Inducibility of Com-PreLie Automorphisms

• Com-PreLie automorphisms inducibility is defined by algebra extensions where a pair of automorphisms lifts if a precise second cohomology obstruction vanishes.
• The framework employs a Wells exact sequence that bridges 1-cocycles and compatible automorphism pairs, fully characterizing when liftability occurs.
• Concrete models using shuffle and symmetric algebras demonstrate that many automorphisms directly originate from underlying linear or preLie structures.

Searching arXiv for the cited papers to ground the article in current arXiv records. Inducibility of Com-PreLie automorphisms concerns the lifting of automorphisms through extensions of Com-PreLie algebras. In the extension-theoretic formulation, one starts from an abelian extension

$$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$

and asks whether a prescribed pair of automorphisms on the kernel and quotient is realized by an automorphism of the total algebra. Recent work places this problem in a cohomological framework: compatible pairs are liftable precisely when an obstruction class in the second cohomology group vanishes, and the resulting structure is organized by a Wells exact sequence (Zhang et al., 30 Sep 2025). In parallel, explicit Com-PreLie Hopf and bialgebra models on shuffle and symmetric algebras show that many automorphisms are induced directly from automorphisms of underlying linear or preLie data, making inducibility computable in concrete families (Foissy, 2013, Foissy, 2015).

1. Com-PreLie structures and the lifting problem

In the formulation used for the cohomological inducibility problem, a Com-PreLie algebra is a vector space $A$ over a field of characteristic zero, endowed with a commutative associative product $\ast$ and a left pre-Lie product $\bullet$ satisfying

$$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$

An abelian extension of a Com-PreLie algebra $A$ by a representation $V$ is a short exact sequence of Com-PreLie algebras

$$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$

with $V$ having trivial Com-PreLie structure and the structure of the extension encoding the representation (Zhang et al., 30 Sep 2025).

The corresponding automorphism problem is formulated in terms of $\operatorname{Aut}_V(\widehat{A})$, the Com-PreLie automorphisms $A$0 of $A$1 with $A$2. Any such $A$3 induces automorphisms on the two visible layers of the extension: $A$4 and $A$5. The inducibility question is then: given a pair $A$6, does there exist $A$7 such that $A$8 (Zhang et al., 30 Sep 2025)?

This formulation separates two distinct issues. First, one must decide whether $A$9 is compatible with the representation encoded by the extension. Second, one must determine whether the residual obstruction vanishes in cohomology. The general theory shows that compatibility is necessary but not by itself sufficient (Zhang et al., 30 Sep 2025).

2. Abelian extensions and second cohomology

The cohomological treatment of inducibility is built on a representation $\ast$0 of $\ast$1, where the operators encode how the commutative product and the pre-Lie product act on the module $\ast$2. The corresponding cohomology theory is defined through a complex $\ast$3 whose coboundary combines Harrison cohomology for the commutative part and Dzhumadil’daev cohomology for the pre-Lie part, extending methods used for Poisson and pre-Lie algebras (Zhang et al., 30 Sep 2025).

At degree two, the relevant objects are 2-cocycles $\ast$4, bilinear maps encoding the non-split part of the extension. The second cohomology group

$\ast$5

classifies equivalence classes of abelian extensions (Zhang et al., 30 Sep 2025). In this setting, inducibility is not an external add-on to extension theory; it is encoded by the same degree-two data that classifies the extensions themselves.

A central feature is the twisted action of automorphisms on the cocycle representing the extension. For a pair $\ast$6, one defines

$\ast$7

The comparison between $\ast$8 and $\ast$9 is precisely what measures whether the pair can be lifted (Zhang et al., 30 Sep 2025).

This gives second cohomology a dual role. It classifies abelian extensions and simultaneously detects the failure of an automorphism pair on the constituents to extend to the whole algebra. The inducibility problem is therefore an obstruction problem internal to the extension class.

3. Compatible pairs and the cohomological criterion

The first layer of the theory is the notion of a compatible pair. A pair $\bullet$0 is compatible if, for all $\bullet$1 and $\bullet$2, \begin{align} \beta(\mu(x)u) &= \mu(\alpha(x))\beta(u), \ \beta(l(x)u) &= l(\alpha(x))\beta(u), \ \beta(r(x)u) &= r(\alpha(x))\beta(u). \end{align} These identities express that $\bullet$3 and $\bullet$4 respect the action of $\bullet$5 on $\bullet$6 determined by the extension (Zhang et al., 30 Sep 2025).

The necessary and sufficient inducibility criterion states that a compatible pair $\bullet$7 is inducible if and only if there exists a linear map $\bullet$8 such that, for all $\bullet$9, \begin{align} \beta(\phi(x,y)) - \phi(\alpha(x), \alpha(y)) &= \mu(\alpha(x))\varphi(y) - \varphi(x \ast y) + \mu(\alpha(y))\varphi(x), \ \beta(\psi(x,y)) - \psi(\alpha(x), \alpha(y)) &= l(\alpha(x))\varphi(y) - \varphi(x \bullet y) + r(\alpha(y))\varphi(x). \end{align} Equivalently, the cohomology classes of $$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$0 and $$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$1 coincide in $$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$2 if and only if $$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$3 is inducible (Zhang et al., 30 Sep 2025).

This criterion is exact in a strong sense. It does not merely provide a sufficient condition or a deformation-theoretic heuristic; it fully characterizes liftability. The obstruction is therefore not an ad hoc invariant but the failure of two cocycles to be cohomologous.

A common misconception is that the existence of automorphisms on the kernel and quotient should automatically imply liftability. The criterion shows otherwise: compatibility is a structural prerequisite, but actual inducibility occurs only when the twisted cocycle differs from the original one by a coboundary (Zhang et al., 30 Sep 2025).

4. Wells map and exact-sequence formulation

The obstruction can be assembled into a canonical cohomological map, the Wells map,

$$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$4

where $$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$5 denotes the subgroup of compatible automorphism pairs. A key property is that $$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$6 does not depend on the choice of section of the extension. The vanishing criterion becomes

$$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$7

(Zhang et al., 30 Sep 2025).

The full structure is expressed by the Wells exact sequence

$$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$8

Here $$x \bullet (y \ast z) = (x \bullet y)\ast z + y\ast(x\bullet z) \qquad \forall x,y,z \in A.$$9 consists of 1-cocycles, identified with automorphisms of the extension that induce the identity on both $A$0 and $A$1; $A$2 sends an extension automorphism to its induced pair on the kernel and quotient; and $A$3 records the obstruction class (Zhang et al., 30 Sep 2025).

Exactness has two immediate interpretations. First, the kernel of $A$4 is precisely the subgroup of extension automorphisms acting trivially on both visible layers. Second, the image of $A$5 is precisely the kernel of $A$6, so inducible pairs are exactly those with trivial obstruction. The sequence therefore packages automorphism structure and cohomological obstruction into a single functorial framework (Zhang et al., 30 Sep 2025).

In this formulation, the inducibility problem becomes structurally transparent: automorphisms of the total extension sit between 1-cocycles and compatible pairs, and second cohomology measures the precise failure of surjectivity from extension automorphisms to compatible constituent automorphisms.

5. Explicit inducing mechanisms in shuffle and symmetric models

Before the general extension-theoretic treatment, inducibility already appeared in concrete Com-PreLie Hopf constructions. Given a linear endomorphism $A$7, one can define a preLie product on the shuffle algebra $A$8 by

$A$9

and the resulting structure $V$0 is a Com-PreLie bialgebra (Foissy, 2015). In the closely related formulation of the shuffle model, if $V$1, $V$2, and $V$3 satisfy $V$4, then the induced algebra morphism

$V$5

is a Com-PreLie algebra morphism (Foissy, 2013).

For automorphisms, this yields a rigid description: any Com-PreLie automorphism of $V$6 is induced from a vector space automorphism $V$7 satisfying $V$8, and the automorphism group is the centralizer of $V$9 in $$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$0 (Foissy, 2013). The same phenomenon is stated in the bialgebra classification framework: for $$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$1, an automorphism corresponds to a vector space automorphism $$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$2 with $$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$3; for $$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$4, automorphisms correspond to preLie automorphisms of $$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$5; and for $$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$6, automorphisms are essentially linear isomorphisms respecting $$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$7 and $$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$8 (Foissy, 2015).

These families show a form of inducibility internal to the construction. In $$0 \rightarrow V \xrightarrow{i} \widehat{A} \xrightarrow{j} A \rightarrow 0$$9, any linear endomorphism $V$0 on $V$1 can appear as $V$2, and every automorphism of the Com-PreLie algebra is induced from an automorphism of the base space commuting with $V$3. In $V$4, any preLie product $V$5 on $V$6 can be lifted uniquely, and automorphisms are induced by preLie automorphisms of $V$7 (Foissy, 2015). This is a more rigid situation than the extension-theoretic problem: the global automorphism is determined directly by the underlying data rather than obstructed by a nontrivial cohomology class.

The classification of connected cocommutative Com-PreLie bialgebras sharpens this point. If $V$8 is connected and cocommutative, then either $V$9 is isomorphic to a symmetric algebra $\operatorname{Aut}_V(\widehat{A})$0 equipped with a preLie product as in $\operatorname{Aut}_V(\widehat{A})$1, or, when $\operatorname{Aut}_V(\widehat{A})$2, $\operatorname{Aut}_V(\widehat{A})$3 is isomorphic to $\operatorname{Aut}_V(\widehat{A})$4 (Foissy, 2015). In these cases, automorphism and isomorphism questions reduce to the corresponding automorphisms of the underlying data, which makes inducibility highly constrained and often completely explicit.

6. Analogies, scope, and structural significance

The general Com-PreLie obstruction theory belongs to a broader pattern in extension theory. A closely related multiplicative Lie algebra result shows that the obstruction to the inducibility of pairs lies in the second cohomology group and establishes a Wells type exact sequence

$\operatorname{Aut}_V(\widehat{A})$5

relating automorphism groups of extensions to second cohomology (Singh et al., 2024). The Com-PreLie development follows the same categorical template, but with the cohomology and compatibility conditions adapted to the simultaneous commutative and pre-Lie structures (Zhang et al., 30 Sep 2025).

This analogy is methodologically important. In both settings, the data of an extension produces cocycles, automorphism pairs act by twisting those cocycles, and inducibility is equivalent to the triviality of the resulting cohomology class. A plausible implication is that Wells-type exact sequences are not accidental artifacts of a single algebraic category but structural features of extension theories equipped with a suitable degree-two cohomology.

Two distinctions are therefore essential. First, inducibility in the abstract extension-theoretic sense is not the same as the direct induction of automorphisms from base vector-space data in shuffle models. The former is governed by compatibility and a cohomological obstruction; the latter is often rigidly controlled by centralizers or automorphism groups of the underlying preLie structure (Foissy, 2013, Foissy, 2015). Second, the existence of functorial constructions in model families does not eliminate the possibility of genuine obstructions in general abelian extensions; the Wells map exists precisely to detect those obstructions (Zhang et al., 30 Sep 2025).

The current theory therefore presents inducibility of Com-PreLie automorphisms as a two-level phenomenon. At the explicit combinatorial level, many automorphisms are induced directly from linear or preLie automorphisms of primitive data. At the extension-theoretic level, lifting is controlled by compatibility and by the vanishing of a degree-two obstruction. Together, these results place Com-PreLie automorphisms within a unified framework linking combinatorial Hopf algebra constructions, representation theory, cohomology, and exact sequences (Zhang et al., 30 Sep 2025).