Abelian Extensions in Com-PreLie Algebras

• Abelian extensions of Com-PreLie algebras are classified by second cohomology, capturing the interplay between commutative and left pre-Lie structures through cocycle conditions.
• The methodology employs module representations and coupled cocycle equations to construct extensions and ensure compatibility between the two algebraic operations.
• Criteria for lifting automorphisms via a Wells exact sequence are established, providing practical computational templates and insights into deformation theory.

Searching arXiv for the cited Com-PreLie extension papers to ground the article in current literature. Abelian extensions of Com-PreLie algebras form the extension theory attached to algebraic systems carrying both a commutative associative product and a left pre-Lie product, together with a compatibility axiom. In the formulation developed by Zhang–Lu, a Com-PreLie algebra is a triple $(A,\cdot,\circ)$ such that $(A,\cdot)$ is commutative associative, $(A,\circ)$ is left pre-Lie, and

$$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$

An abelian extension is a short exact sequence

$0\to V\to E\to A\to 0$

in which $V$ is an abelian ideal, and the recent cohomological theory identifies equivalence classes of such extensions with the second cohomology group $H^2(A,V)$; the same framework also yields a cohomological criterion for lifting automorphisms through the extension and a Wells-type exact sequence governing inducibility (Zhang et al., 30 Sep 2025). Closely related work on compatible pre-Lie algebras shows an analogous classification by second cohomology in a bidifferential graded setting (Liu et al., 2023).

1. Algebraic setting and representation-theoretic data

Let $k$ be a field of characteristic $0$. A Com-PreLie algebra in the sense of Zhang–Lu consists of a $k$-vector space $(A,\cdot)$0 with two bilinear products, written here as $(A,\cdot)$1 and $(A,\cdot)$2, subject to three axioms: $(A,\cdot)$3 is a commutative associative algebra, $(A,\cdot)$4 is a left pre-Lie algebra,

$(A,\cdot)$5

and the compatibility axiom is

$(A,\cdot)$6

This left-derivation identity is the structural condition that couples the commutative and pre-Lie parts (Zhang et al., 30 Sep 2025).

A representation of $(A,\cdot)$7 is a quadruple $(A,\cdot)$8 in which $(A,\cdot)$9 is a representation of the commutative associative algebra $(A,\circ)$0 and $(A,\circ)$1 is a representation of the pre-Lie algebra $(A,\circ)$2, with additional compatibility between these actions. Explicitly,

$(A,\circ)$3

$(A,\circ)$4

$(A,\circ)$5

and

$(A,\circ)$6

These identities are equivalent to the statement that the semi-direct product $(A,\circ)$7 with operations

$(A,\circ)$8

$(A,\circ)$9

is again a Com-PreLie algebra. Thus the module notion is not auxiliary: it is the exact representation-theoretic datum needed to place $$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$0 as an abelian ideal inside a larger Com-PreLie algebra.

2. Low-dimensional cohomology and the coupled cocycle equations

The cohomology introduced for Com-PreLie algebras is low-dimensional and is modeled after Flato–Gerstenhaber–Voronov cohomology for Poisson algebras and the Dzhumadil’daev coboundary for pre-Lie algebras (Zhang et al., 30 Sep 2025). In degree $$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$1,

$$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$2

and in degree $$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$3,

$$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$4

with the convention that the $$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$5-part $$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$6 is symmetric while the $$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$7-part $$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$8 is an arbitrary bilinear map.

For $$a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).$$9, the coboundary has two components:

$0\to V\to E\to A\to 0$0

$0\to V\to E\to A\to 0$1

Accordingly, a $0\to V\to E\to A\to 0$2-cocycle is a linear map $0\to V\to E\to A\to 0$3 satisfying both identities

$0\to V\to E\to A\to 0$4

$0\to V\to E\to A\to 0$5

For $0\to V\to E\to A\to 0$6 with $0\to V\to E\to A\to 0$7 symmetric, the degree-$0\to V\to E\to A\to 0$8 coboundary conditions split into three parts. The Harrison-type equation for the commutative associative component is

$0\to V\to E\to A\to 0$9

The pre-Lie-type equation is

$V$0

The mixed compatibility equation coupling $V$1 and $V$2 is

$V$3

This tripartite form is decisive: the extension problem is not a direct sum of an associative extension problem and a pre-Lie extension problem, because the mixed equation couples the two pieces through $V$4.

A $V$5-coboundary is a pair of the form $V$6, namely

$V$7

$V$8

Two $V$9-cocycles are cohomologous exactly when they differ by such a coboundary. The second cohomology group is

$H^2(A,V)$0

and the low-degree identity $H^2(A,V)$1 implies $H^2(A,V)$2.

3. Abelian extensions and classification by second cohomology

An abelian extension of a Com-PreLie algebra $H^2(A,V)$3 by a vector space $H^2(A,V)$4 is a short exact sequence

$H^2(A,V)$5

of Com-PreLie algebras such that $H^2(A,V)$6 is an abelian ideal: its internal products are trivial,

$H^2(A,V)$7

and the induced $H^2(A,V)$8-action on $H^2(A,V)$9 agrees with a prescribed representation $k$0 (Zhang et al., 30 Sep 2025). If $k$1 is a section, then the module structure is recovered from the extension by

$k$2

and this representation is independent of the choice of section.

The classification theorem proceeds in both directions. From an extension and a section $k$3, one defines

$k$4

The pair $k$5 is a $k$6-cocycle, and its cohomology class is independent of the chosen section and of the chosen representative of the extension isomorphism class. Conversely, given a $k$7-cocycle $k$8, one defines a Com-PreLie structure on $k$9 by

$0$0

$0$1

The cocycle identities are exactly the conditions ensuring that these two operations satisfy commutative associativity, the pre-Lie identity, and the compatibility axiom.

If two cocycles differ by a coboundary $0$2, then the resulting extensions are isomorphic via

$0$3

Hence equivalence classes of abelian extensions are parametrized by second cohomology:

$0$4

The same construction admits an obstruction-theoretic reading: the condition $0$5 is the obstruction-vanishing condition for defining a Com-PreLie algebra structure on $0$6 with the displayed formulas, while changing the section changes $0$7 by a coboundary and therefore preserves the cohomology class.

4. Inducibility of automorphisms and the Wells exact sequence

Given an abelian extension

$0$8

with representation $0$9, let $k$0 denote the group of Com-PreLie automorphisms $k$1 of $k$2 satisfying $k$3 (Zhang et al., 30 Sep 2025). There is a natural group homomorphism

$k$4

where $k$5 is independent of the chosen section. A pair $k$6 is called inducible if it belongs to the image of $k$7.

The necessary and sufficient criterion for inducibility has two components. First, $k$8 must be compatible with the representation:

$k$9

The compatible pairs form a subgroup

$(A,\cdot)$00

Second, if $(A,\cdot)$01 is the cocycle attached to the extension, there must exist a linear map $(A,\cdot)$02 such that

$(A,\cdot)$03

$(A,\cdot)$04

Equivalently, using the natural action of $(A,\cdot)$05 on cocycles,

$(A,\cdot)$06

the pair $(A,\cdot)$07 is inducible if and only if $(A,\cdot)$08 and $(A,\cdot)$09 is cohomologous to $(A,\cdot)$10.

This criterion is encoded by the Wells map

$(A,\cdot)$11

which is well defined and independent of the chosen section. The vanishing criterion is

$(A,\cdot)$12

One obtains the Wells exact sequence

$(A,\cdot)$13

Here $(A,\cdot)$14 is the group of $(A,\cdot)$15-cocycles, and the map into $(A,\cdot)$16 identifies $(A,\cdot)$17-cocycles with automorphisms fixing both $(A,\cdot)$18 and $(A,\cdot)$19 by

$(A,\cdot)$20

Exactness expresses two structural facts: a compatible pair lifts precisely when its Wells image vanishes, and the fiber of $(A,\cdot)$21 over an inducible pair is a torsor under $(A,\cdot)$22.

5. Examples, splitting phenomena, and computational templates

A basic computation is obtained by taking $(A,\cdot)$23 with its usual commutative product and trivial pre-Lie product $(A,\cdot)$24, and letting $(A,\cdot)$25 with $(A,\cdot)$26 and $(A,\cdot)$27 (Zhang et al., 30 Sep 2025). In this case the Harrison-type cocycle condition for

$(A,\cdot)$28

holds trivially, while the compatibility equation for

$(A,\cdot)$29

forces $(A,\cdot)$30, so $(A,\cdot)$31. For a $(A,\cdot)$32-cochain $(A,\cdot)$33,

$(A,\cdot)$34

and therefore every cocycle $(A,\cdot)$35 is cohomologous to $(A,\cdot)$36. Consequently,

$(A,\cdot)$37

The stated consequences are that all abelian extensions of $(A,\cdot)$38 by $(A,\cdot)$39 split, the Wells map is identically zero, and any compatible pair $(A,\cdot)$40 is inducible; in this example $(A,\cdot)$41 and $(A,\cdot)$42 acts on $(A,\cdot)$43 by scaling.

More generally, the construction of an extension from cocycle data is uniform. Given any Com-PreLie algebra $(A,\cdot)$44 and a representation $(A,\cdot)$45, one sets

$(A,\cdot)$46

$(A,\cdot)$47

The three cocycle identities ensure associativity and commutativity of $(A,\cdot)$48, the pre-Lie identity for $(A,\cdot)$49, and the compatibility

$(A,\cdot)$50

This gives a direct computational procedure: specify the module actions, solve the cocycle equations, quotient by coboundaries, and thereby recover the extension classes.

The available examples also indicate a recurring vanishing phenomenon. In many trivial or semitrivial situations, such as $(A,\cdot)$51 and $(A,\cdot)$52 with $(A,\cdot)$53 factoring through a simple algebra, the second cohomology may vanish, implying that all abelian extensions split. The explicit one-dimensional calculation serves as the model for checking this concretely. The same pattern governs inducibility: after verifying compatibility of $(A,\cdot)$54 and $(A,\cdot)$55 with $(A,\cdot)$56, one computes the class

$(A,\cdot)$57

and the vanishing of this class is exactly the lifting condition.

6. Relation to compatible pre-Lie algebras, deformation theory, and terminology

The supplied literature uses closely related but not identical terminology. Zhang–Lu study Com-PreLie algebras as systems with a commutative associative product and a pre-Lie product (Zhang et al., 30 Sep 2025). Liu–Chen, by contrast, study compatible pre-Lie algebras, written $(A,\cdot)$58, in which both $(A,\cdot)$59 and $(A,\cdot)$60 are pre-Lie products and the compatibility condition is that every linear combination

$(A,\cdot)$61

is again a pre-Lie product (Liu et al., 2023). Their extension theory is parallel in form, but the underlying algebraic object is different.

In the compatible pre-Lie setting, a representation is a quadruple $(A,\cdot)$62, one pair for each pre-Lie product, and the extension data are two $(A,\cdot)$63-cochains $(A,\cdot)$64 defined from a section $(A,\cdot)$65 by

$(A,\cdot)$66

The extension on $(A,\cdot)$67 is then

$(A,\cdot)$68

$(A,\cdot)$69

and the cocycle conditions are

$(A,\cdot)$70

As in the Com-PreLie case above, the mixed condition shows that the two pieces of extension data cannot be chosen independently. Abelian extensions are classified by the second cohomology group $(A,\cdot)$71.

This second framework is embedded in a broader bidifferential graded Lie algebra formalism. The graded Lie algebra $(A,\cdot)$72 carries two differentials

$(A,\cdot)$73

satisfying

$(A,\cdot)$74

Maurer–Cartan elements encode compatible pre-Lie structures and their deformations, and the same cohomological apparatus governs infinitesimal deformations, Nijenhuis operators, formal rigidity, and abelian extensions. In particular, if $(A,\cdot)$75, then the compatible pre-Lie algebra is rigid. This suggests a broader structural principle across the adjacent theories: second cohomology controls both extension classes and first-order deformation data, while mixed cocycle identities record the interaction between the two algebraic operations.