Cohomologies, deformations and extensions of pre-LieDer pairs

Derivations play an important role in the study of various algebraic structures. They can be used to construct higher derived brackets and homotopy Lie algebras [36], deformation formulas [10] and differential Galois theory [30]. They are also important in control theory [1, 2] and functional analysis [6, 7]. Moreover, they also play a fundamental role in the study of gauge theories in quantum field theory via the BV formalism [5]. In [16, 29], the authors studied the operad of associative algebras with derivation. Recently, in [33], the authors studied the cohomology, extensions and deformations of Lie algebras with derivations. Using similar ideas, the author studied associative algebras with derivations [13] and Leibniz algebras with derivations [12].

The notion of a pre-Lie algebra (Its other names are left-symmetric algebras, quasi-associative algebras, Vinberg algebras and so on) has been introduced independently by M. Gerstenhaber in the deformation theory of rings and algebras [19]. Pre-Lie algebra arose from the study of affine manifolds and affine structures on Lie groups [24] and homogeneous convex cones [34]. Its defining identity is weaker than associativity. This algebraic structure describes some properties of cochains space in Hochschild cohomology of an associative algebra, rooted trees and vector fields on affine spaces. Moreover, it is playing an increasing role in algebra, geometry and physics due to their applications in nonassociative algebras, combinatorics, numerical analysis and quantum field theory; see also in [3, 4, 8, 11]. There is a close relationship between pre-Lie algebras and Lie algebras: a pre-Lie algebra $(𝔤,\cdot )$ gives rise to a Lie algebra $(𝔤,{[\cdot ,\cdot ]}_C)$ via the commutator bracket, which is called the subadjacent Lie algebra and denoted by ${𝔤}^C\text{.}$ Furthermore, the map $L:𝔤\to 𝔤𝔩(𝔤)\text{,}$ defined by $L_{x}y=x\cdot y$ for all $x,y\in 𝔤\text{,}$ gives rise to a representation of the subadjacent Lie algebra ${𝔤}^C$ on $𝔤\text{.}$

The purpose of the paper is to study the cohomology of a pre-LieDer pair and its applications. The motivation for such a study is due to the study of relative difference Lie algebras and relative Rota–Baxter Lie algebras [21–23, 27]. We use the higher derived bracket to give the controlling algebra of pre-LieDer pairs. Using the Maurer–Cartan element of this controlling algebra, we construct a twist $L_{\mathrm{\infty }}\text{‑}$algebra. We give the cohomology of pre-LieDer pairs by using this twist $L_{\mathrm{\infty }}\text{‑}$algebra. Compared to Lie algebras, associative algebras and Leibniz algebras, this is a different way to constructing cohomologies with derivations. Using this cohomology, we study infinitesimal deformations of pre-LieDer pairs. We also define the cohomology of regular pre-LieDer pairs with coefficients in an arbitrary representation and study abelian extensions of regular pre-LieDer pairs.

The paper is organized as follows. In Section 2, first, we give the notion of pre-LieDer pairs. Then, by a given representation $(V;\rho ,\mu )$ of a pre-Lie algebra $(𝔤,\pi )\text{,}$ we recall the graded Lie algebra whose Maurer–Cartan elements are $\pi +\rho +\mu \text{.}$ Finally, we use this Maurer–Cartan element to give the cohomology of pre-Lie algebras with representations. In Section 3, first, we recall the notion of $L_{\mathrm{\infty }}\text{‑}$algebras and higher derived brackets. Then, we use higher derived brackets to construct an $L_{\mathrm{\infty }}\text{‑}$algebra. Given a pre-LieDer pair, we can obtain the Maurer–Cartan element of this $L_{\mathrm{\infty }}\text{‑}$algebra. Finally, using this Maurer–Cartan element, we construct a twist $L_{\mathrm{\infty }}\text{‑}$algebra, which controls deformations of pre-LieDer pairs. In Section 4, first, using the twist $L_{\mathrm{\infty }}\text{‑}$algebra constructed from the last section, we give the cohomology of pre-LieDer pairs. Then, we study infinitesimal deformations of pre-LieDer pairs by using this cohomology. We show that equivalent infinitesimal deformations are in the same second cohomology group. Finally, we use this cohomology to give the cohomology of regular pre-LieDer pairs. In Section 5, first, we introduce the notion of representations of regular pre-LieDer pairs. Then, we give cohomologies of regular pre-LieDer pairs with coefficients in an arbitrary representation. Finally, we deal with abelian extensions of regular pre-LieDer pairs. We show that the second cohomology group classifies abelian extensions of regular pre-LieDer pairs.

In this paper, we work over an algebraically closed field $𝕂$ of characteristic $0\text{,}$ and all the vector spaces are over $𝕂\text{.}$

The following are cohomologies that appeared in this paper. We made a summary.

In this section, first, we define a pre-LieDer pair, which consists of a pre-Lie algebra $(𝔤,\pi )$ and a derivation $D\text{.}$ Then, we recall the graded Lie algebra whose Maurer–Cartan element is $\pi +\rho +\mu \text{,}$ where $(V;\rho ,\mu )$ is a representation of the pre-Lie algebra $(𝔤,\pi )\text{.}$ Finally, we use this Maurer–Cartan element to give the cohomology of the pre-Lie algebra $(𝔤,\pi )$ with the representation $(V;\rho ,\mu )\text{.}$

Let $(𝔤,\cdot )$ be a pre-Lie algebra. The commutator ${[x,y]}_C=x\cdot y-y\cdot x$ gives a Lie algebra $(𝔤,{[\cdot ,\cdot ]}_C)\text{,}$ which is denoted by ${𝔤}^C$ and called the sub-adjacent Lie algebra of $(𝔤,\cdot )\text{.}$

We denote a representation of a pre-Lie algebra $(𝔤,\cdot )$ by $(V;\rho ,\mu )\text{.}$ Furthermore, let $L,R:𝔤\to 𝔤𝔩(𝔤)$ be linear maps, where $L_{x}y=x\cdot y,R_{x}y=y\cdot x\text{.}$ Then $(𝔤;L,R)$ is also a representation, which is called the regular representation.

Let $(V;\rho ,\mu )$ be a representation of a pre-Lie algebra $(𝔤,\cdot )\text{.}$ The set of $n\text{-}$cochains is given by

For all $f\in C^n(𝔤;V),x_1,\mathrm{\dots },x_{n+1}\in 𝔤\text{,}$ define the coboundary operator $\mathrm{d}:C^n(𝔤;V)\to C^{n+1}(𝔤;V)$ by

A permutation $\sigma \in {𝕊}_n$ is called an $(i,n-i)\text{-}$unshuffle if $\sigma (1)<\mathrm{\cdots }<\sigma (i)$ and $\sigma (i+1)<\mathrm{\cdots }<\sigma (n)\text{.}$ If $i=0$ and $i=n\text{,}$ we assume $\sigma =\mathrm{Id}\text{.}$ The set of all $(i,n-i)\text{-}$unshuffles will be denoted by ${𝕊}_{(i,n-i)}\text{.}$ The notion of an $(i_1,\mathrm{\dots },i_k)\text{-}$unshuffle and the set ${𝕊}_{(i_1,\mathrm{\dots },i_k)}$ are defined similarly.

Let $𝔤$ be a vector space. It was shown in [9, 31, 38] that the graded vector space $C^{\ast }(𝔤;𝔤)={⨁}_{n=1}^{+\mathrm{\infty }}{C}^n(𝔤;𝔤)={⨁}_{n=1}^{+\mathrm{\infty }}\mathrm{Hom}({\wedge }^{n-1}𝔤\otimes 𝔤,𝔤)$ equipped with the Matsushima–Nijenhuis bracket

is a graded Lie algebra, where $P\circ Q\in C^{p+q+1}(𝔤;𝔤)$ is defined by

In particular, $\pi \in \mathrm{Hom}(𝔤\otimes 𝔤,𝔤)$ defines a pre-Lie algebra if and only if ${[\pi ,\pi ]}^{MN}=0\text{.}$ If $\pi$ is a pre-Lie algebra structure, then ${\mathrm{d}}_{\pi }:={[\pi ,\cdot ]}^{MN}$ is a graded derivation of the graded Lie algebra $(C^{\ast }(𝔤;𝔤),{[\cdot ,\cdot ]}^{MN})$ satisfying ${\mathrm{d}}_{\pi }\circ {\mathrm{d}}_{\pi }=0\text{,}$ so that $(C^{\ast }(𝔤;𝔤),{[\cdot ,\cdot ]}^{MN},{\mathrm{d}}_{\pi })$ becomes a differential graded Lie algebra.

Let ${𝔤}_1$ and ${𝔤}_2$ be vector spaces, and let elements of ${𝔤}_1$ be denoted by $x,y,x_i$ and elements of ${𝔤}_2$ be denoted by $u,v,v_i\text{.}$ Let $c:{\wedge }^{n-1}{𝔤}_1\otimes {𝔤}_1\to {𝔤}_2$ be a linear map. We can construct a linear map $\widehat{c}\in C^n({𝔤}_1\oplus {𝔤}_2,{𝔤}_1\oplus {𝔤}_2)$ by

In general, for a given linear map $f:{\wedge }^{k-1}{𝔤}_1\otimes {\wedge }^l{𝔤}_2\otimes {𝔤}_1\to {𝔤}_j$ for $j\in \{1,2\}\text{,}$ we define a linear map $\widehat{f}\in C^{k+l}({𝔤}_1\oplus {𝔤}_2,{𝔤}_1\oplus {𝔤}_2)$ by

Similarly, for $f:{\wedge }^k{𝔤}_1\otimes {\wedge }^{l-1}{𝔤}_2\otimes {𝔤}_2\to {𝔤}_j$ for $j\in \{1,2\}\text{,}$ we define a linear map $\widehat{f}\in C^{k+l}({𝔤}_1\oplus {𝔤}_2,{𝔤}_1\oplus {𝔤}_2)$ by

We call the linear map $\widehat{f}$ a horizontal lift of $f$ or simply a lift. We define ${𝒢}^{k,l}={\wedge }^{k-1}{𝔤}_1\otimes {\wedge }^l{𝔤}_2\otimes {𝔤}_1+{\wedge }^k{𝔤}_1\otimes {\wedge }^{l-1}{𝔤}_2\otimes {𝔤}_2\text{.}$ The vector space ${\wedge }^{n-1}({𝔤}_1\oplus {𝔤}_2)\otimes ({𝔤}_1\oplus {𝔤}_2)$ is isomorphic to the direct sum of ${𝒢}^{k,l},k+l=n\text{.}$

We call a linear map $f$ homogeneous if $f$ has a bidegree. We denote the set of homogeneous linear maps of bidegree $k\mid l$ by $C^{k\mid l}({𝔤}_1\oplus {𝔤}_2,{𝔤}_1\oplus {𝔤}_2)\text{.}$ We have $k+l\ge 0,k,l\ge -1$ because $n\ge 1$ and $k+1,l+1\ge 0\text{.}$

In our later study, the subspaces $C^{k\mid 0}({𝔤}_1\oplus {𝔤}_2,{𝔤}_1\oplus {𝔤}_2)$ and $C^{l\mid -1}({𝔤}_1\oplus {𝔤}_2,{𝔤}_1\oplus {𝔤}_2)$ will be frequently used. By the above lift, we have the following isomorphisms:

By Lemmas 2.9 and 2.10, we obtain that $({⨁}_{k=0}^{+\mathrm{\infty }}{C}^{k\mid 0}({𝔤}_1\oplus {𝔤}_2,{𝔤}_1\oplus {𝔤}_2),{[\cdot ,\cdot ]}^{MN})$ is a graded Lie sub-algebra of the graded Lie algebra $(C^{\ast }({𝔤}_1\oplus {𝔤}_2,{𝔤}_1\oplus {𝔤}_2),{[\cdot ,\cdot ]}^{MN})$ and $({⨁}_{l=0}^{+\mathrm{\infty }}{C}^{l\mid -1}({𝔤}_1\oplus {𝔤}_2,{𝔤}_1\oplus {𝔤}_2),{[\cdot ,\cdot ]}^{MN})$ is an abelian graded Lie sub-algebra of the graded Lie algebra $(C^{\ast }({𝔤}_1\oplus {𝔤}_2,{𝔤}_1\oplus {𝔤}_2),{[\cdot ,\cdot ]}^{MN})\text{.}$

For all $\pi \in \mathrm{Hom}(𝔤\otimes 𝔤,V)\text{,}$ $\rho \in \mathrm{Hom}(𝔤\otimes V,V)$ and $\mu \in \mathrm{Hom}(V\otimes 𝔤,V)\text{,}$ we have the following.

Let $(V;\rho ,\mu )$ be a representation of a pre-Lie algebra $(𝔤,\pi )\text{.}$ By Proposition 2.11, $\pi +\rho +\mu$ is a Maurer–Cartan element of $({⨁}_{k=0}^{+\mathrm{\infty }}{C}^{k\mid 0}(𝔤\oplus V,𝔤\oplus V),{[\cdot ,\cdot ]}^{MN})\text{.}$ It follows from the graded Jacobi identity that ${\mathrm{d}}_{\pi +\rho +\mu }:={[\pi +\rho +\mu ,\cdot ]}^{MN}$ is a graded derivation of the graded Lie algebra $({⨁}_{k=0}^{+\mathrm{\infty }}{C}^{k\mid 0}(𝔤\oplus V,𝔤\oplus V),{[\cdot ,\cdot ]}^{MN})$ satisfying ${\mathrm{d}}_{\pi +\rho +\mu }^2=0\text{.}$ Thus, we have the following.

Let $(V;\rho ,\mu )$ be a representation of the pre-Lie algebra $(𝔤,\pi )\text{.}$ Define the set of $0\text{-}$cochains $C^0(𝔤,\rho ,\mu )$ to be $0\text{.}$ For $n\ge 1\text{,}$ we define the set of $n\text{-}$cochains $C^n(𝔤,\rho ,\mu )$ by

Define the coboundary operator $\partial :C^n(𝔤,\rho ,\mu )\to C^{n+1}(𝔤,\rho ,\mu )$ by

Since ${\mathrm{d}}_{\pi +\rho +\mu }^2=0\text{,}$ we obtain that ${\partial }^2=0\text{.}$ Thus, $({⨁}_{n=0}^{+\mathrm{\infty }}{C}^n(𝔤,\rho ,\mu ),\partial )$ is a cochain complex.

For all $f=(f_{𝔤},f_{\rho },f_{\mu })\in C^n(𝔤,\rho ,\mu )\text{,}$ where $f_{𝔤}\in \mathrm{Hom}({\wedge }^{n-1}𝔤\otimes 𝔤,𝔤)\text{,}$ $f_{\rho }\in \mathrm{Hom}({\wedge }^{n-1}𝔤\otimes V,V)$ and $f_{\mu }\in \mathrm{Hom}({\wedge }^{n-2}𝔤\otimes V\otimes 𝔤,V)\text{.}$ Then we have

By straightforward computation, we obtain that

In this section, first, we recall $L_{\mathrm{\infty }}\text{‑}$algebras and higher derived brackets. Then, we use higher derived brackets to construct an $L_{\mathrm{\infty }}\text{‑}$algebra. Given a pre-LieDer pair, we can obtain the Maurer–Cartan element of this $L_{\mathrm{\infty }}\text{‑}$algebra. Finally, we construct a twist $L_{\mathrm{\infty }}\text{‑}$algebra by this Maurer–Cartan element, which controls deformations of pre-LieDer pairs.