---
title: Abelian Extensions in Com-PreLie Algebras
url: https://www.emergentmind.com/topics/abelian-extensions-of-com-prelie-algebras
type: topic
---

# Abelian Extensions in Com-PreLie Algebras

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 [2510.23611]. Closely related work on compatible pre-Lie algebras shows an analogous classification by second cohomology in a bidifferential graded setting [2302.07178].

## 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 $A$ with two bilinear products, written here as $a\cdot b$ and $a\circ b$, subject to three axioms: $(A,\cdot)$ is a commutative associative algebra, $(A,\circ)$ is a left pre-Lie algebra,
$$
(a\circ b)\circ c-a\circ(b\circ c)=(b\circ a)\circ c-b\circ(a\circ c),
$$
and the compatibility axiom is
$$
a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).
$$
This left-derivation identity is the structural condition that couples the commutative and pre-Lie parts [2510.23611].

A representation of $(A,\cdot,\circ)$ is a quadruple $(V,\mu,l,r)$ in which $\mu:A\to \operatorname{End}(V)$ is a representation of the commutative associative algebra $(A,\cdot)$ and $(l,r)$ is a representation of the pre-Lie algebra $(A,\circ)$, with additional compatibility between these actions. Explicitly,
$$
\mu(x\cdot y)=\mu(x)\mu(y),
$$
$$
l(x\circ y)-l(x)l(y)=l(y\circ x)-l(y)l(x),
$$
$$
r(y)l(x)-l(x)r(y)=r(y)r(x)-r(x\circ y),
$$
and
$$
l(x)\mu(y)=\mu(x\circ y)+\mu(y)l(x), \qquad r(x\cdot y)=\mu(y)r(x)+\mu(x)r(y).
$$
These identities are equivalent to the statement that the semi-direct product $A\ltimes V$ with operations
$$
(x+u)\cdot(y+v)=x\cdot y+\mu(x)v+\mu(y)u,
$$
$$
(x+u)\circ(y+v)=x\circ y+l(x)v+r(y)u
$$
is again a Com-PreLie algebra. Thus the module notion is not auxiliary: it is the exact representation-theoretic datum needed to place $V$ 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 [2510.23611]. In degree $1$,
$$
C^1(A,V)=\operatorname{Hom}(A,V),
$$
and in degree $2$,
$$
C^2(A,V)=\operatorname{Hom}(A\times A,V),
$$
with the convention that the $\cdot$-part $\phi$ is symmetric while the $\circ$-part $\psi$ is an arbitrary bilinear map.

For $N\in C^1(A,V)$, the coboundary has two components:
$$
\delta^1_HN(x,y)=\mu(x)N(y)-N(x\cdot y)+\mu(y)N(x),
$$
$$
\delta^1_DN(x,y)=l(x)N(y)-N(x\circ y)+r(y)N(x).
$$
Accordingly, a $1$-cocycle is a linear map $N:A\to V$ satisfying both identities
$$
\mu(x)N(y)-N(x\cdot y)+\mu(y)N(x)=0,
$$
$$
l(x)N(y)-N(x\circ y)+r(y)N(x)=0.
$$

For $(\phi,\psi)\in C^2(A,V)$ with $\phi$ symmetric, the degree-$2$ coboundary conditions split into three parts. The Harrison-type equation for the commutative associative component is
$$
\phi(x,y\cdot z)+\mu(x)\phi(y,z)-\phi(x\cdot y,z)-\mu(z)\phi(x,y)=0.
$$
The pre-Lie-type equation is
$$
\psi(x\circ y,z)+r(z)\psi(x,y)-\psi(x,y\circ z)-l(x)\psi(y,z)
-\psi(y\circ x,z)-r(z)\psi(y,x)+\psi(y,x\circ z)+l(y)\psi(x,z)=0.
$$
The mixed compatibility equation coupling $\phi$ and $\psi$ is
$$
\psi(x,y\cdot z)+l(x)\phi(y,z)-\phi(x\circ y,z)-\mu(z)\psi(x,y)
-\phi(y,x\circ z)-\mu(y)\psi(x,z)=0.
$$
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 $\mu,l,r$.

A $2$-coboundary is a pair of the form $(\phi,\psi)=\delta^1N$, namely
$$
\phi(x,y)=\mu(x)N(y)-N(x\cdot y)+\mu(y)N(x),
$$
$$
\psi(x,y)=l(x)N(y)-N(x\circ y)+r(y)N(x).
$$
Two $2$-cocycles are cohomologous exactly when they differ by such a coboundary. The second cohomology group is
$$
H^2(A,V)=Z^2(A,V)/B^2(A,V),
$$
and the low-degree identity $D^2\circ D^1=0$ implies $B^2(A,V)\subseteq Z^2(A,V)$.

## 3. Abelian extensions and classification by second cohomology

An abelian extension of a Com-PreLie algebra $A$ by a vector space $V$ is a short exact sequence
$$
0\to V\to E\to A\to 0
$$
of Com-PreLie algebras such that $V$ is an abelian ideal: its internal products are trivial,
$$
u\cdot v=0,\qquad u\circ v=0,
$$
and the induced $A$-action on $V$ agrees with a prescribed representation $(V,\mu,l,r)$ [2510.23611]. If $s:A\to E$ is a section, then the module structure is recovered from the extension by
$$
\mu(x)u=s(x)\cdot u,\qquad l(x)u=s(x)\circ u,\qquad r(x)u=u\circ s(x),
$$
and this representation is independent of the choice of section.

The classification theorem proceeds in both directions. From an extension and a section $s$, one defines
$$
\phi(x,y)=s(x)\cdot s(y)-s(x\cdot y),\qquad
\psi(x,y)=s(x)\circ s(y)-s(x\circ y).
$$
The pair $(\phi,\psi)$ is a $2$-cocycle, and its cohomology class is independent of the chosen section and of the chosen representative of the extension isomorphism class. Conversely, given a $2$-cocycle $(\phi,\psi)$, one defines a Com-PreLie structure on $E=A\oplus V$ by
$$
(a,u)\cdot(b,v)=\big(a\cdot b,\mu(a)v+\mu(b)u+\phi(a,b)\big),
$$
$$
(a,u)\circ(b,v)=\big(a\circ b,l(a)v+r(b)u+\psi(a,b)\big).
$$
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 $\delta^1f$, then the resulting extensions are isomorphic via
$$
F(a,u)=(a,u+f(a)).
$$
Hence equivalence classes of abelian extensions are parametrized by second cohomology:
$$
\operatorname{Ext}(A,V)\cong H^2(A,V).
$$
The same construction admits an obstruction-theoretic reading: the condition $\delta^2(\phi,\psi)=0$ is the obstruction-vanishing condition for defining a Com-PreLie algebra structure on $A\oplus V$ with the displayed formulas, while changing the section changes $(\phi,\psi)$ by a coboundary and therefore preserves the cohomology class.

## 4. Inducibility of automorphisms and the Wells exact sequence

Given an abelian extension
$$
0\to V\to E\to A\to 0
$$
with representation $(V,\mu,l,r)$, let $\operatorname{Aut}_V(E)$ denote the group of Com-PreLie automorphisms $\gamma$ of $E$ satisfying $\gamma(V)\subseteq V$ [2510.23611]. There is a natural group homomorphism
$$
\tau:\operatorname{Aut}_V(E)\to \operatorname{Aut}(V)\times \operatorname{Aut}(A),
\qquad
\tau(\gamma)=(\gamma|_V,\bar\gamma),
$$
where $\bar\gamma(x)=j(\gamma(s(x)))$ is independent of the chosen section. A pair $(\beta,\alpha)\in \operatorname{Aut}(V)\times \operatorname{Aut}(A)$ is called inducible if it belongs to the image of $\tau$.

The necessary and sufficient criterion for inducibility has two components. First, $(\beta,\alpha)$ must be compatible with the representation:
$$
\beta(\mu(x)u)=\mu(\alpha(x))\beta(u),\qquad
\beta(l(x)u)=l(\alpha(x))\beta(u),\qquad
\beta(r(x)u)=r(\alpha(x))\beta(u).
$$
The compatible pairs form a subgroup
$$
C=\{(\beta,\alpha)\in \operatorname{Aut}(V)\times \operatorname{Aut}(A)\mid \text{the above equalities hold}\}.
$$
Second, if $(\phi,\psi)$ is the cocycle attached to the extension, there must exist a linear map $\varphi:A\to V$ such that
$$
\beta(\phi(x,y))-\phi(\alpha(x),\alpha(y))
=\mu(\alpha(x))\varphi(y)-\varphi(x\cdot y)+\mu(\alpha(y))\varphi(x),
$$
$$
\beta(\psi(x,y))-\psi(\alpha(x),\alpha(y))
=l(\alpha(x))\varphi(y)-\varphi(x\circ y)+r(\alpha(y))\varphi(x).
$$
Equivalently, using the natural action of $C$ on cocycles,
$$
\phi_{(\beta,\alpha)}(x,y)=\beta\phi(\alpha^{-1}(x),\alpha^{-1}(y)),\qquad
\psi_{(\beta,\alpha)}(x,y)=\beta\psi(\alpha^{-1}(x),\alpha^{-1}(y)),
$$
the pair $(\beta,\alpha)$ is inducible if and only if $(\beta,\alpha)\in C$ and $(\phi,\psi)^{(\beta,\alpha)}$ is cohomologous to $(\phi,\psi)$.

This criterion is encoded by the Wells map
$$
W:C\to H^2(A,V),\qquad W(\beta,\alpha)=[(\phi,\psi)^{(\beta,\alpha)}-(\phi,\psi)],
$$
which is well defined and independent of the chosen section. The vanishing criterion is
$$
(\beta,\alpha)\text{ is inducible}\iff W(\beta,\alpha)=0.
$$
One obtains the Wells exact sequence
$$
0\to Z^1(A,V)\to \operatorname{Aut}_V(E)\to C\to H^2(A,V).
$$
Here $Z^1(A,V)=\{\varphi:A\to V\mid \delta^1\varphi=0\}$ is the group of $1$-cocycles, and the map into $\operatorname{Aut}_V(E)$ identifies $1$-cocycles with automorphisms fixing both $A$ and $V$ by
$$
\gamma_\varphi(s(x)+u)=s(x)+\varphi(x)+u.
$$
Exactness expresses two structural facts: a compatible pair lifts precisely when its Wells image vanishes, and the fiber of $\tau$ over an inducible pair is a torsor under $Z^1(A,V)$.

## 5. Examples, splitting phenomena, and computational templates

A basic computation is obtained by taking $A=k$ with its usual commutative product and trivial pre-Lie product $\circ=0$, and letting $V=k$ with $\mu(\lambda)=\lambda\,\operatorname{Id}_V$ and $l=r=0$ [2510.23611]. In this case the Harrison-type cocycle condition for
$$
\phi(\lambda,\mu)=c\lambda\mu
$$
holds trivially, while the compatibility equation for
$$
\psi(\lambda,\mu)=d\lambda\mu
$$
forces $d=0$, so $\psi=0$. For a $1$-cochain $N(\lambda)=s\lambda$,
$$
\phi(\lambda,\mu)=\mu(\lambda)N(\mu)-N(\lambda\mu)+\mu(\mu)N(\lambda)=s\lambda\mu,
$$
and therefore every cocycle $\phi$ is cohomologous to $0$. Consequently,
$$
H^2(A,V)=0.
$$
The stated consequences are that all abelian extensions of $A$ by $V$ split, the Wells map is identically zero, and any compatible pair $(\beta,\alpha)$ is inducible; in this example $\alpha=\operatorname{id}_A$ and $\beta\in k^\times$ acts on $V$ by scaling.

More generally, the construction of an extension from cocycle data is uniform. Given any Com-PreLie algebra $A$ and a representation $(V,\mu,l,r)$, one sets
$$
(a,u)\cdot(b,v)=\big(a\cdot b,\mu(a)v+\mu(b)u+\phi(a,b)\big),
$$
$$
(a,u)\circ(b,v)=\big(a\circ b,l(a)v+r(b)u+\psi(a,b)\big).
$$
The three cocycle identities ensure associativity and commutativity of $\cdot$, the pre-Lie identity for $\circ$, and the compatibility
$$
a\circ(b\cdot c)=(a\circ b)\cdot c+b\cdot(a\circ c).
$$
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 $\circ=0$ and $l=r=0$ with $\mu$ 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 $\alpha\in\operatorname{Aut}(A)$ and $\beta\in\operatorname{Aut}(V)$ with $\mu,l,r$, one computes the class
$$
[(\phi,\psi)^{(\beta,\alpha)}-(\phi,\psi)]\in H^2(A,V),
$$
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 [2510.23611]. Liu–Chen, by contrast, study compatible pre-Lie algebras, written $(g,\cdot,*)$, in which both $\cdot$ and $*$ are pre-Lie products and the compatibility condition is that every linear combination
$$
x\diamond y=k_1\,x\cdot y+k_2\,x*y
$$
is again a pre-Lie product [2302.07178]. 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 $(\rho,\mu,\tilde\rho,\tilde\mu)$, one pair for each pre-Lie product, and the extension data are two $2$-cochains $\omega,\widetilde\omega$ defined from a section $s$ by
$$
\omega(x,y)=s(x)\cdot s(y)-s(x\cdot y),\qquad
\widetilde\omega(x,y)=s(x)*s(y)-s(x*y).
$$
The extension on $g\oplus V$ is then
$$
(x+m)\,\widehat{\circ}_1\,(y+n)=x\cdot y+\rho(x)n+\mu(y)m+\omega(x,y),
$$
$$
(x+m)\,\widehat{\circ}_2\,(y+n)=x*y+\tilde\rho(x)n+\tilde\mu(y)m+\widetilde\omega(x,y),
$$
and the cocycle conditions are
$$
d_1\omega=0,\qquad d_2\widetilde\omega=0,\qquad d_2\omega+d_1\widetilde\omega=0.
$$
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 $\mathcal{H}^2(g;V)$.

This second framework is embedded in a broader bidifferential graded Lie algebra formalism. The graded Lie algebra $(C^*(g;g),[\cdot,\cdot]_{MN})$ carries two differentials
$$
d_{T_1}=[T_1,\cdot]_{MN},\qquad d_{T_2}=[T_2,\cdot]_{MN},
$$
satisfying
$$
d_{T_1}\circ d_{T_2}+d_{T_2}\circ d_{T_1}=0.
$$
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 $\mathcal{H}^2(g;g)=0$, 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.

Source: https://www.emergentmind.com/topics/abelian-extensions-of-com-prelie-algebras