Morphisms Cohomology and Deformations of Hom-algebras

Anja Arfa; Nizar Ben Fraj; Abdenacer Makhlouf

doi:10.1080/14029251.2018.1503433

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Volume 25, Issue 4, July 2018, Pages 589 - 603

Morphisms Cohomology and Deformations of Hom-algebras

Authors

Anja Arfa

Université de Sfax, Sfax, Tunisia,arfaanja.mail@gmail.com

Nizar Ben Fraj

Université de Carthage, Nabeul, Tunisia,benfraj_nizar@yahoo.fr

Abdenacer Makhlouf^*

Université de Haute Alsace, IRIMAS, Département de Mathématiques, 6 bis ne des Frères Lumière, 68067 Mulhouse, France,abdenacer.makhlouf@uha.fr

^*Corresponding author

Corresponding Author

Abdenacer Makhlouf

Received 21 April 2018, Accepted 4 May 2018, Available Online 6 January 2021.

DOI: 10.1080/14029251.2018.1503433 How to use a DOI?
Keywords: Hom-associative algebra morphism; Hom-Lie algebra morphism; cohomology; deformation
Abstract: The purpose of this paper is to define cohomology complexes and study deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. We discuss infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide various examples.
Copyright: © 2018 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Introduction

The first instance of Hom-type algebras appeared in physics literature when looking for q-deformations, which consists of replacing the usual derivation by a σ-derivation, of some algebras of vector fields, [1, 3, 13]. The main examples dealt with Witt and Virasoro algebras and used Jackson derivation, defined for a polynomial P by Dq(P(t))=P(qt)−P(t)qt−t . It turns out that the obtained algebras no longer satisfy Jacobi identity, but a modified version involving a homomorphism. These algebras were called Hom-Lie algebras and studied first, from mathematical viewpoint, by Hartwig, Larsson and Silvestrov in [11,12]. Hom-associative algebras play the role of associative algebras in the Hom-Lie setting. They were introduced by the last author and Silvestrov in [14], where it is shown that the commutator bracket defined by the multiplication in a Hom-associative algebra leads naturally to a Hom-Lie algebra. The adjoint functor was considered by D. Yau [19].

The deformation theory was developed by Gerstenhaber for rings and algebras using formal power series in [7]. It is closely related to Hochschild cohomology. Then, it was extended to Lie algebras, using Chevalley-Eilenberg cohomology, by Nijenhuis and Richardson [17]. Deformation theory of associative algebra morphisms have been studied by Gerstenhaber and Schack in a series of papers [8–10], while deformations of Lie algebra morphisms have been considered by Nijenhuis and Richardson in [17], and more recently by Frégier [4], see also [5,6]. Cohomology and deformations of Hom-associative algebras and Hom-Lie algebras were studied in [2,16]

The purpose of this paper is to provide first a Hom-type Hochschild cohomology of Hom-associative algebra morphisms and a Hom-type Chevalley-Eilenberg cohomology of Hom-Lie algebra morphisms, generalizing the cohomology theory associated to deformations of Lie algebra morphisms given by Frégier for Lie algebras in [4] and that introduced by Gerstenhaber and Schack in [9] for associative algebra morphisms. To this end, we need to generalize the algebra valued cohomology theory in [2] to any bimodule. Moreover, we discuss a deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. Furthermore, various examples are presented.

1. Preliminaries

We assume that 𝕂 is an algebraically closed field of characteristic 0, even if most of the results are valid for any field. A Hom-algebra is a triple consisting of a 𝕂 -vector space, or a module, together with a bilinear map (multiplication) and a linear map. These Hom-algebras aim to generalize classical algebraic structures, their main feature is that the identities defining the structures are twisted by homomorphisms. In the sequel, we will write ⊗ for ⊗𝕂,𝒜⊗n≔𝒜⊗⋯⊗𝒜 and 𝒜×n≔𝒜×⋯×𝒜 .

Definition 1.1.

A Hom-associative algebra is a triple (𝒜,μ,α) consisting of a 𝕂 -vector space 𝒜 , a bilinear map μ:𝒜×𝒜→𝒜 and a linear map α:𝒜→𝒜 satisfying

μ(α(x), μ(y,z)) = μ(μ(x,y), α(z)), for all x,y,z∈𝒜 (Hom-associativity).

A Hom-Lie algebra is a triple (𝒧,[⋅,⋅],α) consisting of a 𝕂 -vector space 𝒧, a skew-symmetric bilinear map [⋅,⋅]:𝒧×𝒧→𝒧 and a linear map α:𝒧→𝒧 satisfying

↺x,y,z[α(x),[y,z]]=0 for all x,y,z∈𝒧 (Hom-Jacobi identity),

where ↺x,y,z denotes summation over the cyclic permutation on x, y, z.

A Hom-associative algebra or a Hom-Lie algebra is called multiplicative if α is an algebra morphism.

Definition 1.2.

Let (𝒜,μ,α) and (𝒜′,μ′,α′) (resp. (𝒧,[⋅,⋅],α) and (𝒧′,[⋅,⋅]′,α)) be two Hom-associative (resp. Hom-Lie) algebras. A linear map ϕ:𝒜→𝒜′ (resp. ϕ:𝒧→𝒧′ ) is called a Hom-associative (resp. Hom-Lie) algebra morphism if

μ′∘(ϕ⊗ϕ)=ϕ∘μ (resp. [⋅,⋅]′∘(ϕ⊗ϕ)=ϕ∘[⋅,⋅]) and ϕ∘α=α′∘ϕ.

The following theorem provides an easy way to deform a usual associative algebra morphism (resp. Lie algebra morphism) to a Hom-associative algebra morphism (resp. Hom-Lie algebra morphism).

Theorem 1.3.

Let A = (A, μ) and B = (B, v) be two associative algebras (resp. Lie algebras) and ϕ: A → B be an algebra morphism. Consider two algebra morphisms α: A → A and β: B → B such that ϕ∘α=β∘ϕ . Then ( Aα=(A,μα=α∘μ,α) and Bβ=(B,vβ=β∘v,β) are Hom-associative algebras (resp. Hom-Lie algebras) and ϕ: A_α → B_β is a Hom-associative algebra morphism.

The first assertion was proved in [20] and we have ϕ∘α∘μ=β∘ϕ∘μ=β∘v∘(ϕ⊗ϕ) .

Now, we discuss concepts of module and representation for Hom-associative algebras and Hom-Lie algebras.

Definition 1.4.

Let (𝒜,μ,α) be a Hom-associative algebra. A (left) 𝒜 -module is a triple (M, f, γ), where M is a 𝕂 -vector space, f: M → M and γ:𝒜⊗M→M are 𝕂 -linear maps such that γ∘(μ⊗f)=γ∘(α⊗γ) .

Proposition 1.5 (Left adjoint 𝒜 -module).

Let (𝒜,μ,α) and (𝒜′,μ′,α′) be two Hom-associative algebras and ϕ:𝒜→𝒜′ be a Hom-associative algebra morphism. The triple (M, f, γ), where M=𝒜′,γ=ρl=μ′(ϕ⊗id) and f = α′, is an 𝒜 -module called adjoint representation of 𝒜 induced by the Hom-associative algebra morphism ϕ.

Proof. Straightforward.

Remark 1.6.

We similarly define the right adjoint module by considering the triple (𝒜′,ρr=μ′(id⊗ϕ),α′) . A bimodule structure is given by left and right actions with maps ρ_r and ρ_l that satisfy the following additional condition ρ_r (ρ_l(x,y), α(z)) = ρ_l (α(x,y), ρ_r(y,z)). Left and right modules are special cases of bimodules, one may set ρ_r = 0(resp. ρ_l = 0)

Let (𝒧,[⋅,⋅],α) be a Hom-Lie algebra and β∈𝔤l(V) be an arbitrary linear self map on V, where V is an arbitrary vector space. We denote a left action of 𝒧 on V by the bracket [⋅,⋅]V:𝒧×V→V such that (x,v) → [x,v]_V.

Definition 1.7.

A triple (V, [ ,·]_V, β) is called a left 𝒧 -module with respect to β ∈ 𝔤l(V), if for any x,y∈𝒧 and v ∈ V, we have [α(x), β(v)]_V = β([x,v]_V) and [[x,y], β(v)]_V = [α(x), [y, v]_V] _V − [α(y), [x,v]_V]_V.

We say that (V, [ ,·]_V, β) is a representation of 𝒧 .

Proposition 1.8.

Let (𝒧,[⋅,⋅],α) and (𝒧′,[⋅,⋅]′,α′) be two Hom-Lie algebras and ϕ:𝒧→𝒧′ be a Hom-Lie algebra morphism. Let (Π, β, ρ_l) be a triple, where Π=𝒧′,β=α′,ρl=[⋅,⋅]V=[ϕ,id]′ . Then, Π is a left adjoint 𝒧 -module via ϕ.

2. Cohomology of Hom-associative algebra morphisms

2.1. Cohomology of Hom-associative algebras with values in an adjoint 𝒜 -bimodule

We construct a cochain complex Cα,α′*(𝒜,M) that defines a Hom-type Hochschild cohomology for multiplicative Hom-associative algebras in an adjoint 𝒜 -bimodule M. Let (𝒜,μ,α) and (𝒜′,μ′,α′) be two Hom-associative algebras over 𝕂 and ϕ:𝒜→𝒜′ be a Hom-associative algebra morphism. Let M=(𝒜′,ρl,ρr) be an 𝒜 -bimodule, where ρ_l and ρ_r are defined in Proposition 1.5 (resp. Remark 1.6). Regard 𝒜′ as an 𝒜 -bimodule via the adjoint representation of 𝒜 induced by ϕ.

The set of n-cochains on 𝒜 with values in an 𝒜 -bimodule M, is defined to be the set of n-linear maps which are compatible with α and α′. We denote by Cn(𝒜,M) the set of n-linear maps from 𝒜 to M and let Cα,α′0(𝒜,M)≔M and for n > 0 we set

Cα,α′n(𝒜,M)≔{f∈Cn(𝒜,M):α′∘f=f∘α⊗n}.

Definition 2.1.

For n ≥ 1, n-coboundary operator associated to the triple (𝒜,M,ϕ) is the linear map δHom n:Cα,α′n(𝒜,M)→Cα,α′n+1(𝒜,M) defined by

δHomnφ(x0,x1,…,xn)=μ′(ϕ(αn−1(x0))),φ(x1,x2,…,xn))+∑k=1n(−1)kφ(α(x0),α(x1),…,α(xk−2),μ(xk−1,xk),α(xk+1),…,α(xn))+(−1)n+1μ′(φ(x0,…,xn−1),ϕ(αn−1(xn))). (2.1)

Theorem 2.2.

The pair (Cα,α′*(𝒜,M),δHom *) defines a cohomology complex of Hom-associative algebras with values in the 𝒜 -bimodule M.

Remark 2.3.

In the general case, we let 𝒜,𝒜′ be two Hom-associative algebras and M=(𝒜′,ρl,ρr) be a 𝒜 -bimodule, where ρ_l and ρ_r are left 𝒜 -module and right 𝒜 -module respectively. For φ∈Cα,α′n(𝒜,M) , we set

δHomnφ(x0,x1,…,xn)=ρl(αn−1(x0),φ(x1,x2,…,xn))+∑k=1n(−1)kφ(α(x0),α(x1),…,α(xk−2),μ(xk−1,xk),α(xk+1),…,α(xn))+(−1)n+1ρr(φ(x0,…,xn−1),αn−1(xn)).

In the particular case where M=𝒜 and ρ_l, ρ_r = μ, the Hom-associative algebra is an 𝒜 -bimodule over itself. We recover the coboundary operator defined in [2]. One considers the previous definition with ρ_l = ρ_r = μ and denote Cα,αn(𝒜,M) by CHom n(𝒜,M) .

The space of n-cocycles is ZHom n(𝒜,M)≔{φ∈Cα,α′n(𝒜,M):δHom nφ=0}, and the space of n-coboundaries is BHomn(𝒜,M)≔{ψ=δHomn−1φ:φ∈Cα,α′n−1(𝒜,M)} . The n^th Hochschild cohomology group of the Hom-associative algebra 𝒜 with values in an adjoint 𝒜 -bimodule is the quotient HHomn(𝒜,M)≔ZHomn(𝒜,M)BHomn(𝒜,M) .

2.2. Cohomology Complex of Hom-associative algebra morphisms

The original cohomology theory associated to deformation of associative algebra morphisms was introduced by M. Gerstenhaber in [9]. In this section, we will discuss this theory for Hom-associative algebra morphisms. Let 𝒜,𝒝 be two Hom-associative algebras and ϕ:𝒜→𝒝 be a Hom-associative algebra morphism. Regard 𝒝 as a representation of 𝒜 via ϕ wherever appropriate. Define the module of n-cochains of ϕ by

CHomn(ϕ,ϕ)=CHomn(𝒜,𝒜)×CHomn(𝒝,𝒝)×Cα,α′n−1(𝒜,𝒝).

The coboundary operator δn:CHom n(ϕ,ϕ)→CHom n+1(ϕ,ϕ) is defined by

δn(φ1,φ2,φ3)=(δHomnφ1,δHomnφ2,ϕ∘φ1−φ2∘ϕ⊗n−δHomn−1φ3),

where δHom nφ1 and δHom nφ2 are defined in Remark 2.3 and δHom nφ3 is defined by (2.1).

Theorem 2.4.

We have δn+1∘δn=0 . Hence (CHom *(ϕ,ϕ),δ*) is a cochain complex.

Proof. The most-right component of (δn+1∘δn)(φ1,φ2,φ3) is ϕ∘(δHomnφ1)−(δHom nφ2)∘ϕ−δHomn(ϕ∘φ1−φ2∘ϕ⊗n−δHomn−1φ3)=ϕ∘(δHomnφ1)−(δHomnφ2)∘ϕ⊗n+1−δHomn(ϕ∘φ1)−δHomn(φ2∘ϕ⊗n) . To finish the proof, one checks that ϕ∘(δHom nφ1)=δHom n(ϕ∘φ1) and (δHom nφ2)∘ϕ⊗n+1=δHom n(φ2∘ϕ⊗n) . Indeed, ϕ∘φ1 is defined as follows: (ϕ∘φ1)(x0,…,xn)=ϕ∘(φ1(x0,…,xn)) and φ2∘ϕ⊗n as φ2∘ϕ(x0,…,xn)=φ2∘(ϕ(x1),…,ϕ(xn)) .

Remark 2.5.

If HHom n(𝒜,𝒜),HHom n(𝒝,𝒝) and Hα,α′n−1(𝒜,𝒝) are all trivial, then so is HHom n(ϕ,ϕ) . The proof is similar to that of Proposition 3.3 in [18].

We define the n^th Hochschild cohomology group of a Hom-associative algebra morphism ϕ: 𝒜→𝒝 to be

HHomn(ϕ,ϕ)≔HHomn(𝒜,𝒜)×HHomn(𝒝,𝒝)×Hα,α′n−1(𝒜,𝒝).

The corresponding cohomology modules of the cochain complex (CHom*(ϕ,ϕ),δ*) are denoted by HHom n(ϕ,ϕ)≔HHom n(CHom *(ϕ,ϕ),δ) .

Example 2.6.

We consider a 3-dimensional Hom-associative algebra 𝒜 defined in [15], with respect to a basis {e₁, e₂, e₃}, by the multiplication μ_A and the linear map α_A such that

μA(e1,e1)=ae1, μA(e1,e2)=μA(e2,e1)=ae2, μA(e1,e3)=μA(e3,e1)=be3,μA(e2,e2)=ae2, μA(e2,e3)=be3, μA(e3,e2) =μA(e3,e3)=0, αA(e1)=ae1,αA(e2)=ae2,αA(e3)=be3,

where a, b are parameters.

We consider also a 2-dimensional Hom-associative algebra 𝒝 defined, with respect to a basis {f1,f2}, by the multiplication μ𝒝 and the linear map α_B such that

μ𝒝(f1,f1)=f1, μ𝒝(fi,fj)=f2 for (i,j)≠(1,1), αB(f1)=βf1−βf2, αB(f2)=0,

where β is a parameter.

Let ϕ:𝒜→𝒝 be a Hom-associative algebra morphism. It is defined, when, α = β = 1, as ϕ(e1)=f1−f2,ϕ(e2)=f1−f2,ϕ(e3)=0 .

In the following, we compute the second cohomology spaces HHom 2(𝒜,𝒜) and HHom 2(𝒝,𝒝) . The 2-cocycles ψ:𝒜⊗𝒜→𝒜 are of the form

ψ(e1,e1)=p1e1+(p2b−p1)e2,ψ(e2,e1)=p2be2,ψ(e3,e1)=bp1e3,ψ(e1,e2)=p2be2,ψ(e2,e2)=p3e1+p4e2,ψ(e3,e2)=−bp3e3ψ(e1,e3)=p2e3,ψ(e2,e3)=b(p3+p4)e3,ψ(e3,e3)=0.

where p₁, p₂, p₃, p₄ are parameters.

It turns out that they are all coboundaries. Moreover, we get

HHom 2(𝒜,𝒜)={0},HHom 2(𝒝,𝒝)={ψ∣ψ(f1,f1)=cf1+df2,ψ(fi,fj)=pf2,p≠(c+d),(i,j)≠(1,1)}HHom 1(𝒜,𝒝)={ϕ1∣ϕ1(e1)=(p1−c)f1+(c−p1)f2, ϕ1(e2)=(p2−c)f1+(c−p2)f2;ϕ1(e3)=0}.

3. Cohomology Complex of Hom-Lie algebra morphisms

In this section, we deal with a cohomology of Hom-Lie algebra morphisms. We construct first a cochain complex CHL*(𝒧,Π) that defines a Chevalley-Eilenberg cohomology for multiplicative Hom-Lie algebras with values in a left 𝒧 -module Π, then a cohomology complex of Hom-Lie algebra morphisms.

3.1. Cohomology complex of multiplicative Hom-Lie algebras with values in a left 𝒧 -module

Let 𝒧 and 𝒧′ be two Hom-Lie algebras and ϕ:𝒧→𝒧′ be a Hom-Lie morphism. Regard 𝒧′ as a representation Π of 𝒧 via ϕ defined by (1.8). The set CHLn(𝒧,Π) of n-cochains on 𝒧 with values in Π is the set of skewsymmetric 𝕂 -linear maps from 𝒧×n to Π. We set C˜α,α′n(𝒧,Π)≔{f∈CHLn(𝒧,Π):α′∘f=f∘α⊗n} . For n = 0 we have C˜α,α′0(𝒧,Π)=Π .

Definition 3.1.

Let (𝒧,[⋅,⋅],α) and (𝒧′,[⋅,⋅]′,α′) be two Hom-Lie algebras. Let ϕ:𝒧→𝒧′ be a Hom-Lie algebra morphism. Regard 𝒧′ as a representation of 𝒧 via ϕ wherever appropriate. For n ≥ 1, the n-coboundary operator associated to the triple (𝒧,Π,ϕ) is the linear map δHIn:CHLn(𝒧,Π)→CHLn+1(𝒧,Π) defined by

δHLnφ(x0,…,xn)=∑i=0n(−1)i[ϕ(αn−1(xi)),φ(x0,…,x^i,…,xn)]′ +∑0≤i<j≤n(−1)i+jφ([xi,xj],α(x0),…,x^i,…,x^j,…,α(xn)). (3.1)

Theorem 3.2.

The pair (C˜α,α′*(𝒧,Π),δHL) defines a cochain complex. The corresponding cohomology denoted by HHL*(𝒧,Π) , is called the cohomology of the Hom-Lie algebra 𝒧 with coefficients in the representation Π.

Proof. The proof of δHLn+1∘δHLn=0 is straightforward and lengthy. One may view it in the arxiv version of this paper, see arXiv: 1710.07599.

Remark 3.3.

In the case where Π=𝒧 and [,] = [,]′, we recover the coboundary operator defined in [2]. The space of n-cochains is denoted by CHLn(𝒧,𝒧) .

The space of n-cocycles is ZHLn(𝒧,Π)≔{φ∈ C˜α,α′n(𝒧,Π):δHLnφ=0} , and the space of n-coboundaries is BHLn(𝒧,Π)≔{ψ=δHLn−1φ:φ∈ C˜α,α′n−1(𝒧,Π)} . The n^th cohomology group of the Hom-Lie algebra 𝒧 with coefficients in Π is the quotient HHLn(𝒧,Π)≔ZHn(𝒧,Π)BHI (𝒧,Π) .

3.2. Cohomology complex of Hom-Lie algebra morphisms

In this section, we generalize the cohomology theory of Lie algebra morphisms developed by Frégier in [4] to Hom-Lie algebras. We adopt the same notations. Consider the product ⋄ defined, for λ∈CHom n(𝒧′,𝒧′) and ϕ∈Hom(𝒧,𝒧′) , as λ⋄ϕ∈CHLn(𝒧,𝒧′) such that λ⋄ϕ(x1,…,xn)≔λ(ϕ(x1),…,ϕ(xn)), for any x1,…,xn∈𝒧 .

Let ϕ:𝒧→𝒧′ be a Hom-Lie algebra morphism. Regard 𝒧′ as a representation of 𝒧 via ϕ wherever appropriate. Define the module of n-cochains of ϕ by

CHLn(ϕ,ϕ)≔CHLn(𝒧,𝒧)×CHLn(𝒧′,𝒧′)×C˜α,α′n−1(𝒧,𝒧′).

The coboundary operator δ(ϕ,ϕ)n:CHLn(ϕ,ϕ)→CHLn+1(ϕ,ϕ) is defined by

δ(ϕ,ϕ)n(φ1,φ2,φ3)=(δHLnφ1,δHLnφ2,δHLn−1φ3+(−1)n−1(ϕ∘φ1−φ2⋄ϕ)),

where δHLnφ1 and δHLnφ2 are defined in Remark 3.3 and δHLnφ3 by formula (3.1).

Theorem 3.4.

We have δ(ϕ,ϕ)n+1∘δ(ϕ,ϕ)n=0 . Hence (CHL*(ϕ,ϕ),δ*) is a cohomology complex.

Proof. The proof is similar to the proof of Theorem 2.4.

Remark 3.5.

The corresponding cohomology modules of the cochain complex (CHL*(ϕ,ϕ),δ*) are denoted by HHLn(ϕ,ϕ)≔HHLn(CHL*(ϕ,ϕ),δ*) . If HHLn(𝒧,𝒧),HHLn(𝒧′,𝒧′) and HHLn−1(𝒧,𝒧′) are all trivial then so is HHLn(ϕ,ϕ) . The proof is similar to that of Proposition 3.3 in [18].

4. Deformations of Hom-associative algebra morphisms

In this section, we study 1-parameter formal deformations of Hom-associative algebra morphisms and Hom-Lie algebra morphisms using the approach introduced by Gerstenhaber [7]. Recall that the main idea is to change the scalar field 𝕂 to a formal power series ring 𝕂⟦t⟧ , in one indeterminate t, and the main results provide us with cohomological interpretations. Let A ⟦t⟧ be the set of formal power series whose coefficients are elements of A.

4.1. Deformation of Hom-associative algebra morphisms

First, we recall the definition of a formal deformation of a Hom-associative algebra, then discuss deformations of Hom-associative algebra morphisms.

Definition 4.1.

A 1-parameter formal deformation of a Hom-associative algebra (𝒜,μ0,α) is a Hom-associative 𝕂⟦t⟧ -algebra (𝒜⟦t⟧,μt,α) , where μt=∑i≥0μiti , which is a 𝕂⟦t⟧ -bilinear map satisfying the condition μt∘(μt⊗α)=μt∘(α⊗μt) .

The deformation is said to be of order N if μt=∑i≥0Nμiti and infinitesimal if N = 1.

Let (𝒜,μ𝒜,α) and (𝒝,μ𝒝,β) be two Hom-associative algebras and ϕ:𝒜→𝒝 be a Hom-associative algebra morphism. A deformation of ϕ is given by a triple Θt=(μ𝒜,t,μ𝒝,t,ϕt) where

• μ𝒜,t=∑n≥0μ𝒜,ntn is a deformation of 𝒜 ,

• μ𝒝,t=∑n≥0μ𝒝,ntn is a deformation of 𝒝 ,

• ϕt:𝒜[t]→𝒝[t] is a Hom-associative algebra morphism of the form ϕt=∑n≥0ϕntn , where each ϕn:𝒜→𝒝 is a 𝕂 -linear map and ϕ₀ = ϕ.

Proposition 4.2.

The linear coefficient Θ1=(μ𝒜,1,μ𝒝,1,ϕ1) , called the infinitesimal part of the deformation Θ_t of ϕ, is a 2-cocycle in CHom 2(ϕ,ϕ) .

Proof. Straightforward.

4.2. Equivalent deformations and Rigidity

Let μ𝒜,t and μ˜𝒜,t be two formal deformations of a Hom-associative algebra 𝒜 . A formal automorphism ψt:𝒜⟦t⟧→𝒜⟦t⟧ is a power series ψt=∑n≥0ψntn in which each ψ_n ∈ End (𝒜) and ψ0=Id𝒜 such that ψt(μ𝒜,t(x,y))=μ˜𝒜,t(ψt(x),ψt(y)) and ψt∘α=α˜∘ψt for all x,y∈𝒜 . Two deformations μ𝒜,t and μ˜𝒜,t are said to be equivalent if and only if there exists a formal automorphism which transforms μ𝒜,t to μ˜𝒜,t .

Definition 4.3.

Let Θt=(μ𝒜,t,μ𝒝,t,ϕt) and → Θ˜t=(μ˜𝒜,t,μ˜𝒝,t,ϕ˜t) be two deformations of a Hom-associative algebra morphism ϕ:𝒜→𝒝. A formal automorphism ϕt:Θt→ Θ˜t is a pair (ψ𝒜,t,ψ𝒝,t) , where ψ𝒜,t:𝒜⟦t⟧→𝒜⟦t⟧ and ψ𝒝,t:𝒝⟦t⟧→𝒝⟦t⟧ are formal automorphisms such that ϕ˜t=ψ𝒝,t∘ϕt∘ψ𝒜,t−1 . Two deformations Θt and Θ˜t are said to be equivalent if and only if there exists a formal automorphism such that Θt→ Θ˜t .

Given a deformation Θt and a pair of power series

ψt=(ψ𝒜,t=∑nψ𝒜,ntn,ψ𝒝,t=∑nψ𝒝,ntn),

one can define a deformation Θ˜t which is automatically equivalent to Θt .

In [2], it was shown that if μ𝒜,t and μ˜𝒜,t are, thanks to the automorphism ϕ𝒜,t:𝒜⟦t⟧→𝒜⟦t⟧, equivalent deformations of 𝒜 , then the infinitesimals of μ𝒜,t and μ˜𝒜,t belong to the same cohomology class.

Theorem 4.4.

The infinitesimal part of a deformation Θ_t of ϕ is a 2-cocycle in CHom 2(ϕ,ϕ) whose cohomology class is determined by the equivalence class of the first term of Θ_t.

Proof. In view of Proposition 4.2, it remains to show that if ψt:Θt→Θ˜t is a formal automorphism, then 2-cocycles Θ₁ and Θ˜1 differ by a 2-coboundary. Write ψt=(ψ𝒜,t,ψ𝒝,t) and Θ˜t=(μ˜𝒜,t,μ˜𝒝,t,ϕ˜t) . According to [2], we have δ1ψ*,1=μ*,1−μ˜*,1∈CHom2(*,*) for * denoting either 𝒜 or 𝒝 . To finish the proof, we develop both sides of ϕ˜t=ψ𝒝,tϕtψ𝒜,t−1 and collecting the coefficients of tⁿ yield for n = 1 the equality ϕ1−ϕ˜1=ϕψ𝒜,1−ψ𝒝,1ϕ . It follows that a 1-cochain α=(ψ𝒜,1,ψ𝒝,1,0)∈CHom1(ϕ,ϕ) satisfies δϕ,ϕ1α=Θ1−Θ˜1 .

Definition 4.5.

Let (𝒜,μ,α) be a Hom-associative algebra and Θ1=(μ𝒜,1,μ𝒝,1,ϕ1) be an element of ZHom 2(ϕ,ϕ) . The 2-cocycle Θ₁ is said to be integrable if there exists a family (μ𝒜,t=∑nμ𝒜,n,μ𝒝,t=∑nμ𝒝,n,ϕt=∑nϕn) defining a formal deformation Θ_t of ϕ.

The integrability of Θ₁ depends only on its cohomology class.

Theorem 4.6.

Let (𝒜,μ𝒜,α) and (𝒝,μ𝒝,β) be two Hom-associative algebras and Θt=(μ𝒜,t,μ𝒝,t,ϕt) be a deformation of a Hom-associative algebra morphism ϕ:𝒜→𝒝 . Then, there exists an equivalent deformation Θ˜t=(μ˜𝒜,t,μ˜𝒝,t,ϕ˜) such that Θ˜t∈ZHom2(ϕ,ϕ) and Θ˜1∉BHom2(ϕ,ϕ) . Hence, if HHom2(ϕ,ϕ)=0, then every formal deformation is equivalent to a trivial deformation.

Proof. Define a pair of power series ψt=(ψ𝒜,t,ψ𝒝,t) . According to Definition 4.3, we define an equivalent deformation Θ˜t=(μ˜𝒜,t,μ˜𝒝,t,ϕ˜t)=∑nΘ˜ntn .

We have μ*,1∈ZHom 2(*,*) and also μ*,1−μ˜*,1∈ZHom 2(*,*) for *∈{𝒜,𝒝}. Moreover, since ϕ1∈ZHom1(𝒜,𝒝), we have ϕ1−ϕ˜∈ZHom1(𝒜,𝒝) . If Θ˜1∈BHom2(ϕ,ϕ), then Θ1−Θ˜1=δϕ,ϕ1φ for some φ∈CHom1(ϕ,ϕ) .

Remark 4.7.

Let Θt=∑i≥0Θiti be a deformation of a Hom-associative algebra morphism, in which Θ_i = 0 for i =1,…, n and Θ_n₊₁ is a coboundary in CHom2(ϕ,ϕ), then there exists a deformation Θ˜t equivalent to Θ_t and a formal automorphism ψt:Θt→Θ˜t such that Θ˜i=0 for i = 1,…, n + 1.

A morphism of Hom-associative algebras for which every formal deformation is equivalent to a trivial deformation (μ𝒜,0,μ𝒝,0,ψ) is said to be analytically rigid. The vanishing of the second cohomology group, HHom2(ϕ,ϕ)=0, gives a sufficient criterion for rigidity.

4.2.1. Obstructions

A deformation of order N of ϕ is a triple Θt=(μ𝒜,t,μ𝒝,t,ϕt) satisfying ϕt(μ𝒜,t,(a,b))=μ𝒝,t(ϕt(a),ϕt(b)) or equivalently

∑i=0nϕi(μ𝒜,n−i(a,b))=∑i+j+k=nμ𝒝,i(ϕj(a),ϕk(b)) for n≤N.

Given a deformation Θ_t of order N, it can be extended to order N + 1 if and only if there exists a 2-cochain ΘN+1=(μ𝒜,N+1,μ𝒝,N+1,ϕN+1)∈CHom 2(ϕ,ϕ) such that Θ¯t=Θt+tN+1ΘN+1 is a deformation of order N + 1.

The primary obstruction of a deformation μ𝒜,t=∑i=0N+1μiti is

−∑p+q=N+1,p>0,q>0μp∘αμq.

An analogous obstruction to an infinitesimal deformation of a morphism ϕ:𝒜→𝒝 is obtained by calculating the second order term in the deformation equation of ϕ. Then

μ𝒝,2(ϕ(a),ϕ(b))+μ𝒝,1(ϕ1(a),ϕ(b))+μ𝒝,1(ϕ(a),ϕ1(b))+μ𝒝,0(ϕ1(a),ϕ1(b))+μ𝒝,0(ϕ2(a),ϕ(b))+μ𝒝,0(ϕ(b),ϕ2(b))=ϕ2μ𝒜,0(a,b)+ϕ1μ1,𝒜(a,b)+ϕμ𝒜,2(a,b).

Therefore, we get

(μ𝒜,1∘¯μ𝒜,1,μ𝒝,1∘¯μ𝒝,1,μ1,𝒝∘¯ϕ1−ϕ1∘μ𝒜,1+ϕ1⌣ϕ1)=δ2(μ𝒜,2,μ𝒝,2,ϕ2),

where for a p-cochain φ₁ (resp. q-cochain φ₂), we have

φ1∘¯φ2(x1,…,xp+q−1)=∑i=1q(−1)i(p−1)φ1(ϕ(x1),…,ϕ(xi−1),φ2(xi,…,xi+q−1),ϕ(xi+q),…,ϕ(xp+q−1)).

Following [2], we express the obstructions for the algebras using the Gerstenhaber bracket. For φ∈CHomp(𝒜,𝒜) and ψ∈CHomq(𝒜,𝒜), where p,q ≥ 0, we define jφαψ∈CHomp+q+1(𝒜,𝒜) to be the composition product given by the operator

jφα(ψ)(x0,…,xp+q)=∑k=0q(−1)pkψ(αp(x0),…,αp(xk−1),φ(xk,…,xk+p),αp(xp+k+1),…,αp(xp+q)).

and set [φ,ψ]αΔ≔jψα(φ)−(−1)abjφα(ψ) and let

𝒪b𝒜=∑p+q=N+1p>0,q>012[μ𝒜,p,μ𝒜,q]αΔ,𝒪b𝒝=∑p+q=N+1p>0,q>012[μ𝒝,p,μ𝒝,q]αΔ

for the obstruction of a deformation of any Hom-associative algebra 𝒜 . Set

𝒪bϕ≔∑p+q=N+1p>0,q>0μ𝒝,p∘¯ϕq−∑p+q=N+1p>0,q>0ϕp∘μ𝒜,q+∑p+q=N+1p>0,q>0ϕp⌣ϕq+∑p+q=N+1p>0,q>0,k>0μ𝒝,q∘(ϕq,ϕk)

for the obstruction of the extension of the Hom-associative algebra morphism ϕ.

Theorem 4.8.

Let (𝒜,μ𝒜,0,α𝒜) and (𝒝,μ𝒝,0,α𝒝) be two Hom-associative algebras. Let ϕ:𝒜→𝒝 be a Hom-associative algebra morphism and Θt=(μ𝒜,t,μ𝒝,t,ϕt) be an order k 1-parameter formal deformation of ϕ. Then 𝒪b=(𝒪b𝒜,𝒪b𝒝,𝒪bϕ)∈ZHom 3(ϕ,ϕ) and Θ_t extends to a deformation of order k + 1 if and only if 𝒪bϕ is a coboundary.

Proof. The proof is straightforward but lengthy. One may view it in the arXiv version arXiv: 1710.07599.

Corollary 4.9.

If HHom 3(ϕ,ϕ)=0 , then every infinitesimal deformation can be extended to a formal deformation of larger order.

4.3. Deformations of Hom-Lie algebra morphisms

In this section, we discuss deformations of Hom-Lie algebra morphisms. We obtain similar results as in previous section. For proofs, see arXiv: 1710.07599.

Definition 4.10.

Let (𝒧,[⋅,⋅],α) be a Hom-Lie algebra. A 1-parameter formal Hom-Lie deformation of 𝒧 is given by a 𝕂⟦t⟧ -bilinear map [⋅,⋅]t:𝒧⟦t⟧×𝒧 ⟦t⟧→𝒧⟦t⟧ of the form [⋅,⋅]t=∑i≥0ti[⋅,⋅]i , where each [⋅,⋅]i is a skewsymmetric bilinear map [⋅,⋅]i:𝒧×𝒧→𝒧 (extended to 𝕂⟦t⟧ -bilinear map), such that [⋅,⋅]=[⋅,⋅]0 and satisfying

↺x,y,z[α(x),[y,z]t]t=0 (Hom−Jacobi identity).

Let ϕ:𝒧→𝒧′ be a Hom-Lie algebra morphism. A deformation of ϕ is a triple Θt=([⋅,⋅]t;[⋅,⋅]t′;ϕt) in which:

•
[⋅,⋅]t=∑i≥0ti[⋅,⋅]i is a deformation of 𝒧 ,
•
[⋅,⋅]′t=∑i≥0ti[⋅,⋅]i′ is a deformation of 𝒧′ ,
•
ϕt:𝒧⟦t⟧→𝒧′⟦t⟧ is a Hom-Lie algebra morphism of the form ϕt=∑n≥0ϕntn, where each ϕn:𝒧→𝒧′ is a 𝕂 -linear map and ϕ₀ = ϕ.

Proposition 4.11.

The linear coefficient, Θ1=([⋅,⋅]1,[⋅,⋅]1′,ϕ1), which is called the infinitesimal part of the deformation Θ_t, is a 2-cocycle in CHL2(ϕ,ϕ) .

Let (𝒧,[⋅,⋅],α) be a multiplicative Hom-Lie algebra. Let 𝒧t=(𝒧,[⋅,⋅]t,α) and 𝒧′t=(𝒧,[⋅,⋅)′t,α) be two deformations of 𝒧, where [⋅,⋅]t=∑i≥0ti[⋅,⋅]i and [⋅,⋅]′t=∑i≥0ti[⋅,⋅]′i with [⋅,⋅]0=[⋅,⋅]0′=[⋅,⋅] . We say that 𝒧t and 𝒧′t are equivalent if there exists a formal automorphism ψt:𝒧[t]→𝒧 ⟦t⟧ , that may be written in the form ψt=∑i≥0ψiti , where ψi∈End(𝒧) and ψ₀ = id, such that ψt([x,y]t)=[ψt(x),ψt(y)]t′ .

A deformation 𝒧t is said to be trivial if and only if 𝒧t is equivalent to 𝒧 .

Definition 4.12.

Let (𝒧,[⋅,⋅]𝒧,α),(𝒢,[⋅,⋅]𝒢,β) be two Hom-Lie algebras and ϕ:𝒧→𝒢 be Hom-Lie algebra morphism. Let Θt=([⋅,⋅]𝒧,t,[⋅,⋅]𝒢,t,ϕt) and Θ˜t=([⋅,⋅]′𝒧,t,[⋅,⋅]′𝒢,t,ϕ˜t) be two deformations of a Hom-Lie algebra morphism ϕ.

A formal automorphism ψt:Θt→ Θ˜t is a pair (ψ𝒧,t,ψ𝒢,t) , where ψ𝒧,t:𝒧⟦t⟧→𝒧⟦t⟧ and ψ𝒢,t:𝒢⟦t⟧→𝒢⟦t⟧ are formal automorphisms, such that ϕ˜t=ψ𝒧,t∘ϕt∘ψ𝒢,t−1 . Two deformations Θ_t and Θ˜t are equivalent if and only if there exists a formal automorphism that transforms Θ_t in to Θ˜t .

Remark 4.13.

Given a deformation Θ_t and a pair of power series ψt=(ψ𝒧,t=∑nψ𝒧,ntn,ψ𝒢,t=∑nψ𝒢,ntn) , one can define a deformation Θ˜t . The deformation Θ˜t is automatically equivalent to Θ_t.

Theorem 4.14.

The infinitesimal of a deformation Θ_t of ϕ is a 2-cocycle in CHL2(ϕ,ϕ) whose cohomology class is determined by the equivalence class of the first term of Θ_t.

Theorem 4.15.

Let (𝒧,[⋅,⋅∣𝒧,α) and (𝒢,[⋅,⋅∣𝒢,β) be two Hom-Lie algebras. Let Θt=([⋅,⋅]𝒧,t,[⋅,⋅]𝒢,t,ϕt) be a deformation of a Hom-Lie algebra morphism ϕ Then there exists an equivalent deformation Θ˜t=([⋅,⋅]′𝒧,t,[⋅,⋅]𝒢,t,ϕ˜) such that Θ˜1∈ZHL2(ϕ,ϕ) and Θ˜1∉BHL2(ϕ,ϕ) . Hence, if HHL2(ϕ,ϕ)=0, then every formal deformation is equivalent to a trivial deformation.

4.3.1. Obstructions

A deformation of order N of ϕ is a triple, Θt=([⋅,⋅]𝒧,t,[⋅,⋅]𝒢,t,ϕt) satisfying ϕt([x,y]𝒧,t)=[ϕi(x),ϕt(y)]𝒢,t or equivalently

∑i=0Nϕi([x,y]𝒧,N−i)=∑i+j+k=N[ϕi(x),ϕj(y)]𝒢,k.

Given a deformation Θ_t of order N, it extends to a deformation of order N + 1 if and only if there exists a 2-cochain ΘN+1=([⋅,⋅]𝒧,N+1,[⋅,⋅]𝒢,N+1,ϕN+1)∈CHL2(ϕ,ϕ) such that Θ¯t=Θt+tN+1ΘN+1 is a deformation of order N + 1. Then Θ¯t is said to be an extension of Θ_t of order N + 1.

Let

𝒪b𝒧=12∑p+q=N+1p>0,q>0[[⋅,⋅]𝒧,p,[⋅,⋅]𝒧,q]a^

be the obstruction of a deformation of the Hom-Lie algebra 𝒧, where [,]α∧ is the Gerstenhaber bracket defined in [2]. Let

𝒪bϕ=∑i=0N+1ϕi([x,y]𝒧,N+1−i)−∑′[ϕi(x),ϕj(y)]𝒢,k

be the obstruction of the extension of the Hom-Lie algebra morphism ϕ, where

∑′=∑i+j=N+1 i,j>0 k=0+∑i+k=N+1 i,k>0 j=0+∑j+k=N+1 j,k>0 i=0+∑i+k+j=N+1 i,k,j>0.

Theorem 4.16.

Let (𝒧,[⋅,⋅]𝒧,0,α) and (𝒢,[⋅,⋅]𝒢,0,β) be two Hom-Lie algebras and ϕ:𝒧→𝒢 be a Hom-Lie algebra morphism. Let Θt=([⋅,⋅]𝒧,t,[⋅,⋅]𝒢,t,ϕt) be an order k 1-parameter formal deformation of ϕ. Then 𝒪b=(𝒪b𝒜,𝒪b𝒝,𝒪bϕ)∈ZHL3(ϕ,ϕ) . Therefore, the deformation extends to a deformation of order k + 1 if and only if 𝒪b is a coboundary.

5. Example

We compute in this section a cohomology of a given Hom-Lie algebra morphism and discuss some deformations. Let 𝔤1=(𝔤1,[⋅,⋅]1,α1) and 𝔤2=(𝔤2,[⋅,⋅]2,α2) be two Hom-Lie algebras defined with respect to the basis {e₁, e₂, e₃} (resp. {f₁, f₂, f₃} ) by the following relations

[e1,e2]1=e3,[e2,e3]1=0,[e1,e3]1=0,α1(e1)=p1e1,α1(e2)=p2e2,α1(e3)=p1p2e3;[f1,f2]2=f1+f3,[f2,f3]2=f2,[f1,f3]2=f1+2f3,α2(f1)=f1,α2(f2)=2f2,α2(f3)=2f3,

where p₁, p₂ are parameters.

Let ϕ1,2:𝔤1→𝔤2 be a Hom-Lie algebra morphism. We have the following cases

(1)
If p₁ = p₂ = 2: ϕ1,21(e1)=λ2,1f2+λ3,1f3,ϕ1,21(e2)=λ2,2f2+λ2,2λ3,1λ2,1f3,ϕ1,21(e3)=0 .
(2)
If p₁ = 2 and p₂ = 0: ϕ1,22(e1)=λ2,1f2+λ3,1f3,ϕ1,22(e2)=0,ϕ1,22(e3)=0 .

A 2-cochain is given by a triple (ψ, φ, ρ) where ψ:𝔤1×𝔤1→𝔤1,φ:𝔤2×𝔤2→𝔤2,ρ:𝔤1→𝔤2 . The 2-cochains of the Hom-Lie algebra 𝔤1 are defined by ψ₁ for i = 1,…, 8

{ψ1(e1,e2)=a1e2ψ1(e2,e3)=0ψ1(e1,e3)=0 {ψ2(e1,e2)=a2e3ψ2(e2,e3)=0ψ2(e1,e3)=0 {ψ3(e1,e2)=0ψ3(e2,e3)=a3e3ψ3(e1,e3)=0 {ψ4(e1,e2)=0ψ4(e2,e3)=0ψ4(e1,e3)=a4e2{ψ5(e1,e2)=0ψ5(e2,e3)=0ψ5(e1,e3)=a5e3 {ψ6(e1,e2)=a6e1ψ6(e2,e3)=0ψ6(e1,e3)=0 {ψ7(e1,e2)=0ψ7(e2,e3)=a7e1ψ7(e1,e3)=0 {ψ8(e1,e2)=0ψ8(e2,e3)=−p2p1a8e2ψ8(e1,e3)=a8e1

where a₁,…, a₈ are parameters.

We obtain the following results

(1)
If p₁ = 0, then ZHL2(𝔤1,𝔤1)=〈ψ2,ψ3,ψ5,ψ6,ψ7〉. Hence, dimZHL2(𝔤1,𝔤1)=5 .
(2)
If p₁ = 1, and p₂ ∉ {−1, 0, 1}, then ZHL2(𝔤1,𝔤1)=〈ψ1,ψ2,ψ4,ψ5〉 .
1. (a)
  If p₂ = 1, then ZHL2( 𝔤1,𝔤1) is generated in addition by {ψ3,ψ6,ψ7,ψ8} .
2. (b)
  If p₂ = 0, then ZHL 2(𝔤1,𝔤1) is generated in addition by {ψ3} .
3. (c)
  If p₂ = − 1, then ZHL2(𝔤1,𝔤1) is generated in addition by {ψ7} .
(3)
If p₁ = − 1, and p₂ ∉ {−1, 0, 1}, then ZHL 2(𝔤1,𝔤1)=〈ψ2,ψ4〉 .
1. (a)
  If p₂ = 1, then ZHL2(𝔤1,𝔤1) is generated in addition by {ψ3,ψ6,ψ7} .
2. (b)
  If p₂ = 0, then ZHI 2(𝔤1,𝔤1) is generated in addition by {ψ1,ψ3,ψ5} .
3. (c)
  If p₂ = − 1, then ZHL2(𝔤1,𝔤1) is generated in addition by {ψ7,ψ8} .
(4)
If p₁ ∉ {−1, 0, 1}, and p2=1p1, then ZHL2(𝔤1,𝔤1)=〈ψ2,ψ8〉 .
1. (a)
  If p₂ = 1, then ZHL2(g1,𝔤1) is generated by {ψ2,ψ3,ψ6,ψ7} .
2. (b)
  If p₂ = 0, then ZHL2(𝔤1,𝔤1) is generated by {ψ1,ψ2,ψ3,ψ4,ψ5} .
3. (c)
  If p₂ = − 1, then ZHL2(𝔤1,𝔤1) is generated by {ψ2,ψ7}.
(5)
If p₁ ∉ {−1, 0, 1}, and p₂ ∉ {−1, 0, 1}, such that p2≠1p1 , then ZHL2(𝔤1,𝔤1)=〈ψ2〉 .

All cocycles are not coboundaries except ψ₂.

Now, we consider the Lie algebra 𝔤2 . The 2-cocycles are defined to be

φ(f1,f2)=k1f2+k2f3,φ(f2,f3)=0,φ(f1,f3)=k3f2−k1f3,

where k₁, k₂, k₃ are parameters. Therefore, dimHHL 2(𝔤2,𝔤2)=1 and it is generated by

φ(f1,f2)=f3,φ(f2,f3)=0,φ(f1,f3)=0.

Now, we consider the third component. For the first morphism ϕ1,21, there is only one 1-cocycle corresponding to ψ = ψ₂ and φ. It is given by

ρ(e1)=(−a3,2λ2,12λ2,2λ3,1+a2,2λ2,1λ2,2+a3,1λ2,1λ3,1)f2+a3,1f3,ρ(e2)=a2,2f2+a3,2f3,ρ(e3)=0.

Therefore HHL1(𝔤1,𝔤2) is 3-dimensional.

For the second morphism ϕ1,22, we have

ρ(e1)=a2,1f2+a3,1f3, ρ(e2)=0, ρ(e3)=0.

Therefore HHL1(𝔤1,𝔤2) is 2-dimensional.

Now, we provide examples of deformations. For 𝔤1 , we consider the deformed bracket

[e1,e2]′1=e3;[e2,e3]′1=0;[e1,e3]′1=wte2,

where w is a parameter, or

[e1,e2]′1=e3+te2;[e2,e3]′1=0;[e1,e3]′1=te3.

For 𝔤2 , we consider the deformed bracket defined by

[f1,f2]′2=af1+(b+tk2)f3; [f2,f3]′2=cf2; [f2,f3]′2=df1+2af3.

Let ϕ˜ be a deformation of the second morphism given by

ϕ˜(e1)=(λ2,1+a2,1t)f2+(λ3,1+a3,1t)f3;ϕ˜(e2)=0;ϕ˜(e3)=0.

Then ([,]′1,[,]′2, ϕ˜) is an infinitesimal deformation of ϕ1,22 .

Let ϕ˜ be an infinitesimal deformation of the first morphism, where

ϕ˜(e1)=(λ21+(−a3,2λ2,12λ2,2λ3,1+a22λ2,1λ2,2+a3,1λ2,1λ3,1)t)f2+(λ3,1+ta3,1)f3;ϕ˜(e2)=(λ2,2+ta2,2)f2+(λ2,2λ3,1λ2,1+ta3,2)f3;ϕ˜(e3)=0.

Then ([,]′1,[,]′2, ϕ˜) is an infinitesimal deformation of ϕ1,21 .

References

[1]N Aizawa and H Sato, q-Deformation of the Virasoro algebra with central extension, Phys. Lett. B, Vol. 256, No. 1, 1991, pp. 185-190.

[2]F Ammar, Z Ejbehi, and A Makhlouf, Cohomology and Deformations of Hom-algebras, Journal of Lie Theory, Vol. 21, No. 4, 2011, pp. 813-836.

[3]M Chaichian, P Kulish, and J Lukierski, q-Deformed Jacobi identity, q-oscillators and q-deformed infinite-dimensional algebras, Phys. Lett. B, Vol. 237, No. 3–4, 1990, pp. 401-406.

[4]Y Frégier, A new cohomology theory associated to deformation of Lie algebra morphisms, Lett. Math. Phys., Vol. 70, 2004, pp. 97-107.

[5]Y Frégier, M Markl, and D Yau, The L∞-deformation complex of diagrams of algebras, N.Y. J. Math, Vol. 15, 2009, pp. 353-392.

[6]Y Frégier and M Zambon, Simultaneous deformations of algebras and morphisms via derived brackets, J. Pure Appl. Algebra, Vol. 219, No. 12, 2015, pp. 5344-5362.

[7]M Gerstenhaber, On the deformation of rings and algebras, Ann. Math., Vol. 79, 1964, pp. 59-103.

[8]M Gerstenhaber and SD Schack, On the deformation of algebra morphisms and diagrams, Trans. Amer. Math. Soc., Vol. 279, 1983, pp. 1-50.

[9]M Gerstenhaber and S.D Schack, On the cohomology of an algebra morphism, J. Algebra, Vol. 95, 1985, pp. 245-262.

[10]M Gerstenhaber and S.D Schack, Sometimes H1 is H2 and discrete groups deform, Geometry of group representation, Boulder, CO, 1987, pp. 149-168. Contemp. Math. 74, Amer. Math. Soc., Providence, RI, 1988.

[11]J.T Hartwig, D Larsson, and SD Silvestrov, Deformations of Lie algebras using σ -derivations, J. Algebra, Vol. 295, No. 2, 2006, pp. 314-361.

[12]D Larsson and SD Silvestrov, Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra, Vol. 288, No. 2, 2005, pp. 321-344.

[13]Ke Qin Liu, Characterizations of the quantum Witt algebra, Lett. Math. Phys., Vol. 24, No. 4, 1992, pp. 257-265.

[14]A Makhlouf and S.D Silvestrov, Hom-algebras structures, J. Gen. Lie Theory Appl., Vol. 2, No. 2, 2008, pp. 51-64.

[15]A Makhlouf and SD Silvestrov, Hom-algebras and Hom-coalgebras, J. Algebra Appl., Vol. 09, 2010, pp. 553.

[16]A Makhlouf and SD Silvestrov, Notes on formal deformations of Hom-associative and Hom-Lie algebras, Forum Mathematicum, Vol. 22, No. 4, 2010, pp. 715-739.

[17]A Nijenhuis and RW Richardson, Deformation of homomorphisms of Lie group and Lie algebras, Bull. Amer. Math. Soc., Vol. 73, 1967, pp. 175-179.

[18]D Yau, Deformation theory of dialgebra morphisms, Algebra Colloquium, Vol. 15, 2008, pp. 279-292.

[19]D Yau, Enveloping algebras of Hom-Lie algebras, J. Gen. Lie Theory Appl., Vol. 2, No. 2, 2008, pp. 95-108.

[20]D Yau, Hom-algebras and homology, Journal of Lie Theory, Vol. 19, 2009, pp. 409-421.

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 25 - 4
Pages: 589 - 603
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2018.1503433 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

ris enw bib

TY  - JOUR
AU  - Anja Arfa
AU  - Nizar Ben Fraj
AU  - Abdenacer Makhlouf
PY  - 2021
DA  - 2021/01/06
TI  - Morphisms Cohomology and Deformations of Hom-algebras
JO  - Journal of Nonlinear Mathematical Physics
SP  - 589
EP  - 603
VL  - 25
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1503433
DO  - 10.1080/14029251.2018.1503433
ID  - Arfa2021
ER  -

download .riscopy to clipboard