Chen Lie algebra is the maximal metabelian quotient of a holonomy Lie algebra. In the group-theoretic setting of a finitely generated group $G$, the holonomy Lie algebra is the quadratic Lie algebra

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$

where $$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$ is the cup product and $\mu_G^\vee$ is its dual; the holonomy Chen Lie algebra is then

$\mathfrak h(G)/\mathfrak h(G)''.$

An analogous construction exists for matroids via the Orlik–Solomon algebra and for connected CDGAs via quadratic-linear holonomy, and in each case the resulting metabelian quotient organizes Chen-type invariants in a form amenable to explicit computation, resonance-theoretic analysis, and comparison with lower-central-series graded Lie algebras (Suciu et al., 2017, Suciu, 28 Sep 2025, Suciu, 27 Apr 2026).

1. Definition and conceptual position

For a finitely generated group $G$, the holonomy Lie algebra is functorial in $G$ and depends only on $H^1(G;\mathbb Q)$, $H^2(G;\mathbb Q)$, and the degree-$1$ cup product. Its defining ideal is generated in degree $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$0, so $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$1 is a quadratic Lie algebra. If $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$2 is a connected CW-complex with $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$3, then $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$4, which allows one to compute holonomy from a convenient presentation $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$5-complex rather than from a classifying space (Suciu et al., 2017).

The metabelian quotient

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$6

is called the holonomy Chen Lie algebra. It is the Lie-theoretic analogue of replacing a group by its maximal metabelian quotient before taking the associated graded object. In the same spirit, for a matroid $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$7 with holonomy Lie algebra $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$8, the holonomy Chen Lie algebra is $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$9, and its graded ranks are the holonomy Chen ranks $$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$0 (Suciu, 28 Sep 2025).

For a connected CDGA $$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$1, the relevant holonomy object is no longer purely quadratic. The holonomy Lie algebra is

$$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$2

where $$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$3. This filtered, quadratic-linear Lie algebra refines the quadratic holonomy algebra of $$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$4, and its infinitesimal Alexander invariant produces the corresponding holonomy Chen ranks (Suciu, 27 Apr 2026).

2. Presentation-theoretic construction for finitely presented groups

A central development in the finitely presented group setting is an explicit computation of the cup product on the cohomology of a presentation $$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$5-complex by means of a relative Magnus expansion. If $$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$6 is generated by $$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$7, with $$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$8 the free group on the same generators and $$\mu_G\colon H^1(G;\mathbb Q)\wedge H^1(G;\mathbb Q)\to H^2(G;\mathbb Q)$$9, the rational Magnus expansion relative to $\mu_G^\vee$0 is

$\mu_G^\vee$1

and for $\mu_G^\vee$2 its degree-$\mu_G^\vee$3 truncation has the form

$\mu_G^\vee$4

The coefficients $\mu_G^\vee$5 are computable from Fox derivatives through

$\mu_G^\vee$6

where $\mu_G^\vee$7 is the matrix of the map $\mu_G^\vee$8 induced on rational abelianizations (Suciu et al., 2017).

After replacing an arbitrary finite presentation by an echelon presentation, one obtains bases $\mu_G^\vee$9 and $\mathfrak h(G)/\mathfrak h(G)''.$0 for the presentation $\mathfrak h(G)/\mathfrak h(G)''.$1-complex $\mathfrak h(G)/\mathfrak h(G)''.$2, and the cup product becomes

$\mathfrak h(G)/\mathfrak h(G)''.$3

This chain-level computation uses the Fenn–Sjerve chain transformation to the normalized bar resolution and turns Fox-derivative data into explicit cup-product coefficients (Suciu et al., 2017).

The consequence for holonomy is immediate. If $\mathfrak h(G)/\mathfrak h(G)''.$4 is the basis of $\mathfrak h(G)/\mathfrak h(G)''.$5 dual to $\mathfrak h(G)/\mathfrak h(G)''.$6, then

$\mathfrak h(G)/\mathfrak h(G)''.$7

and therefore

$\mathfrak h(G)/\mathfrak h(G)''.$8

This gives the holonomy Chen Lie algebra by adding the second derived ideal of the free Lie algebra to the quadratic Magnus relations (Suciu et al., 2017).

3. Relation to lower central series and to the actual Chen Lie algebra

Let

$\mathfrak h(G)/\mathfrak h(G)''.$9

be the graded Lie algebra associated to the lower central series. There is a natural epimorphism

$G$0

which is an isomorphism in degrees $G$1 and $G$2. The group $G$3 is graded-formal exactly when $G$4 is an isomorphism, equivalently when $G$5 is quadratic, or equivalently when

$G$6

Thus the holonomy Lie algebra is always a quadratic approximation to $G$7, and it is exact precisely in the graded-formal case (Suciu et al., 2017).

The actual Chen Lie algebra of $G$8 is

$G$9

Its relation to holonomy is controlled by formality. For $G$0-formal groups,

$G$1

and in particular

$G$2

Hence the Chen ranks

$G$3

coincide with the holonomy Chen ranks

$G$4

for all $G$5 whenever $G$6 is $G$7-formal (Suciu et al., 2017).

This exactness mechanism is particularly clean for arrangement groups. In the realizable matroid case, with $G$8 and $G$9, one has

$H^1(G;\mathbb Q)$0

This identifies the holonomy Chen Lie algebra of the matroid with the rational Chen Lie algebra of the arrangement group (Suciu, 28 Sep 2025).

4. Matroids, Orlik–Solomon algebras, and resonance

For a simple matroid $H^1(G;\mathbb Q)$1 on ground set $H^1(G;\mathbb Q)$2, the Orlik–Solomon algebra $H^1(G;\mathbb Q)$3 is extracted from the circuit structure, and the holonomy Lie algebra is defined by dualizing its degree-$H^1(G;\mathbb Q)$4 multiplication: $H^1(G;\mathbb Q)$5 where $H^1(G;\mathbb Q)$6 and $H^1(G;\mathbb Q)$7 is the natural projection. A concrete presentation is

$H^1(G;\mathbb Q)$8

Accordingly, $H^1(G;\mathbb Q)$9 depends only on the truncated lattice $H^2(G;\mathbb Q)$0, that is, on the rank-$H^2(G;\mathbb Q)$1 and rank-$H^2(G;\mathbb Q)$2 flats (Suciu, 28 Sep 2025).

The holonomy Chen Lie algebra is

$H^2(G;\mathbb Q)$3

with graded ranks

$H^2(G;\mathbb Q)$4

These satisfy

$H^2(G;\mathbb Q)$5

where $H^2(G;\mathbb Q)$6 are the holonomy ranks, and equality holds for $H^2(G;\mathbb Q)$7. Thus the metabelian quotient preserves the first three degrees and discards higher non-metabelian commutators (Suciu, 28 Sep 2025).

The key structural device is the infinitesimal Alexander invariant

$H^2(G;\mathbb Q)$8

of a graded Lie algebra $H^2(G;\mathbb Q)$9. For $1$0, the generating function for holonomy Chen ranks is

$1$1

Moreover, the first Koszul module satisfies

$1$2

so the holonomy Chen Lie algebra can be studied equivalently through its metabelian quotient, its infinitesimal Alexander invariant, or the first Koszul module of the Orlik–Solomon algebra (Suciu, 28 Sep 2025).

Resonance controls this module. If $1$3 denotes degree-$1$4 resonance, then

$1$5

whenever $1$6. Positive-dimensional irreducible components of $1$7 correspond bijectively to multinets on submatroids, and this leads to the Chen ranks conjecture for matroids: $1$8 This suggests that, asymptotically, each $1$9-dimensional resonance component contributes like a rank-$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$00 uniform piece $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$01 (Suciu, 28 Sep 2025).

5. CDGA models, Koszul modules, and linearization

The CDGA formulation places holonomy Chen theory in a model-categorical setting. For a connected CDGA $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$02 over a field of characteristic $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$03, the holonomy Lie algebra

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$04

retains not only the degree-$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$05 product structure but also the linear part of the differential. This distinguishes $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$06 from the purely quadratic holonomy algebra of $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$07 (Suciu, 27 Apr 2026).

Let $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$08, and let $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$09 be the homological Koszul complex. Its first homology

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$10

is the first Koszul module. A basic theorem identifies it with the infinitesimal Alexander invariant of the holonomy Lie algebra: $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$11 Consequently, the holonomy Chen ranks of $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$12 are encoded by the associated graded Hilbert series: $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$13 This gives a direct route from CDGA data to Chen-type invariants without passing through a group presentation (Suciu, 27 Apr 2026).

The same framework compares infinitesimal and classical Alexander invariants. If a finitely generated group $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$14 admits a $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$15-finite $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$16-model $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$17, then

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$18

and

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$19

Thus Chen ranks are determined by the model $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$20, and this comparison does not require a formality assumption (Suciu, 27 Apr 2026).

A further refinement is Koszul linearization. The associated graded of the Koszul complex of $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$21 is the Koszul complex of $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$22, yielding spectral sequences whose higher differentials are expressed by iterated Massey products. This implies that cohomology controls the first-order behavior of resonance at the origin, while higher-order Chen data may still depend on the full differential $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$23. A plausible implication is that holonomy Chen invariants are especially sensitive detectors of non-formality precisely because they sit between cohomology-level quadratic structure and the full CDGA model (Suciu, 27 Apr 2026).

6. Examples, exactness regimes, and limitations

The finitely presented group case already shows the range of behavior. For a one-relator group $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$24, if $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$25, then

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$26

whereas if $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$27, then

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$28

a free Lie algebra on $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$29 generators. A one-relator group is graded-formal iff the relator has weight $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$30. For orientable surface groups,

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$31

and the Chen ranks are explicitly computable. For link groups with connected linking graph, holonomy Chen and actual Chen coincide; by contrast, for orientable Seifert manifolds with $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$32, the discrepancy between holonomy Chen ranks and actual Chen ranks detects non-$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$33-formality and non-graded-formality (Suciu et al., 2017).

In the matroid setting, the rank-$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$34 uniform matroid $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$35 has

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$36

while for graphic matroids

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$37

For the complete graph $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$38, local lower bounds for $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$39 can be strict, and for the non-Fano arrangement the rational comparison between holonomy Chen and group Chen holds whereas the mod-$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$40 comparison can fail already in degree $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$41. This demonstrates that rational exactness and positive-characteristic exactness are genuinely different phenomena (Suciu, 28 Sep 2025).

CDGA examples exhibit the same sensitivity to higher-order structure. For the free $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$42-step nilpotent Lie algebra $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$43,

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$44

and for pure elliptic braid groups

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$45

By contrast, for the $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$46-dimensional Heisenberg Lie algebra,

$$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$47

whereas $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$48. This shows that the full model can drastically suppress resonance and alter holonomy Chen data relative to the cohomology algebra (Suciu, 27 Apr 2026).

Several limitations are structural. In the presentation-theoretic group setting, the explicitly computed cup product is that of the presentation $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$49-complex $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$50, not necessarily of a $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$51, and $$\mathfrak h(G)=\operatorname{lie}\bigl(H_1(G;\mathbb Q)\bigr)\big/\bigl(\operatorname{im}\mu_G^\vee\bigr),$$52 in general, although the construction still recovers the correct holonomy Lie algebra after passage to echelon approximation. The framework is rational for arbitrary finitely presented groups, and holonomy remains only a quadratic approximation unless graded-formality holds (Suciu et al., 2017). In the matroid and arrangement setting, the rational comparison with group Chen invariants is established in the realizable case, but the survey literature emphasizes open problems in positive characteristic, higher resonance, and the full Chen ranks conjecture for non-realizable matroids (Suciu, 28 Sep 2025). Recent work on middle convolution for Lie algebra representations acts on holonomy Lie algebras of suitable arrangement complements and is compatible with logarithmic connections and local systems, but it does not develop Chen Lie algebras, Chen ranks, or metabelian quotients as independent objects. This suggests a current division between a highly developed representation theory on the holonomy side and a distinct metabelian theory on the Chen side (Hiroe, 11 May 2026).