aa r X i v : . [ m a t h . R A ] J u l On hom-algebras with surjective twisting
Aron Gohr
University of LuxembourgMathematics Research Unit162A, avenue de la faiencerieL-1511, LuxembourgGrand-Duchy of Luxembourg.
October 29, 2018
Abstract
A hom-associative structure is a set A together with a binary oper-ation ⋆ and a selfmap α such that an α -twisted version of associativityis fulfilled. In this paper, we assume that α is surjective. We showthat in this case, under surprisingly weak additional conditions on themultiplication, the binary operation is a twisted version of an asso-ciative operation. As an application, an earlier result [1] on weaklyunital hom-algebras is recovered with a different proof. In the secondsection, consequences for the deformation theory of hom-algebras withsurjective twisting map are discussed. Introduction
The study of hom-algebras originates with [3], who introduced a notion ofhom-Lie algebra in the context of deformation theory of Witt and Vira-soro algebras. Later, this notion was generalized and transferred to othercategories in [5]. Deformation theory of hom-associative algebras was firstexplored in [4], where a Gerstenhaber-type notion of formal deformation forhom-associative algebras is introduced and the beginnings of a cohomologytheory appropriate for studying such deformations are developed. In [7], us-ing a different notion of deformation, new examples of hom-associative and1om-Lie algebras were constructed from associative respectively Lie alge-bras. In [1], it was shown that in the case of unital hom-associative algebrasrelatively innocent-looking conditions on the twisting map can force a hom-associative algebra to be associative. For instance, unital hom-associativealgebras with surjective twisting are associative. Also a weakened notionof unitality for hom-associative algebras was investigated and it was foundthat under the assumption of weak unitality and bijective twisting, a hom-associative algebra, while not being necessarily associative, can always beconstructed from an associative algebra by a generalization of one of theconstruction procedures in [7].This paper is divided in two sections after this introduction. The first sectionextends and improves upon some of the findings of [1]. We prove a very gen-eral result (Proposition 1) about hom-associative structures with surjectivetwisting, which says essentially that either the multiplication on such a struc-ture is in some way degenerate or it can be constructed from an associativestructure as in [7]. The notion of nondegeneracy of an algebraic structureused here will be made precise in the first section. The previous result onweakly unital hom-structures with bijective twisting from [1] is obtained as aspecial case. A second theorem is subsequently proven which shows that theassumption of surjective instead of bijective twisting map in Proposition 1 isno real advantage over what was obtained in [1], because in our situation thetwisting map can be shown to be bijective anyway. However, the replacementof weak unitality with nondegeneracy as well as the different method of proofemployed provide real progress over [1].The second section is devoted to a treatment of hom-deformation theory inthe sense of [4], in the special case where the hom-associative algebra tobe deformed had a surjective twisting map and a nondegenerate multiplica-tion. The key observation here is that both nondegeneracy and surjectiv-ity of the twisting map are preserved under hom-associative deformation ofhom-associative algebras. We use this observation to relate hom-associativedeformations of an algebra B which arises from a surjective twisting of anassociative algebra A to associative deformations of A .We use similar conventions and notations as in [1]. Specifically, k will alwaysdenote a commutative ring with unit, K will be a field.2 Hom-structures with surjective twisting
A hom-associative structure is a set A together with a multiplication ⋆ : A × A → A and a self-map α : A → A such that the condition α ( x ) ⋆ ( y ⋆ z ) = ( x ⋆ y ) ⋆ α ( z )is fulfilled. Depending on the category under consideration, α is expectedto satisfy other conditions as well. In general, the philosophy is that α should be a homomorphism for all functions and relations on our algebraicstructure, except possibly for the multiplication ⋆ . For example, in the caseof hom-rings α is supposed an abelian group endomorphism, in the case ofhom- k -algebras it is linear over the commutative base ring k . However, inthe present section we need no such additional structures on A .If we consider A and α fixed, it is clear that the extent to which such anassociativity condition restricts the possible choices of ⋆ is highly dependenton the choice of α . In the case of hom-algebras for instance, it is possibleto choose α = 0 and obtain a hom-associative structure with every bilinear ⋆ : A × A → A . On the other hand, if α = id , we obtain the usual notionof associative algebras. In this section, we will study some aspects of whathappens when α is assumed to be surjective . We recall in the following the definition of a twist from [1]. We also introducethe notion of an untwist : Definition 1.
Let ( A, ⋆, α ) be a hom-associative structure. Then A is calleda twist if there is an associative multiplication · : A × A → A such that ( A, ⋆, α ) arises from ( A, · ) by setting x ⋆ y := α ( x · y ) . The structure ( A, · ) iscalled an untwist of ( A, ⋆, α ) . Note that any such multiplication must due to the hom-associativity of A satisfy the hom-associativity-like condition α ( α ( x ) · α ( y · z )) = α ( α ( x · y ) · α ( z )) . If α is bijective and if there is an untwist of of A , then this untwist is obvi-ously uniquely defined. We will therefore in this case talk of the untwist of( A, ⋆, α ). 3n [1] it was shown that weakly unital hom-associative structures with bi-jective twisting are always twists, and indeed twists of unital structures.Equivalently, one could say that any unital structure ( A, · , α,
1) such that α is bijective and satisfies the hom-associativity-like relation α ( α ( x ) · α ( y · z )) = α ( α ( x · y ) · α ( z )) . is associative.In the sequel, our goal is to show that the condition of weak unitality is notessential here. Our first goal is to prove a technical lemma about hom-associative structureswith surjective twisting. The motivation behind the introduction of thislemma is as follows: suppose that (
A, ⋆, α ) is a hom-associative structure with α surjective. Then ideally, we would like to be able to write the multiplication ⋆ as a twisting of an associative multiplication by α , i.e. in the form x ⋆ y := α ( x · y ), where · : A × A → A is associative. Since by surjectivity of α there exist β : A → A with α ◦ β = id A , a natural ansatz is to simply set x · y := β ( x ⋆ y )with such a β . The associativity condition ( x · y ) · z = x · ( y · z ) is then thesame as x ⋆ β ( y ⋆ z ) = β ( x ⋆ y ) ⋆ z , since β is necessarily injective.In general, β cannot be chosen such that this associativity condition is ful-filled. However, the following weaker statement can be shown: Lemma 1.
Let ( A, ⋆ ) be a hom-associative structure with α surjective andlet β : A → A be a map with α ◦ β = id A . Then, the following associativityconditions are satisfied for all a, b, x, y, z ∈ A : a ⋆ ( b ⋆ ( x ⋆ β ( y ⋆ z ))) = a ⋆ ( b ⋆ ( β ( x ⋆ y ) ⋆ z )) , (1) a ⋆ (( x ⋆ β ( y ⋆ z )) ⋆ b ) = a ⋆ (( β ( x ⋆ y ) ⋆ z ) ⋆ b ) , (2)(( x ⋆ β ( y ⋆ z )) ⋆ b ) ⋆ a = (( β ( x ⋆ y ) ⋆ z ) ⋆ b ) ⋆ a, (3)( b ⋆ ( x ⋆ β ( y ⋆ z ))) ⋆ a = ( b ⋆ ( β ( x ⋆ y ) ⋆ z )) ⋆ a. (4) Proof.
Equations 3 and 4 can be obtained from (Eq. 1, 2) by passing to theopposite hom-structure, i.e. to the hom-associative structure with multipli-cation a ⋆ op b := b ⋆ a . 4e first remark that we have( β ( x ) ⋆ y ) ⋆ z = x ⋆ ( y ⋆ β ( z )) (5)as was already shown in [1]. We also have for any x, y, z, u ∈ A the identity α ( x ) ⋆ (( y ⋆ z ) ⋆ u ) = α ( x ⋆ y ) ⋆ ( α ( z ) ⋆ u ) (6)because of α ( x ) ⋆ (( y ⋆ z ) ⋆ u ) = ( α ( x ) ⋆ ( y ⋆ z )) ⋆ α ( u )= (( x ⋆ y ) ⋆ α ( z )) ⋆ α ( u )= α ( x ⋆ y ) ⋆ ( α ( z ) ⋆ u ) . With this said, we are ready for a proof of (Eq. 1): a ⋆ ( b ⋆ ( x ⋆ β ( y ⋆ z ))) Eq. 5 = a ⋆ (( β ( b ) ⋆ x ) ⋆ ( y ⋆ z )) α ◦ β = id A = α ( β ( a )) ⋆ (( β ( b ) ⋆ x ) ⋆ ( y ⋆ z )) Eq. 6 = α ( β ( a ) ⋆ β ( b )) ⋆ (( x ⋆ y ) ⋆ α ( z )) hom-ass. = (( β ( a ) ⋆ β ( b )) ⋆ ( x ⋆ y )) ⋆ α ( z ) Eq. 5 = ( β ( a ) ⋆ ( β ( b ) ⋆ β ( x ⋆ y ))) ⋆ α ( z ) α ◦ β = id A = a ⋆ (( β ( b ) ⋆ β ( x ⋆ y )) ⋆ α ( z )) α ◦ β = id A = a ⋆ ( b ⋆ ( β ( x ⋆ y ) ⋆ z )) . The proof of (Eq. 2) follows the same method, starting from a ⋆ (( β ( x ⋆ y ) ⋆ z ) ⋆ b ) = a ⋆ (( x ⋆ y ) ⋆ ( z ⋆ β ( b ))) . The same trick as above of replacing a by α ( β ( a )) and using (Eq. 6) isapplied to obtain a sub-term of the form α ( y ) ⋆ ( z ⋆ β ( b )), which simplifies to( y ⋆ z ) ⋆ b . The same simplifying steps as above then yield (Eq. 2). Our first main result will be that a hom-associative structure (
A, ⋆, α ) withnondegenerate multiplication and surjective α is always a twist. Obviously,to properly understand this statement, a definition of the concept of nonde-generacy used is needed: 5 efinition 2. Let ( A, ⋆ ) be a set together with a binary operation ⋆ : A × A → A . Then ( A, ⋆ ) is called left-degenerate if there exist a = b ∈ A such that x ⋆ a = x ⋆ b for all x ∈ A . Right degeneracy is defined accordingly. A iscalled two-sided degenerate if it is both right and left degenerate. It is called strongly degenerate if there exist a = b ∈ A such that both x ⋆ a = x ⋆ b and a ⋆ x = b ⋆ x for all x ∈ A . If for example (
A, ⋆, +) is a nonassociative ring without unit, left degen-eracy in the sense defined above means that there exists some 0 = c ∈ A such that x ⋆ c = 0 for all x ∈ A .We are now ready to state and prove: Proposition 1.
Let ( A, ⋆, α ) be a hom-associative structure with α surjec-tive. Then either:1. A is a twist.2. A is strongly degenerate.Proof. Take some β : A → A such that α ◦ β = id A and define x · y := β ( x ⋆ y )for x, y ∈ A . Assume that a ⋆ β ( b ⋆ c ) = β ( a ⋆ b ) ⋆ c for some a, b, c ∈ A ,i.e. ( A, · ) not associative and suppose that ( A, ⋆ ) is not strongly degenerate.Then setting r := a ⋆ β ( b ⋆ c ) , s := β ( a ⋆ b ) ⋆ c we can find some b ∈ A suchthat b ⋆ r = b ⋆ s or r ⋆ b = s ⋆ b . Assume without loss of generality theformer. Then by repeating the same argument, we find a ∈ A such thateither a ⋆ ( b ⋆ r ) = a ⋆ ( b ⋆ s ) or ( b ⋆ r ) ⋆ a = ( b ⋆ s ) ⋆ a . But both inequalitiesare in contradiction to Lemma 1, so the proposition follows.Using similar ideas, it is possible to show also the following observationon properties of the twisting map: Proposition 2.
Suppose ( A, ⋆, α ) hom-associative, not strongly degenerate,and α surjective. Then α is in fact bijective.Proof. Define β as in the proof of the previous proposition and suppose thatthere is ξ ∈ A with β ( α ( ξ )) = ξ . As before, we can assume without loss ofgenerality that there exists b ∈ A with b ⋆ β ( α ( ξ )) = b ⋆ ξ and can then find an a ∈ A with either a ⋆ ( b ⋆ β ( α ( ξ ))) = a ⋆ ( b ⋆ ξ ) or ( b ⋆ β ( α ( ξ ))) ⋆ a = ( b ⋆ ξ ) ⋆ a .The first of these possibilities leads to a contradiction due to the generalidentity x ⋆ ( y ⋆ β ( α ( z ))) = ( β ( x ) ⋆ y ) ⋆ α ( z ) = x ⋆ ( y ⋆ z ) . (7)6he second case requires application of the same line of reasoning again. Wecan find c ∈ A such that either c ⋆ (( b ⋆ β ( α ( ξ ))) ⋆ a ) = c ⋆ (( b ⋆ ξ ) ⋆ a ) (8)or (( b ⋆ β ( α ( ξ ))) ⋆ a ) ⋆ c = (( b ⋆ ξ ) ⋆ a ) ⋆ c. (9)We will show that both of these possibilities are in contradiction to generalidentities on A . As far as (8) is concerned, we find that c ⋆ (( b ⋆ ξ ) ⋆ a ) = ( β ( c ) ⋆ ( b ⋆ ξ )) ⋆ α ( a ) Eq. 7 = ( β ( c ) ⋆ ( b ⋆ β ( α ( ξ )))) ⋆ α ( a )= c ⋆ (( b ⋆ β ( α ( ξ ))) ⋆ a ) . To dispose of (9), we calculate(( b ⋆ ξ ) ⋆ a ) ⋆ c = ( α ( b ) ⋆ ( ξ ⋆ β ( a ))) ⋆ c Eq. 7 = ( β ( α ( b )) ⋆ ( ξ ⋆ β ( a ))) ⋆ c = α ( b ) ⋆ (( ξ ⋆ β ( a )) ⋆ β ( c ))= α ( b ) ⋆ (( ξ ⋆ β ( a )) ⋆ α ( β ( c )))= α ( b ) ⋆ ( α ( ξ ) ⋆ ( β ( a ) ⋆ β ( c )))= α ( b ) ⋆ (( β ( α ( ξ )) ⋆ β ( a )) ⋆ β ( c ))= ( α ( b ) ⋆ ( β ( α ( ξ )) ⋆ β ( a ))) ⋆ c = (( b ⋆ β ( α ( ξ ))) ⋆ a ) ⋆ c. So both (8) and (9) lead to a contradiction. This concludes the proof.It is clear that in general the assumption of surjectivity can not be weak-ened. For instance, N with addition as binary operation and α ( x ) := x + 1is clearly hom-associative, but does not arise by the construction describedfrom anything else. The reason is that 0 = 0 + 0 is outside the image of α , soaddition can not be written as an α -twisted version of any other operation.Also non-associative examples of this situation can be constructed.If ( A, ⋆, α, c ) is left weakly unital, and if α is bijective, we know that ( A, ⋆ )is nondegenerate since c acts bijectively by left multiplication. Hence, in thiscase, Proposition 1 recovers the result from [1] that A is a twisting of anassociative structure. 7 Hom-associative deformation theory
We will now explore some applications of our results in the previous sectionto the deformation theory of Hom-associative algebras. The basic idea wewill follow is that both bijectivity of a twisting map and nondegeneracy ofa multiplication are properties which are preserved under formal deforma-tion. This enables us to partially “pull back” the deformation problem forhom-associative algebras to the deformation problem for associative algebras,which is much better understood.The notion of formal deformations of associative algebras goes back to [2]. Itwas extended to hom-algebras in [4]. It is well-known that infinitesimal de-formations and obstruction theory of associative deformations are controlledby second and third Hochschild cohomology respectively. Equivalence classesof hom-associative deformations of hom-associative algebras have similarlybeen identified with elements of a second cohomology module [4], but sofar no cohomology theory for hom-associative algebras has been constructedthat would allow a cohomological description of obstruction theory.Throughout this section, k is a commutative ring and ( A, ⋆, α ) is a hom-associative k -algebra, unless explicitly stated otherwise with nondegeneratemultiplication and surjective α . By a “nondegenerate” multiplication, wewill in the sequel always mean a not strongly degenerate one.This section is divided into two subsections. The first one briefly recallsthe notion of hom-associative formal deformation as given in [4]. In thesecond subsection, we show that hom-associative deformation preserves non-degeneracy and surjectivity of the twisting map. We use this fact to deducethat deformations of ( A, ⋆, α ) have an associative untwist. We prove thatthis untwist is, in turn, an associative formal deformation of the untwist of A . Let (
A, ⋆, α ) be an arbitrary hom-associative algebra. Then [4] give thefollowing definition of a formal deformation of A : Definition 3.
Let A [[ t ]] be the module of formal power series over A in onevariable. Consider a k [[ t ]] -bilinear extension µ t of a k -bilinear map of type A ⊗ A → A [[ t ]] of the form µ t = X i ≥ t i µ i ith µ ( a, b ) = a ⋆ b for all a, b ∈ A and µ i : A ⊗ A → A a bilinear mapfor every i ∈ N . Suppose further that we have given a k [[ t ]] -linear map α t arising by k [[ t ]] -linear extension of a k -linear map of the form P i ≥ X i α i ,with α = α . Then ( A [[ t ]] , µ t , α t ) is called a formal deformation of ( A, ⋆, α ) if ( A [[ t ]] , µ t , α t ) is hom-associative. In [4], also a notion of formal equivalence for deformations of hom-associativealgebras is defined:
Definition 4.
Suppose that ( A [[ t ]] , µ t , α t ) and ( A [[ t ]] , µ ′ t , α ′ t ) are hom-associativedeformations of the hom-associative algebra ( A, µ, α ) . Then both deforma-tions are called equivalent if there exists a formal isomorphism between them,i.e. a k [[ t ]] -linear map ϕ t , compatible with both the deformed multiplicationsand the deformed twisting maps, of the form ϕ t = X i ≥ t i ϕ i where the ϕ i are linear maps A → A and ϕ = id A . Compatibility with thedeformed multiplications means that ϕ t ◦ µ t = µ ′ t ◦ ( ϕ t ⊗ ϕ t ) , compatibility tothe twisting maps means ϕ t ◦ α t = α ′ t ◦ ϕ t . Now if (
A, ⋆, α ) is a nondegenerate hom-associative algebra with surjectivetwisting, we know by the results of the first section that α is in fact a bijection.Consider then a hom-associative deformation ( A, µ t , α t ) of ( A, ⋆, α ). Since in α t = X i ≥ t i α i we have α = α by definition, the usual arguments on invertibility of formalpower series yield bijectivity of α t immediately.Nondegeneracy of the multiplication is also preserved under hom-associativedeformation. To see this, let a := P i ≥ t i a i be a nonzero element of A [[ t ]].Choose n ∈ N such that a i = 0 for all i < n and a n = 0. Denote by ⋆ t aformal deformation of the original product. Since A was nondegenerate, wecan find a b ∈ A such that a n ⋆ b = 0 or b ⋆ a n = 0. Assume without loss ofgenerality b ⋆ a n = 0. Then since b ⋆ t a = t n b ⋆ a n + [terms of order ≥ n + 1]we have also b ⋆ t a = 0.We have therefore proven: 9ormal hom-associative deformations of not strongly degeneratehom-associative algebras with surjective twisting map are twists.Consider now a formal deformation ( A [[ t ]] , µ t , α t ) of ( A, µ, α ) under theseconditions. Then the untwist of the deformed algebra has α − t ◦ µ t as multi-plication. This can be expressed as a k [[ t ]]-linear extension of a formal powerseries with order zero term α − ◦ µ , which means that the untwist of thedeformed algebra is an associative formal deformation of ( A, α − ◦ µ ), theuntwist of A . Set the following: Definition 5.
Let ( A, µ ) be an associative algebra, let ( A [[ t ]] , ⋆ t ) be an asso-ciative formal deformation of A and let α t : A [[ t ]] → A [[ t ]] be a k [[ t ]] -linearmap of the form α t = X i ≥ t i α i , with the α i being module endomorphisms of A and such that α t ( α t ( x ) ⋆ t α t ( y ⋆ t z )) = α t ( α t ( x ⋆ t y ) ⋆ t α t ( z )) . (10) Then α t is called a formal twisting of A [[ t ]] with respect to µ t . Three remarks about Eq. 10 are in order. First, Eq. 10 is obviouslydesigned in such a way as to give rise to a hom-associative twisting of theformal deformation (
A, ⋆ t ). Second, using the k [[ t ]]-linearity of all mapsappearing in a standard way, one can check that it is sufficient to verify Eq. 10for x, y, z ∈ A . Third, if ( A, µ, α ) is hom-associative and nondegenerate withsurjective twisting, then our previous observation that any hom-associativedeformations of A can be obtained as twists of associative deformations of( A, α − ◦ µ ) is immediately refined to Proposition 3.
Any formal hom-associative deformation ( A [[ t ]] , µ t , α t ) of ( A, µ, α ) is obtained from a formal twisting with degree zero component α ofan associative deformation of ( A, α − ◦ µ ) . These findings suggest treating the deformation problem for a nonde-generate hom-associative algebra (
A, ⋆, α ) with surjective α in the followingway:1. Compute ( A, α − ◦ µ ). 10. Use associative deformation theory to classify deformations of this al-gebra.3. Finally, find all formal twistings with degree zero component α of thesedeformations.There are two problems standing in the way of this program:1. One needs to verify that any formal twisting with degree zero compo-nent α of an associative formal deformation of ( A, α − ◦ µ ) gives riseto a hom-associative formal deformation of ( A, µ, α ). This is not hard,since hom-associativity of any such formal twisting is true by construc-tion and because verification of the rest of the “formal deformation”condition involves only calculations in degree zero terms.2. One should check that, in order to find all hom-associative formal de-formations of the original algebra, it suffices to carry out the last stepfor one member of each equivalence class of formal deformations of(
A, α − ◦ µ ).We will deal with the second problem now. We start with the following: Remark 1.
Let ( A, · , α ) be an associative algebra together with a k -modulehomomorphism α satisfying α ( α ( x ) α ( yz )) = α ( α ( xy ) α ( z )) . Assume that ( A, · ′ ) is another k -algebra structure isomorphic to ( A, · ) via ϕ : A → A .Then α ′ := ϕ ◦ α ◦ ϕ − is a twisting for ( A, · ′ ) and the hom-associative algebrasinduced by α on ( A, · ) and by α ′ on ( A, · ′ ) are isomorphic as hom-algebras.Proof. It is clear that ϕ ◦ α = α ′ ◦ ϕ . Next, we need to prove that α ′ actuallyinduces a twisted multiplication on ( A, · ′ ) which with respect to α ′ is hom-associative. To do this, we calculate α ′ ( α ′ ( x ) · ′ α ′ ( y · ′ z )) = α ′ ( ϕ ( α ( ϕ − ( x ))) · ′ ϕ ( α ( ϕ − ( y · ′ z ))))= α ′ ( ϕ ( α ( ϕ − ( x )) · α ( ϕ − ( y ) · ϕ − ( z ))))= ϕ ( α ( α ( ϕ − ( x )) · α ( ϕ − ( y ) · ϕ − ( z ))))= ϕ ( α ( α ( ϕ − ( x ) · ϕ − ( y )) · α ( ϕ − ( z ))))= ϕ ( α ( α ( ϕ − ( x · ′ y )))) · ′ ϕ ( α ( ϕ − ( z )))= α ′ ( ϕ ( α ( ϕ − ( x · ′ y )))) · ′ α ′ ( z )= α ′ ( α ′ ( x · ′ y ) · ′ α ′ ( z )) . ϕ with thehom-associative multiplications x ⋆ y := α ( x · y ) and x ⋆ ′ y := α ′ ( x · ′ y ). Thisis done by calculating ϕ ( x ⋆ y ) = ϕ ( α ( x · y )) = α ′ ( ϕ ( x · y )) = α ′ ( ϕ ( x ) · ′ ϕ ( y )) = ϕ ( x ) ⋆ ′ ϕ ( y ) . It is clear that the previous remark holds also when we do everything onthe formal level, i.e. replace isomorphisms with formal isomorphisms andtwistings with formal twistings. Only one thing still needs to be checked.Assume that ( A, · , α ) is an associative algebra together with a twisting α satisfying α ( α ( x ) · α ( y · z )) = α ( α ( x · y ) · α ( z )). Suppose further that ( A [[ t ]] , ⋆ )and ( A [[ t ]] , ⋆ ′ ) are associative formal deformations of A , that ϕ t is a formalisomorphism between them and that α t is a deformation compatible with( A [[ t ]] , ⋆ ) of the twisting α . Then for our deformation program to work, wemust verify that ϕ t ◦ α t ◦ ϕ − t has α as degree zero contribution. But thisfollows from the fact that ϕ is by definition a deformation of the identitymap. The author is grateful to Yael Fregier for useful discussions and for proof-reading of parts of an earlier version of this paper. He also wishes to thankthe University of Luxembourg for providing excellent working conditions.In the research leading to this work, we were aided by the computer pro-grams Prover9 and Mace4 written by William McCune [6]. Specifically, theproofs of Propositions 1 and 2 were obtained by generalisation, structuringand (minor) simplification of proof objects provided by Prover9 for the caseof a hom-associative structure A containing some c ∈ A such that c ⋆ x = c ⋆ y only if x = y . Prover9 is also capable of proving suitable formalisations ofthese propositions directly, but the proofs given in this paper are in our viewmuch easier to understand than the ones obtained in this way. Mace4 wasvery helpful in producing finite examples of hom-associative structures whichguided our search for general structure theorems.12 eferences [1] Yael Fregier and Aron Gohr. On unitality conditions for hom-associativealgebras. arXiv:0904.4874 , 2009.[2] Murray Gerstenhaber. On the deformations of rings and algebras. Ann.math. , 79:59–103, 1964.[3] Jonas Hartwig, Daniel Larsson, and Sergei Silvestrov. Deformations ofLie algebras using σ -derivations. J. Algebra , 295(2):314–361, 2006.[4] Abdenacer Makhlouf and Sergei Silvestrov. Notes on formal deformationsof Hom-associative and Hom-Lie algebras. arXiv:0712.3130v1 , 2007.[5] Abdenacer Makhlouf and Sergei Silvestrov. Hom-algebra structures.
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