Nearly associative and nearly Hom-associative algebras and bialgebras
aa r X i v : . [ m a t h . R A ] J a n Nearly associative and nearly Hom-associative algebrasand bialgebras
Mafoya Landry Dassoundo , Sergei Silvestrov Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, Chinae-mail: [email protected] Division of Mathematics and Physics, School of Education, Culture and Communication,M¨alardalen University, Box 883, 72123 V¨asteras, Sweden.e-mail: [email protected]
Abstract
Basic definitions and properties of nearly associative algebras are described. Nearlyassociative algebras are proved to be Lie-admissible algebras. Two-dimensional nearlyassociative algebras are classified, and its main classes are derived. The bimodules,matched pairs and Manin triple of a nearly associative algebras are derived and theirequivalence with nearly associative bialgebras is proved. Basic definitions and proper-ties of nearly Hom-associative algebras are described. Related bimodules and matchedpairs are given, and associated identities are established.
An algebra A with a bilinear product · : A × A → A is not necessarily associative orpossibly non-associative if possibly there exist x, y, z ∈ A such that ( x · y ) · z − x · ( y · z ) = 0 . If such x, y, z ∈ A exist, then algebra is not associative. The term non-associativealgebras is used often to mean all possibly non-associative algebras, including also theassociative algebras. Associative algebras, Lie algebras, and Jordan algebras are well-known sub-classes of non-associative algebras in the sense of possibly not associativealgebras [60].Hom-algebraic structures originated from quasi-deformations of Lie algebras of vec-tor fields which gave rise to quasi-Lie algebras, defined as generalized Lie structuresin which the skew-symmetry and Jacobi conditions are twisted. Hom-Lie algebras andmore general quasi-Hom-Lie algebras where introduced first by Silvestrov and his stu-dents Hartwig and Larsson in [27], where the general quasi-deformations and discretiza-tions of Lie algebras of vector fields using general twisted derivations, σ -derivations, Mathematics subject classification : 17B61, 17D30, 17D25, 17B62
Keywords : nearly Hom-associative algebra, nearly associative algebra, bialgebra, bimodule.
Corresponding author : Sergei Silvestrov, [email protected] nd a general method for construction of deformations of Witt and Virasoro typealgebras based on twisted derivations have been developed. The initial motivationcame from examples of q -deformed Jacobi identities discovered in q -deformed ver-sions and other discrete modifications of differential calculi and homological algebra, q -deformed Lie algebras and other algebras important in string theory, vertex mod-els in conformal field theory, quantum mechanics and quantum field theory, such asthe q -deformed Heisenberg algebras, q -deformed oscillator algebras, q -deformed Witt, q -deformed Virasoro algebras and related q -deformations of infinite-dimensional alge-bras [1, 16–22, 33, 34, 41–43].Possibility of studying, within the same framework, q -deformations of Lie algebrasand such well-known generalizations of Lie algebras as the color and super Lie alge-bras provided further general motivation for development of quasi-Lie algebras andsubclasses of quasi-Hom-Lie algebras and Hom-Lie algebras. The general abstractquasi-Lie algebras and the subclasses of quasi-Hom-Lie algebras and Hom-Lie alge-bras, as well as their color (graded) counterparts, color (graded) quasi-Lie algebras,color (graded) quasi-Hom-Lie algebras and color (graded) Hom-Lie algebras, includingin particular the super quasi-Lie algebras, super quasi-Hom-Lie algebras, and superHom-Lie algebras, have been introduced in [27, 37–39, 63, 64]. In [48], Hom-associativealgebras were introduced, generalizing associative algebras by twisting the associativitylaw by a linear map. Hom-associative algebra is a triple ( A, · , α ) consisting of a linearspace A , a bilinear product · : A × A → A and a linear map α : A → A , satisfying a α, · ( x, y, z ) = ( x · y ) · α ( z ) − α ( x ) · ( y · z ) = 0 , for any x, y, z ∈ A . In [48], alongsideHom-associative algebras, the Hom-Lie admissible algebras generalizing Lie-admissiblealgebras, were introduced as Hom-algebras such that the commutator product, definedusing the multiplication in a Hom-algebra, yields a Hom-Lie algebra, and also Hom-associative algebras were shown to be Hom-Lie admissible. Moreover, in [48], moregeneral G -Hom-associative algebras including Hom-associative algebras, Hom-Vinbergalgebras (Hom-left symmetric algebras), Hom-pre-Lie algebras (Hom-right symmetricalgebras), and some other Hom-algebra structures, generalizing G -associative algebras,Vinberg and pre-Lie algebras respectively, have been introduced and shown to be Hom-Lie admissible, meaning that for these classes of Hom-algebras, the operation of takingcommutator leads to Hom-Lie algebras as well. Also, flexible Hom-algebras have beenintroduced, connections to Hom-algebra generalizations of derivations and of adjointmaps have been noticed, and some low-dimensional Hom-Lie algebras have been de-scribed. The enveloping algebras of Hom-Lie algebras were considered in [67] usingcombinatorial objects of weighted binary trees. In [29], for Hom-associative algebrasand Hom-Lie algebras, the envelopment problem, operads, and the Diamond Lemmaand Hilbert series for the Hom-associative operad and free algebra have been studied.Strong Hom-associativity yielding a confluent rewrite system and a basis for the freestrongly hom-associative algebra has been considered in [28]. An explicit constructiveway, based on free Hom-associative algebras with involutive twisting, was developedin [25] to obtain the universal enveloping algebras and Poincar´e-Birkhoff-Witt type the-orem for Hom-Lie algebras with involutive twisting map. Free Hom-associative coloralgebra on a Hom-module and enveloping algebra of color Hom-Lie algebras with in-volutive twisting and also with more general conditions on the powers of twisting mapwas constructed, and Poincar´e-Birkhoff-Witt type theorem was obtained in [4, 5]. It isworth noticing here that, in the subclass of Hom-Lie algebras, the skew-symmetry is ntwisted, whereas the Jacobi identity is twisted by a single linear map and containsthree terms as in Lie algebras, reducing to ordinary Lie algebras when the twistinglinear map is the identity map.Hom-algebra structures include their classical counterparts and open new broadpossibilities for deformations, extensions to Hom-algebra structures of representations,homology, cohomology and formal deformations, Hom-modules and hom-bimodules,Hom-Lie admissible Hom-coalgebras, Hom-coalgebras, Hom-bialgebras, Hom-Hopf al-gebras, L -modules, L -comodules and Hom-Lie quasi-bialgebras, n -ary generalizationsof biHom-Lie algebras and biHom-associative algebras and generalized derivations,Rota-Baxter operators, Hom-dendriform color algebras, Rota-Baxter bisystems andcovariant bialgebras, Rota-Baxter cosystems, coquasitriangular mixed bialgebras, coas-sociative Yang-Baxter pairs, coassociative Yang-Baxter equation and generalizations ofRota-Baxter systems and algebras, curved O -operator systems and their connectionswith tridendriform systems and pre-Lie algebras, BiHom-algebras, BiHom-Frobeniusalgebras and double constructions, infinitesimal biHom-bialgebras and Hom-dendriform D -bialgebras, Hom-algebras has been considered from a category theory point of view[3, 8–15, 24, 26, 30, 31, 35–37, 40, 44–46, 49–52, 56, 57, 61, 62, 65–70].This paper is organized as follows. In Section 2, basic definitions and fundamentalidentities and some elementary examples of nearly associative algebras are given. InSection 3, we derive the classification of the two-dimensional nearly associative algebrasand main classes are provided. In Section 4, bimodules, duals bimodules and matchedpair of nearly associative algebras are established and related identities are derivedand proved. In Section 5, Manin triple of nearly associative algebras is given andits equivalence to the nearly associative bialgebras is derived. In Section 6, Hom-Lie-admissible, G -Hom-associative, flexible Hom-algebras, the result on Lie-admissibility of G -Hom-admissible algebras and subclasses of G -Hom-admissible algebras are reviewed.In Section 7, main definitions and fundamental identities of Hom-nearly associativealgebras are given. Furthermore, the bimodules, and matched pair of the Hom-nearlyassociative algebras are derived and related properties are obtained. Throughout this paper, for simplicity of exposition, all linear spaces are assumed to beover field K of characteristic is 0, even though many results hold in general for otherfields as well unchanged or with minor modifications. An algebra is a couple ( A, µ )consisting of a linear space A and a bilinear product µ : A × A → A . Definition 2.1.
An algebra ( A, · ) is called nearly associative if, for all x, y, z ∈ A , x · ( y · z ) = ( z · x ) · y. (1) Example 2.2.
Consider a two-dimensional linear space A with basis { e , e } . • Then, ( A, · ) is a nearly associative algebra, where e · e = e + e and for all ( i, j ) = (1 , with i, j ∈ { , } , e i · e j = 0 . • The linear product defined on A by: e · e = e , e · e = e = e · e and e · e = e , is such that ( A, · ) is a nearly associative algebra. xample 2.3. Consider a three-dimensional linear space A with basis { e , e , e } . • The linear space A equipped with the linear product defined on A by: e · e = e + e , e · e = e + e − e , e · e = − e + e and for all i = j, e i · e j = 0 ,where i, j ∈ { , , } , is a nearly associative algebra. • The linear space A equipped with the linear product defined on A by: e · e = e − e , e · e = e + e , e · e = e − e + e and for all i = j, e i · e j = 0 , where i, j ∈ { , , } , is a nearly associative algebra. • The linear space A equipped with the linear product defined on A by: e · e = e + e , e · e = e + e + e , e · e = e + e and for all i = j, e i · e j = 0 , where i, j ∈ { , , } , is a nearly associative algebra. Definition 2.4 ( [2, 23, 53–55, 58, 59]) . An algebra ( A, · ) is called Lie admissible if ( A, [ ., . ]) is a Lie algebra, where [ x, y ] = x · y − y · x for all x, y ∈ A . For a Lie admissible algebra ( A, · ), the Lie algebra G ( A ) = ( A, [ ., . ]) is called anunderlying Lie algebra of ( A, · ).It is known that associative algebras, left-symmetric algebras and anti-flexible al-gebras (center-symmetric algebras) are Lie-admissible [6, 7, 30]. Proposition 2.5.
Any nearly associative algebra is Lie-admissible.Proof.
For [ ., . ] : ( v, w ) v · w − w · v and x, y, z in a nearly associative algebra ( A, · ),[ x, [ y, z ]] + [ y, [ z, x ]] + [ z, [ x, y ]]= [ x, y · z − z · y ] + [ y, z · x − x · z ] + [ z, x · y − y · x ]= x · ( y · z ) − x · ( z · y ) − ( y · z ) · x + ( z · y ) · x + y · ( z · x ) − y · ( x · z ) − ( z · x ) · y + ( x · z ) · y + z · ( x · y ) − z · ( y · x ) − ( x · y ) · z + ( y · x ) · z = { x · ( y · z ) − ( z · x ) · y } + { ( y · x ) · z − x · ( z · y ) } + { y · ( z · x ) − ( x · y ) · z } + { z ∗ ( x · y ) − ( y · z ) · x } + { ( z · y ) · x − y · ( x · z ) } + { ( x · z ) · y − z · ( y · x ) } = 0 . Therefore, ( A, [ ., . ]) is a Lie algebra. Remark 2.6.
In a nearly associative algebra ( A, · ) , for x, y ∈ A , L ( x ) L ( y ) = R ( y ) R ( x ) , (2a) L ( x ) R ( y ) = L ( y · x ) , (2b) R ( x ) L ( y ) = R ( x · y ) , (2c) where L, R : A → End( A ) are the operators of left and right multiplications. Definition 2.7.
An anti-flexible algebra is a couple ( A, · ) where A is a linear space,and · : A × A → A is a bilinear product such that for all x, y, z ∈ A , ( x · y ) · z − ( z · y ) · x = x · ( y · z ) − z · ( y · x ) . (3)Using associator a ( x, y, z ) = ( x · y ) · z − x · ( y · z ), the equality (3) is equivalent to a ( x, y, z ) = a ( z, y, x ) . (4)In view of (4), anti-flexible algebras were called center-symmetric algebras in [30]. roposition 2.8. Any commutative nearly associative algebra is anti-flexible.Proof.
For all x, y, z ∈ A in a commutative nearly associative algebra ( A, · ), by usingnearly associativity, commutativity and again nearly associativity, a ( x, y, z ) = ( x · y ) · z − x · ( y · z ) = y · ( z · x ) − ( z · x ) · y = [ y, z · x ] = [ y, x · z ] = y · ( x · z ) − ( x · z ) · y = ( z · y ) · x − z · ( y · x ) = a ( z, y, x )proves (4) meaning that ( A, · ) is anti-flexible. Theorem 3.1.
Any two-dimensional algebra ( A, · ) is nearly associative if and only if e · ( e · e ) = ( e · e ) · e , e · ( e · e ) = ( e · e ) · e ,e · ( e · e ) = ( e · e ) · e , e · ( e · e ) = ( e · e ) · e ,e · ( e · e ) = ( e · e ) · e , e · ( e · e ) = ( e · e ) · e ,e · ( e · e ) = ( e · e ) · e , e · ( e · e ) = ( e · e ) · e , where { e , e } is a basis of A. Theorem 3.2.
Any two-dimensional nearly associative algebra is isomorphic to oneof the following nearly associative algebras: • For all ( α, β ) ∈ K \{ (0 , } , e · e = αe , e · e = βe = e · e , e · e = βe . • For all ( α, β ) ∈ K \{ (0 , } , e · e = αe + βe , e · e = βe + αe = e · e ,e · e = αe + βe . • For all ( α, β, γ ) ∈ K , such that γ + 4 αβ ≥ , e · e = αe , e · e = βe + γe ,e · e = (cid:16) γ + p γ + 4 αβ (cid:17) e = e · e . Proof.
Equip the linear space A with the basis { e , e } , and for all i ; j ∈ {
1; 2 } , set e i · e j = a ij e + b ij e , where a ij ∈ K and b ij ∈ K . In addition, for all i, j, k ∈ { , } , jk a i + b jk a i = a ki a j + b ki a j , a jk b i + b jk b i = a ki b j + b ki b j . By Theorem 3.1, e · ( e · e ) = ( e · e ) · e e · ( e · e ) = ( e · e ) · e e · ( e · e ) = ( e · e ) · e e · ( e · e ) = ( e · e ) · e e · ( e · e ) = ( e · e ) · e e · ( e · e ) = ( e · e ) · e e · ( e · e ) = ( e · e ) · e e · ( e · e ) = ( e · e ) · e ⇐⇒ a a + b a = a a + b a ,a b + b b = a b + b b ,a a + b a = a a + b a ,a b + b b = a b + b b ,a a + b a = a a + b a ,a b + b b = a b + b b ,a a + b a = a a + b a ,a b + b b = a b + b b ,a a + b a = a a + b a ,a b + b b = a b + b b ,a a + b a = a a + b a ,a b + b b = a b + b b ,a a + b a = a a + b a ,a b + b b = a b + b b ,a a + b a = a a + b a ,a b + b b = a b + b b ⇐⇒ e ( b − c ) = 0 , e ( f − g ) = 0 ,h ( b − c ) = 0 , d ( b − c ) = 0 ,d ( f − g ) = 0 , a ( f − g ) = 0( b − c )( b + c ) = 0 , ( f − g )( f + g ) = 0 e ( b − c ) + f ( g − a ) = 0 d ( a − g ) + b ( h − c ) = 0 a ( b − c ) = 0 , h ( f − g ) = 0( bf − cg ) = 0 , bg = de = f c ⇒ (cid:26) a = r , b = r , c = r , d = r ,e = r , f = r , g = r , h = r or (cid:26) a = r , b = r , c = r , d = r ,e = r , f = r , g = r , h = r or (cid:26) a = r , b = 0 , c = 0 , d = 0 ,e = r , f = r , g = r , h = r or (cid:26) a = r , b = 0 , c = 0 , d = r ,e = 0 , f = r , g = r , h = r or a = r , b = | r | + r ,c = | r | + r , d = 0 ,e = r , f = 0 , g = 0 , h = r or a = r , b = √ r r + r + r ,c = √ r r + r + r , d = r ,e = 0 , f = 0 , g = 0 , h = r or a = r , b = r − √ r ,c = r − √ r , d = 0 ,e = r , f = 0 , g = 0 , h = r or a = r , b = r − √ r +4 r r ,c = r − √ r +4 r r , d = r ,e = 0 , f = 0 , g = 0 , h = r or (cid:26) a = r , b = 0 , c = 0 , d = 0 ,e = r , f = 0 , g = 0 , h = 0 or (cid:26) a = r , b = √ r r , c = √ r r ,d = r , e = 0 , f = 0 , g = 0 , h = 0or (cid:26) a = r , b = 0 , c = 0 , d = 0 ,e = r , f = 0 , g = 0 , h = 0 or a = r , b = −√ r r ,c = −√ r r , d = r ,e = 0 , f = 0 , g = 0 , h = 0or (cid:26) a = 0 , b = 0 , c = 0 , d = 0 ,e = r , f = 0 , g = 0 , h = r or (cid:26) a = 0 , b = 0 , c = 0 , d = r ,e = 0 , f = 0 , g = 0 , h = r or (cid:26) a = 0 , b = r , c = r , d = 0 ,e = r , f = 0 , g = 0 , h = r or (cid:26) a = 0 , b = r , c = r , d = r ,e = 0 , f = 0 , g = 0 , h = r or (cid:26) a = r , b = 0 , c = 0 , d = 0 ,e = 0 , f = 0 , g = 0 , h = 0with a = a, a = b, a = c, a = d, b = e, b = f, b = g, b = h. Therefore, the non-isomorphic algebras generated by these constants structures are: • For all ( α, β ) ∈ K \{ (0 , } , e · e = αe , e · e = βe = e · e , e · e = βe . • For all ( α, β ) ∈ K \{ (0 , } , e · e = αe + βe , e · e = βe + αe = e · e ,e · e = αe + βe . • For all ( α, β, γ ) ∈ K , such that γ + 4 αβ ≥ , e · e = αe , e · e = βe + γe ,e · e = (cid:16) γ + p γ + 4 αβ (cid:17) e = e · e . Bimodules and matched pairs nearly associa-tive algebras
Definition 4.1.
Let ( A, · ) be a nearly associative algebra. Consider the linear maps l ; r : A → End( V ) , where V is a linear space. A triple ( l, r, V ) is a bimodule of ( A, · ) if for all x, y ∈ A , the following relations l ( x ) l ( y ) = r ( y ) r ( x ) , (5a) l ( x ) r ( y ) = l ( y · x ) , (5b) r ( x ) l ( y ) = r ( x · y ) (5c) are satisfied. Example 4.2.
Let ( A, · ) be a nearly associative algebra. The triple ( L, R, A ) is abimodule of ( A, · ) , where for any x, y ∈ A , L ( x ) y = x · y = R ( y ) x . Proposition 4.3.
Let ( l, r, V ) be a bimodule of a nearly associative algebra ( A, · ) ,where l ; r : A → End( V ) are two linear maps and V a linear space. There is a nearlyassociative algebra defined on A ⊕ V by, for any x, y ∈ A and any u, v ∈ V, ( x + u ) ∗ ( y + v ) = x · y + l ( x ) v + r ( y ) u. (6) Proof.
Consider the bimodule ( l, r, V ) of the nearly associative algebra ( A, · ). For all x, y, z ∈ A and u, v, w ∈ V we have:( x + u ) ∗ (( y + v ) ∗ ( z + w )) = x · ( y · z ) + l ( x ) l ( y ) w + l ( x ) r ( z ) v + r ( y · z ) u (7a)(( z + w ) ∗ ( x + u )) ∗ ( y + v ) = ( z · x ) · y + l ( z · x ) v + r ( y ) l ( z ) u + r ( y ) r ( x ) w (7b)Using (5a) - (5c) in (7a) and (7b) we easily deduce that ( A ⊕ V, ∗ ) is a nearly associativealgebra. Corollary 4.4.
Let ( l, r, V ) be a bimodule of a nearly associative algebra ( A, · ) , where l, r : A → End(V) are two linear maps and V a linear space. Then there is a Liealgebra product on A ⊕ V given by [ x + u, y + v ] = [ x, y ] · + ( l ( x ) − r ( x )) v − ( l ( y ) − r ( y )) u (8) for all x, y ∈ A and for any u, v ∈ V .Proof. It is simple to remark that the commutator of the product defined in (6) is theproduct defined in (8). By taking into account Proposition 2.5, the Jacobi identity ofthe product given in (8) is satisfied.
Definition 4.5.
Let ( G , [ ., . ] G ) be a Lie algebra. A representation of ( G , [ ., . ] G ) over thelinear space V is a linear map ρ : G → End( V ) satisfying ρ ([ x, y ] G ) = ρ ( x ) ◦ ρ ( y ) − ρ ( y ) ◦ ρ ( x ) (9) for all x, y ∈ G . roposition 4.6. Let ( A, · ) be a nearly associative algebra and let V be a finite-dimen-sional linear space over the field K such that ( l, r, V ) is a bimodule of ( A, · ) , where l, r : A → End( V ) are two linear maps. Then, the linear map l − r : A → End( V ) , x l ( x ) − r ( x ) is a representation of the underlying Lie algebra G ( A ) underlying ( A, · ) .Proof. Let ( l, r, V ) be a bimodule of the nearly associative algebra ( A, · ). For x, y ∈ A ,( l ( x ) − r ( x ))( l ( y ) − r ( y )) − ( l ( y ) − r ( y ))( l ( x ) − r ( x )) = l ( x ) l ( y ) − l ( x ) r ( y ) − r ( x ) l ( y ) + r ( x ) r ( y ) − l ( y ) l ( x ) + l ( y ) r ( x ) + r ( y ) l ( x ) − r ( y ) r ( x )= − l ( x ) r ( y ) − r ( x ) l ( y ) + l ( y ) r ( x ) + r ( y ) l ( x ) = − l ( y · x ) − r ( x · y ) + l ( x · x ) + r ( y · x )= ( l − r )( x · y − y · x ) = ( l − r )([ x, y ]) . Therefore, (9) is satisfied for l − r = ρ . Definition 4.7.
Let ( A, · ) be a nearly associative algebra and ( l, r, V ) its associateda bimodule, where V is a finite-dimensional linear space. The dual maps l ∗ , r ∗ oflinear maps l, r, respectively are defined as l ∗ , r ∗ : A → End( V ∗ ) such that for any x ∈ A, u ∗ ∈ V ∗ , v ∈ V, h l ∗ ( x ) u ∗ , v i = h u ∗ , l ( x ) v i , (10a) h r ∗ ( x ) u ∗ , v i = h u ∗ , r ( x ) v i . (10b) Proposition 4.8.
Let ( A, · ) be a nearly associative algebra and ( l, r, V ) be its bimodule.The following relations are equivalent: (i) ( r ∗ , l ∗ , V ∗ ) is a bimodule of ( A, · ) , (ii) l ( x ) r ( y ) = r ( y ) l ( x ) , for all x, y ∈ A , (iii) ( l ∗ , r ∗ , V ∗ ) is a bimodule of ( A, · ) .Proof. Let ( A, · ) be a nearly associative algebra and ( l, r, V ) be its associated bimodulei.e. the linear maps l, r : A → End( V ) satisfying (5a) - (5c) and V is a finite-dimensionallinear space. • Suppose that ( r ∗ , l ∗ , V ∗ ) is a bimodule of ( A, · ), i.e. with correspondences l → r ∗ and r → l ∗ , (5a) - (5c) are satisfied. For any x, y ∈ A , v ∈ V , u ∗ ∈ V ∗ : h l ( x ) r ( y ) v, u ∗ i = h v, r ∗ ( y ) l ∗ ( x ) u ∗ i = h v, r ∗ ( y · x ) u ∗ i = h r ( y · x ) v, u ∗ i = h r ( y ) l ( x ) v, u ∗ i . Therefore, the relation l ( x ) r ( y ) = r ( y ) l ( x ) is satisfied. • Suppose l ( x ) r ( y ) = r ( y ) l ( x ) for any x, y ∈ A . For any x, y ∈ A , v ∈ V , u ∗ ∈ V ∗ : h l ∗ ( x ) l ∗ ( y ) u ∗ , v i = h u ∗ , l ( y ) l ( x ) v i = h u ∗ , r ( x ) r ( y ) v i = h r ∗ ( y ) r ∗ ( x ) u ∗ , v i , yields l ∗ ( x ) l ∗ ( y ) = r ∗ ( y ) r ∗ ( x ); l ∗ ( x ) r ∗ ( y ) u ∗ , v i = h u ∗ , r ( y ) l ( x ) v i = h u ∗ , l ( x ) r ( y ) v i = h u ∗ , l ( y · x ) v i = h l ∗ ( y · x ) u ∗ , v i , yields l ∗ ( x ) r ∗ ( y ) = l ∗ ( y · x ); h r ∗ ( y ) l ∗ ( x ) u ∗ , v i = h u ∗ , l ( x ) r ( y ) v i = h u ∗ , r ( y ) l ( x ) v i = h u ∗ , r ( y · x ) v i = h r ∗ ( y · x ) u ∗ , v i , yields r ∗ ( y ) l ∗ ( x ) = r ∗ ( y · x ) . Thus, with correspondences r ∗ → l and l ∗ → r ,(5a) - (5c) are satisfied.Similarly, one obtains the equivalence between l ( x ) r ( y ) = r ( y ) l ( x ), for any x, y ∈ A ,and ( l ∗ , r ∗ , V ∗ ) being a bimodule of ( A, · ). Remark 4.9.
It is clear that ( L ∗· , R ∗· , A ∗ ) and ( R ∗· , L ∗· , A ∗ ) are bimodules of the nearlyassociative algebra ( A, · ) if and only if L and R commute. Theorem 4.10.
Let ( A, · ) and ( B, ◦ ) be two nearly associative algebras. Supposethat ( l A , r A , B ) and ( l B , r B , A ) are bimodules of ( A, · ) and ( B, ◦ ) , respectively, where l A , r A : A → End( B ) , l B , r B : B → End( A ) are four linear maps satisfying for all x, y ∈ A , a, b ∈ B the following relations r B ( l A ( x ) a ) y + y · ( r B ( a ) x ) − ( l B ( a ) y ) · x − l B ( r A ( y ) a ) x = 0 , (11a) r B ( a )( x · y ) − y · ( l B ( a ) x ) − r B ( r A ( x ) a ) y = 0 , (11b) l B ( a )( x · y ) − ( r B ( a ) y ) · x − l B ( l A ( y ) a ) x = 0 , (11c) r A ( l B ( a ) x ) b + b ◦ ( r A ( x ) a ) − ( l A ( x ) b ) ◦ a − l A ( r B ( b ) x ) a = 0 , (11d) r A ( x )( a ◦ b ) − b ◦ ( l A ( x ) a ) − r A ( r B ( a ) x ) b = 0 , (11e) l A ( x )( a ◦ b ) − ( r A ( x ) b ) ◦ a − l A ( l B ( b ) x ) a = 0 . (11f) Then, ( A ⊕ B, ∗ ) is a nearly associative algebra, where ( x + a ) ∗ ( y + b ) = ( x · y + l B ( a ) y + r B ( b ) x ) + ( a ◦ b + l A ( x ) b + r A ( y ) a ) . (12) for all x, y ∈ A, a, b ∈ B .Proof. For any x, y, z ∈ A , and for any a, b, c ∈ B , we have( x + a ) ∗ (( y + b ) ∗ ( z + c )) = x · ( y · z ) + { x · ( l B ( b ) z ) + r B ( r A ( z ) b ) x } + l B ( a )( y · z )+ { x · ( r B ( c ) y ) + r B ( l A ( y ) c ) x } + l B ( a )( l B ( b ) z ) + r B ( b ◦ c ) x + l B ( a )( r B ( c ) y ) + a ◦ ( b ◦ c ) + { a ◦ ( l A ( y ) c ) + r B ( r B ( c ) y ) a } + { a ◦ ( r A ( z ) b ) + r B ( l B ( b ) z ) a } + l A ( x )( l A ( y ) c ) + l A ( x )( b ◦ c ) + l A ( x )( r A ( z ) b ) + r A ( y · z ) a ;(( z + c ) ∗ ( x + a )) ∗ ( y + b ) = ( z · x ) · y + { ( l B ( c ) x ) · y + l B ( r A ( x ) c ) y } + r B ( x )( z · x )+ { ( r A ( a ) z ) · y + l B ( l A ( z ) a ) y } + l B ( c ◦ a ) y + r B ( b )( l B ( c ) x )+ r B ( b )( r B ( a ) z )( c ◦ a ) ◦ y + { ( l A ( z ) a ) ◦ b + l A ( r B ( a ) z ) b } + { ( r A ( x ) c ) ◦ b + l A ( l B ( c ) x ) b } + r A ( y )( r A ( x ) c )+ r A ( y )( c ◦ a )+ r A ( y )( l A ( z ) a )+ l A ( z · x ) b. Using (11a) - (11f) and that ( l A , r A , B ) and ( l B , r B , A ) are bimodules of ( A, · ) and( B, ◦ ), respectively, we derive that ( A ⊕ B, ∗ ) is a nearly associative algebra. efinition 4.11 ( [47]) . Let ( G , [ ., . ] G ) and ( H , [ ., . ] H ) be two Lie algebras such that ρ : G → End( H ) and µ : H → End( G ) are representations of G and H , respectively. Amatched pair of Lie algebras G and H is ( G , H , ρ, µ ) such that ρ and µ are satisfyingthe following relations, for all x, y ∈ G and a, b ∈ H , ρ ( x )[ a, b ] G − [ ρ ( x ) a, b ] H − [ a, ρ ( x ) b ] H + ρ ( µ ( a ) x ) b − ρ ( µ ( b ) x ) a = 0 , (13a) µ ( a )[ x, y ] G − [ µ ( a ) x, y ] G − [ x, µ ( a ) y ] G + µ ( ρ ( x ) a ) y − µ ( ρ ( y ) a ) x = 0 . (13b) Corollary 4.12.
Let ( A, B, l A , r A , l B , r B ) be a matched pair of the nearly associativealgebras ( A, · ) and ( B, ◦ ) . Then ( G ( A ) , G ( B ) , l A − r A , l B − r B ) is a matched pair of Liealgebras G ( A ) and G ( B ) .Proof. Let (
A, B, l A , r A , l B , r B ) be a matched pair of the nearly associative algebras( A, · ) and ( B, ◦ ). In view of Proposition 4.6, the linear maps l A − r A : A −→ End( B )and l B − r B : B −→ End( A ) are representations of the underlying Lie algebras G ( A )and G ( B ), respectively. Therefore, by direct calculation we have (13a) is equivalent to(11a) - (11c) and similarly, (13b) is equivalent to (11d) - (11f). Proposition 4.13.
Let ( A, · ) be a nearly associative algebra. Suppose that there isa nearly associative algebra structure ◦ on its the dual space A ∗ . If in addition, thelinear maps L and R commute then ( A, A ∗ , R ∗· , L ∗· , R ∗◦ , L ∗◦ ) is a matched pair of thenearly associative algebras ( A, · ) and ( A ∗ , ◦ ) if and only if the following relations aresatisfied for any x, y ∈ A and a ∈ A ∗ L ∗◦ ( R ∗· ( x ) a ) y − y · ( L ∗◦ ( a ) x ) − ( R ∗◦ ( a ) y ) · x − R ∗◦ ( L ∗· ( y ) a ) x = 0 , (14a) L ∗◦ ( a )( x · y ) − y · ( R ∗◦ ( a ) x ) − L ∗◦ ( L ∗· ( x ) a ) y = 0 , (14b) R ∗◦ ( a )( x · y ) − ( L ∗◦ ( a ) y ) · x − R ∗◦ ( R ∗· ( y ) a ) x = 0 . (14c) Proof.
Since L and R commute, according to Remark 4.9 and Proposition 4.8, both( R ∗· , L ∗· , A ∗ ) and ( L ∗· , R ∗· , A ∗ ) are bimodules of ( A, · ). Setting l A = R ∗· , r A = L ∗· , l B = R ∗◦ and r B = L ∗◦ in Theorem 4.10 the equivalences among Eq (11a) and (14a), (11b)and (14b), and finally (11c) and (14c) are straightforward. Besides, for any x, y ∈ A and any a, b ∈ A ∗ , we have h L ∗◦ ( R ∗· ( x ) a ) y, b i = h y, L ◦ ( R ∗· ( x ) a ) b i = h y, ( R ∗· ( x ) a ) ◦ b i , h y · ( L ∗◦ ( a ) x ) , b i = h R · ( L ∗◦ ( a ) x ) y, b i = h y, R ∗· ( L ∗◦ ( a ) x ) b i , h ( R ∗◦ ( a ) y ) · x, b i = h R ∗◦ ( a ) y, R ∗· ( x ) b i = h y, ( R ∗· ( x ) b ) ◦ a i , h R ∗◦ ( L ∗· ( y ) a ) x, b i = h L ∗◦ ( b ) x, L ∗· ( y ) a i = h y · ( L ∗◦ ( b ) x ) , a i = h y, R ∗· ( L ∗◦ ( b ) x ) a i , h L ∗◦ ( a )( x · y ) , b i = h R · ( y ) x, a ◦ b i = h x, R ∗· ( y )( a ◦ b ) i , h y · ( R ∗◦ ( a ) x ) , b i = h R ∗◦ ( a ) x, L ∗· ( y ) b i = h x, ( L ∗· ( y ) b ) ◦ a i , h L ∗◦ ( L ∗· ( x ) a ) y, b i = h R ∗◦ ( b ) y, L ∗◦ ( x ) a i = h x · ( R ∗◦ ( b ) y ) , a i = h x, R ∗· ( R ∗◦ ( b ) y ) a i , h R ∗◦ ( a )( x · y ) , b i = h L · ( x ) y, b ◦ a i = h y, L · ( x ) ∗ ( b ◦ a ) i , h ( L ∗◦ ( a ) y ) · x, b i = h L ∗◦ ( a ) y, R ∗· ( b ) i = h y, a ◦ ( R ∗· ( x ) b ) i , h R ∗◦ ( R ∗· ( y ) a ) x, b i = h L ∗◦ ( b ) x, R ∗· ( y ) a i = h ( L ∗◦ ( b ) x ) · y, a i = h y, L ∗· ( L ∗◦ ( b ) x ) a i Then, Eq (11a) holds if and only if (11d) holds, Eq (11b) holds if and only if (11e)holds, and finally Eq (11c) holds if and only if (11f) holds. Manin triple and bialgebra of nearly associa-tive algebras
Definition 5.1.
A bilinear form B on a nearly associative algebra ( A, · ) is called left-invariant if f orallx, y, z ∈ A , B ( x · y, z ) = B ( x, y · z ) . (15) Proposition 5.2.
Let ( A, · ) be a nearly associative algebra. If there is a nondegen-erate symmetric invariant bilinear form B defined on A , then as bimodules of thenearly associative algebra ( A, · ) , ( L, R, A ) and ( R ∗ , L ∗ , A ∗ ) are equivalent. Conversely,if ( L, R, A ) and ( R ∗ , L ∗ , A ∗ ) are equivalent bimodules of a nearly associative algebra ( A, · ) , then there exists a nondegenerate invariant bilinear form B on A . Definition 5.3.
A Manin triple of nearly associative algebras is a triple of nearly asso-ciative algebras ( A, A , A ) together with a nondegenerate symmetric invariant bilinearform B on A such that the following conditions are satisfied. (i) A and A nearly associative subalgebras of A ; (ii) as linear spaces, A = A ⊕ A ; (iii) A and A are isotropic with respect to B , i.e. for any x , y ∈ A and any x , y ∈ A , B ( x , y ) = 0 = B ( x , y ) = 0 . Definition 5.4.
Let ( A, · ) be a nearly associative algebra. Suppose that ◦ is a nearlyassociative algebra structure on the dual space A ∗ of A and there is a nearly associativealgebra structure on the direct sum A ⊕ A ∗ of the underlying linear spaces of A and A ∗ such that ( A, · ) and ( A ∗ , ◦ ) are subalgebras and the natural symmetric bilinear form on A ⊕ A ∗ given by ∀ x, y ∈ A ; ∀ a ∗ , b ∗ ∈ A ∗ , B d ( x + a ∗ , y + b ∗ ) := h a ∗ , y i + h x, b ∗ i , (16) is left-invariant, then ( A ⊕ A ∗ , A, A ∗ ) is called a standard Manin triple of nearly asso-ciative algebras associated to B d . Obviously, a standard Manin triple of nearly associative algebras is a Manin tripleof nearly associative algebras. By symmetric role of A and A ∗ , we have Proposition 5.5.
Every Manin triple of nearly associative algebras is isomorphic toa standard one.
Proposition 5.6.
Let ( A, · ) be a nearly associative algebra. Suppose that there isa nearly associative algebra structure ◦ on the dual space A ∗ . There exists a nearlyassociative algebra structure on the linear space A ⊕ A ∗ such that ( A ⊕ A ∗ , A, A ∗ ) is astandard Manin triple of nearly associative algebras associated to B d defined by (16) ifand only if ( A, A ∗ , R ∗· , L ∗· , R ∗◦ , L ∗◦ ) is a matched pair of nearly associative algebras. Theorem 5.7.
Let ( A, · ) be a nearly associative algebra such that the left and rightmultiplication operators commute. Suppose that there is a nearly associative alge-bra structure ◦ on its the dual space A ∗ given by ∆ ∗ : A ∗ ⊗ A ∗ → A ∗ . Then, A, A ∗ , R ∗· , L ∗· , R ∗◦ , L ∗◦ ) is a matched pair of the nearly associative algebras ( A, · ) and ( A ∗ , ◦ ) if and only if ∆ : A → A ⊗ A satisfies the following relations ( R · ( x ) ⊗ id − σ ( R · ( x ) ⊗ id))∆( y ) + (id ⊗ L · ( y ) − σ (id ⊗ L · ( y )))∆( x ) = 0 , (17a)( L · ( x ) ⊗ id)∆( y ) + σ ( L · ( y ) ⊗ id)∆( x ) = ∆( x · y )= σ (id ⊗ R · ( x ))∆( y ) + (id ⊗ R · ( y ))∆( x ) . (17b) Proof.
For any a, b ∈ A ∗ and any x, y ∈ A we have h ( R · ( x ) ⊗ id)∆( y ) , a ⊗ b i = h y, ( R ∗· ( x ) a ) ◦ b i = h L ∗◦ ( R ∗· ( x ) a ) y, b i , h σ ( R · ( x ) ⊗ id)∆( y ) , a ⊗ b i = h y, ( R ∗· ( x ) b ) ◦ a i = h R ∗◦ ( a ) y, R ∗· ( x ) b i = h ( R ∗◦ ( a ) y ) · x, b i , h (id ⊗ L · ( y ))∆( x ) , a ⊗ b i = h x, a ◦ ( L ∗· ( y ) b ) i = h y · ( L ∗◦ ( a ) x ) , b i , h σ (id ⊗ L · ( y ))∆( x ) , a ⊗ b i = h x, b ◦ ( L ∗· ( y ) a ) i = h R ∗◦ ( L ∗· ( y ) a ) x, b i . Hence (14a) is equivalent to (17a).Similarly, we have for any x, y ∈ A and any a, b ∈ A ∗ h ∆( x · y ) , a ⊗ b i = h x · y, a ◦ b i = h L ∗◦ ( a )( x · y ) , b i = h R ∗◦ ( b )( x · y ) , a i , h ( L · ( x ) ⊗ id)∆( y ) , a ⊗ b i = h y, ( L ∗· ( x ) a ) ◦ b i = h L ∗◦ ( L ∗· ( x ) a ) y, b i , h σ ( L · ( y ) ⊗ id)∆( x ) , a ⊗ b i = h x, ( L ∗· ( y ) b ) ◦ a i = h y · ( R ∗◦ ( a ) x ) , b i , h σ (id ⊗ R · ( x ))∆( y ) , a ⊗ b i = h y, b ◦ ( R ∗· ( x ) a ) i = h R ∗◦ ( R ∗· ( x ) a ) y, b i , h (id ⊗ R · ( y ))∆( x ) , a ⊗ b i = h x, a ◦ ( R ∗· ( y ) b ) i = h ( L ∗◦ ( a ) x ) · y, b i , Therefore, (14b) and (14c) and is equivalent to (17b).
Remark 5.8.
Obviously, if L and R commute, then L ∗ and R ∗ commute too and if inaddition γ : A ∗ → A ∗ ⊗ A ∗ is a linear maps such that its dual γ ∗ : A ⊗ A → A definesa nearly associative algebra structure · on A , then ∆ satisfies (17a) and (17b) if andonly if γ satisfies for all a, b ∈ A ∗ , ( R ◦ ( a ) ⊗ id − σ ( R ◦ ( a ) ⊗ id)) γ ( b ) + (id ⊗ L ◦ ( b ) − σ (id ⊗ L ◦ ( b ))) γ ( a ) = 0 , ( L ◦ ( x ) ⊗ id) γ ( b ) + σ ( L ◦ ( b ) ⊗ id) γ ( a ) = γ ( a ◦ b ) = σ (id ⊗ R ◦ ( a )) γ ( b ) + (id ⊗ R ◦ ( b )) γ ( a ) . Definition 5.9.
Let ( A, · ) be a nearly associative algebra in which the left ( L ) andright ( R ) multiplication operators commute. A nearly anti-flexible bialgebra structureis a linear map ∆ : A → A ⊗ A such that • ∆ ∗ : A ∗ ⊗ A ∗ → A ∗ defines a nearly associative algebra structure on A, • ∆ satisfies (17b) and (17b) . Theorem 5.10.
Let ( A, · ) be a nearly associative algebra in which the left and rightmultiplication operators commute. Suppose that there is a nearly associative algebrastructure on A ∗ denoted by ◦ which defined a linear map ∆ : A → A ⊗ A . Then thefollowing conditions are equivalent: (i) ( A ⊕ A ∗ , A, A ∗ ) is a standard Manin triple of nearly associative algebras ( A, · ) and ( A ∗ , ◦ ) such that its associated symmetric bilinear form B d is defined by (16) . (ii) ( A, A ∗ , R ∗· , L ∗· , R ∗◦ , L ∗◦ ) is a matched pair of nearly associative algebras ( A, · ) and ( A ∗ , ◦ ) . (iii) ( A, A ∗ ) is a nearly associative bialgebra. Hom-Lie admissible, G-Hom-associative, flex-ible and anti-flexible Hom-algebras
Hom-Lie admissible algebras along with Hom-associative algebras and more general G -Hom-associative algebras were first introduced, and Hom-associative algebras and G -Hom-associative algebras were shown to be Hom-Lie admissible in [48].Hom-algebra is a triple ( A, µ, α ) consisting of a linear space A over a field K , abilinear product µ : A × A → A and a linear map α : A → A . Definition 6.1 ( [48]) . Hom-Lie, Hom-Lie admissible, Hom-associative and G -Hom-associative Hom-algebras (over a field K ) are defined as follows: Hom-Lie algebras are triples ( A, [ ., . ] , α ) , consisting of a linear space A over a field K , bilinear map ( bilinear product ) [ ., . ] : A × A → A and a linear map α : A → A satisfying, for all x, y, z ∈ A , [ x, y ] = − [ y, x ] , (Skew-symmetry) (18)[ α ( x ) , [ y, z ]] + [ α ( y ) , [ z, x ]] + [ α ( z ) , [ x, y ]] = 0 . (Hom-Jacobi identity) (19)2) Hom-Lie admissible algebras are Hom-algebras ( A, µ, α ) consisting of possibly non-associative algebra ( A, µ ) and a linear map α : A → A , such that ( A, [ ., . ] , α ) is aHom-Lie algebra, where [ x, y ] = µ ( x, y ) − µ ( y, x ) for all x, y ∈ A . Hom-associative algebras are triples ( A, · , α ) consisting of a linear space A over afield K , a bilinear product µ : A × A → A and a linear map α : A → A , satisfyingfor all x, y, z ∈ A , µ ( µ ( x, y ) , α ( z )) = µ ( α ( x ) , µ ( y, z )) . (Hom-associativity) (20)4) Let G be a subgroup of the permutations group S . Hom-algebra ( A, µ, α ) is saidto be G -Hom-associative if X σ ∈ G ( − ε ( σ ) ( µ ( µ ( x σ (1) , x σ (2) ) , α ( x σ (3) )) − µ ( α ( x σ (1) ) , µ ( x σ (2) , x σ (3) )) = 0 , (21) where x i ∈ A, i = 1 , , and ( − ε ( σ ) is the signature of the permutation σ . For any Hom-algebra (
A, µ, α ), the Hom-associator, called also α -associator of µ , isa trilinear map (ternary product) a α,µ : A × A × A → A defined by a α,µ ( x , x , x ) = µ ( µ ( x , x ) , α ( x )) − µ ( α ( x ) , µ ( x , x ))for all x , x , x ∈ A . The ordinary associator a µ ( x , x , x ) = a id ,µ ( x , x , x ) = µ (( x , x ) , ( x )) − µ (( x ) , µ ( x , x ))on an algebra ( A, µ ) is α -associator for the Hom-algebra ( A, µ, α ) = (
A, µ, id) with α = id : A → A , the identity map on A .Using Hom-associator a α,µ and notation σ ( x , x , x ) = ( x σ (1) , x σ (2) , x σ (3) ), theHom-associativity (20) can be written as a α,µ ( x, y, z ) = µ ( µ ( x, y ) , α ( z )) − µ ( α ( x ) , µ ( y, z )) = 0 , (Hom-associativity) (22) r as a α,µ = 0, and the G -Hom-associativity (21) as X σ ∈ G ( − ε ( σ ) a α,µ ◦ σ = 0 . (23)If µ is the multiplication of a Hom-Lie admissible Lie algebra, then (21) is equivalentto [ x, y ] = µ ( x, y ) − µ ( y, x ) satisfying the Hom-Jacobi identity, or equivalently, X σ ∈ S ( − ε ( σ ) ( µ ( µ ( x σ (1) , x σ (2) ) , α ( x σ (3) )) − µ ( α ( x σ (1) ) , µ ( x σ (2) , x σ (3) ))) = 0 , (24)which may be written as X σ ∈ S ( − ε ( σ ) a α,µ ◦ σ = 0 . (25)Thus, Hom-Lie admissible Hom-algebras are S -associative Hom-algebras. In general,for all subgroups G of the permutations group S , all G -Hom-associative Hom-algebrasare Hom-Lie admissible, or in other words, all Hom-algebras from the six classes of G -Hom-associative Hom-algebras, corresponding to the six subgroups of the symmetricgroup S , are Hom-Lie admissible [48, Proposition 3.4]. All six subgroups of S are G = S (id) = { id } , G = S ( τ ) = { id , τ } , G = S ( τ ) = { id , τ } ,G = S ( τ ) = { id , τ } , G = A , G = S where A is the alternating group and τ ij is the transposition of i and j . Table 1: G -Hom-associative algebras Subgroupof S Hom-algebrasclass names Defining Identity(Notation: µ ( a, b ) = ab ) G = S (id) Hom-associative α ( x )( yz ) = ( xy ) α ( z ) G = S ( τ ) Hom-left symmetricHom-Vinberg α ( x )( yz ) − α ( y )( xz ) = ( xy ) α ( z ) − ( yx ) α ( z ) G = S ( τ ) S ( τ )-Hom-associativeHom-right symmetricHom-pre-Lie α ( x )( yz ) − α ( x )( zy ) = ( xy ) α ( z ) − ( xz ) α ( y ) G = S ( τ ) S ( τ )-Hom-associativeHom-anti-flexibleHom-center symmetric α ( x )( yz ) − α ( z )( yx ) = ( xy ) α ( z ) − ( zy ) α ( x ) G = A A -Hom-associative α ( x )( yz ) + α ( y )( zx ) + α ( z )( xy ) =( xy ) α ( z ) + ( yz ) α ( x ) + ( zx ) α ( y ) G = S Hom-Lie admissible X σ ∈ S ( − ε ( σ ) (cid:0) ( x σ (1) x σ (2) ) α ( x σ (3) ) − α ( x σ (1) )( x σ (2) x σ (3) ) (cid:1) = 0 he skew-symmetric G -Hom-associative Hom-algebras and Hom-Lie algebras formthe same class of Hom-algebras for linear spaces over fields of characteristic differentfrom 2, since then the defining identity of G -Hom-associative algebras is equivalent tothe Hom-Jacobi identity of Hom-Lie algebras when the product µ is skew-symmetric.A Hom-right symmetric (Hom-pre-Lie) algebra is the opposite algebra of a Hom-left-symmetric algebra.Hom-flexible algebras introduced in [48] is a generalization to Hom-algebra contextof flexible algebras [2, 53, 55]. Definition 6.2 ( [48]) . A Hom-algebra ( A, µ, α ) is called flexible if µ ( µ ( x, y ) , α ( x )) = µ ( α ( x ) , µ ( y, x ))) (26) for any x, y in A . Using the α -associator a α,µ ( x, y, z ) = µ ( µ ( x, y ) , α ( z )) − µ ( α ( x ) , µ ( y, z )) , the condi-tion (26) may be written as a α,µ ( x, y, x ) = 0 . (27)Since Hom-associator map a α,µ is a trilinear map, a α,µ ( z − x, y, z − x ) = a α,µ ( z, y, z ) + a α,µ ( x, y, x ) − a α,µ ( x, y, z ) − a α,µ ( z, y, x ) , and hence (27) yields a α,µ ( x, y, z ) = − a α,µ ( z, y, x ) (28)in linear spaces over any field, whereas setting x = z in (28) gives 2 a α,µ ( x, y, x ) = 0,implying that (27) and (28) are equivalent in linear spaces over fields of characteristicdifferent from 2. The equality (28) written in terms of the Hom-algebra producs µ is µ ( µ ( x, y ) , α ( z )) − µ ( α ( x ) , µ ( y, z )) = µ ( α ( z ) , µ ( y, x )) − µ ( µ ( z, y ) , α ( x )) . (29) Definition 6.3.
A Hom-algebra ( A, µ, α ) is called anti-flexible if µ ( µ ( x, y ) , α ( z )) − µ ( µ ( z, y ) , α ( x )) = µ ( µ ( α ( x ) , µ ( y, z )) − µ ( µ ( α ( z ) , µ ( y, x )) (30) for all x, y, z ∈ A . The equality (30) can be written as a α,µ ( x, y, z ) = a α,µ ( z, y, x ) , (31)in terms of the Hom-associator a α,µ ( x, y, z ).Hom-anti-flexible algebras were first introduced in [48] as S ( τ )-Hom-associativealgebras, the subclass of G -Hom-associative algebras corresponding to the subgroup G = S ( τ ) ⊂ S (see Table 1). In view of (31), anti-flexible algebras have been calledHom-center symmetric in [32].Note that (31) differs from (28) by absence of the minus sign on the right hand side,meaning that for any y , the bilinear map a α,µ ( ., y, . ) is symmetric on Hom-anti-flexiblealgebras and skew-symmetric on Hom-flexible algebras. Unlike (26) and (28) in Hom-flexible algebras, in Hom-anti-flexible algebras, (31) is generally not equivalent to therestriction of (31) to z = x trivially identically satisfied for any x and y . In view of(31), Hom-anti-flexible algebras are called Hom-center-symmetric algebras in [32]. Nearly Hom-associative algebras, bimodulesand matched pairs
Definition 7.1.
A nearly Hom-associative algebra is a triple ( A, ∗ , α ) , where A is alinear space endowed to the bilinear product ∗ : A × A → A and α : A → A is a linearmap such that for all x, y, z ∈ A , α ( x ) ∗ ( y ∗ z ) = ( z ∗ x ) ∗ α ( y ) . (32)Nearly Hom-associative algebras are Hom-Lie admissible. Proposition 7.2.
Any nearly Hom-associative algebra ( A, ∗ , α ) is Hom-Lie admissible,that is ( A, [ ., . ] , α ) is a Hom-Lie algebra, where [ x, y ] = x ∗ y − y ∗ x for all x, y ∈ A .Proof. Let ( A, ∗ , α ) be a nearly Hom-associative algebra. The commutator is skew-symmetric since [ x, y ] = x ∗ y − y ∗ x = − ( y ∗ x − x ∗ y ) = − [ y, x ] . For all x, y, z ∈ A ,[ α ( x ) , [ y, z ]] + [ α ( y ) , [ z, x ]] + [ α ( z ) , [ x, y ]]= [ α ( x ) , y ∗ z − z ∗ y ] + [ α ( y ) , z ∗ x − x ∗ z ] + [ α ( z ) , x ∗ y − y ∗ x ]= α ( x ) ∗ ( y ∗ z ) − α ( x ) ∗ ( z ∗ y ) − ( y ∗ z ) ∗ α ( x )+ ( z ∗ y ) ∗ α ( x ) + α ( y ) ∗ ( z ∗ x ) − α ( y ) ∗ ( x ∗ z ) − ( z ∗ x ) ∗ α ( y ) + ( x ∗ z ) ∗ α ( y ) + α ( z ) ∗ ( x ∗ y ) − α ( z ) ∗ ( y ∗ x ) − ( x ∗ y ) ∗ α ( z ) + ( y ∗ x ) ∗ α ( z )= { α ( x ) ∗ ( y ∗ z ) − ( z ∗ x ) ∗ α ( y ) } + { ( y ∗ x ) ∗ α ( z ) − α ( x ) ∗ ( z ∗ y ) } + { α ( y ) ∗ ( z ∗ x ) − ( x ∗ y ) ∗ α ( z ) } + { α ( z ) ∗ ( x ∗ y ) − ( y ∗ z ) ∗ α ( x ) } + { ( z ∗ y ) ∗ α ( x ) − α ( y ) ∗ ( x ∗ z ) } + { ( x ∗ z ) ∗ α ( y ) − α ( z ) ∗ ( y ∗ x ) } = 0 . Therefore, ( A, [ ., . ] , α ) is a Hom-Lie algebra.Commutative nearly Hom-associative algebras are Hom-anti-flexible. Proposition 7.3. If ( A, ∗ , α ) is a commutative nearly Hom-associative algebra, then ( A, ∗ , α ) is a Hom-anti-flexible algebra.Proof. In a commutative nearly Hom-associative algebra ( A, ∗ , α ). a α, ∗ ( x, y, z ) = ( x ∗ y ) ∗ α ( z ) − α ( x ) ∗ ( y ∗ z )= α ( y ) ∗ ( z ∗ x ) − ( z ∗ x ) ∗ α ( y ) (nearly Hom-associativity)= α ( y ) ∗ ( x ∗ z ) − ( x ∗ z ) ∗ α ( y ) (commutativity)= ( z ∗ y ) ∗ α ( x ) − α ( z ) ∗ ( y ∗ x ) (nearly Hom-associativity)= a α, ∗ ( z, y, x ) . So any commutative nearly Hom-associative algebra is a Hom-anti-flexible algebra. efinition 7.4. A bimodule of a nearly Hom-associative algebra ( A, ∗ , α ) is a quadru-ple ( l, r, V, ϕ ) , where V is a linear space, l, r : A → End( V ) are two linear maps and ϕ ∈ End( V ) satisfying the relations, for all x, y ∈ A , ϕ ◦ l ( x ) = l ( α ( x )) ◦ ϕ, ϕ ◦ r ( x ) = r ( α ( x )) ◦ ϕ, (33a) l ( α ( x )) ◦ l ( y ) = r ( α ( y )) ◦ r ( x ) , (33b) l ( α ( x )) ◦ r ( y ) = l ( y ∗ x ) ◦ ϕ, (33c) r ( α ( x )) ◦ l ( y ) = r ( x ∗ y ) ◦ ϕ. (33d) Proposition 7.5.
Consider a nearly Hom-associative ( A, ∗ , α ) . Let l, r : A → End( V ) be two linear maps such that V is a linear space and ϕ ∈ End( V ) . The quadruple ( l, r, V, ϕ ) is a bimodule of ( A, ∗ , α ) if and only if there is a structure of a nearly Hom-associative algebra ⋆ on A ⊕ V given by, for all x, y ∈ A and all u, v ∈ V , ( α ⊕ ϕ )( x + u ) = α ( x ) + ϕ ( u ) , (34)( x + u ) ⋆ ( y + v ) = ( x ∗ y ) + ( l ( x ) v + r ( y ) u ) . (35) Definition 7.6.
A representation of a Hom-Lie algebra ( G , [ ., . ] G , α G ) on a linear space V with respect to ψ ∈ End( V ) is a linear map ρ G : G → End( V ) obeying for all x, y ∈ G , ρ G ( α G ( x )) ◦ ψ = ψ ◦ ρ G ( x ) , (36) ρ G ([ x, y ] G ) ◦ ψ = ρ G ( α G ( x )) ◦ ρ G ( y ) − ρ G ( α G ( y )) ◦ ρ G ( x ) . (37) Proposition 7.7.
Let ( A, · , α ) be a nearly Hom-associative algebra and V be a finite-dimensional linear space over the field K such that ( l, r, ϕ, V ) is a bimodule of ( A, · , α ) ,where l, r : A → End( V ) are two linear maps and ϕ ∈ End( V ) . Then the linear map l − r : A → End( V ) , x l ( x ) − r ( x ) is a representation of the underlying Hom-Liealgebra ( G ( A ) , α ) associated to the nearly Hom-associative algebra ( A, · , α ) .Proof. Let ( A, · , α ) be a nearly Hom-associative algebra and V a finite-dimensionallinear space over the field K such that ( l, r, ϕ, V ) is a bimodule of ( A, · , α ), where l, r : A → End( V ) are two linear maps and ϕ ∈ End( V ). For all x, y ∈ A ,( l − r )( α ( x )) ◦ ϕ = l ( α ( x )) ◦ ϕ − r ( α ( x )) ◦ ϕ = ϕ ◦ l ( x ) − ϕ ◦ r ( x ) = ϕ ◦ ( l − r )( x ) , ( l − r )(( α ( x ))) ◦ ( l − r )( y ) − ( l − r )(( α ( y ))) ◦ ( l − r )( x )= l ( α ( x )) ◦ l ( y ) − l ( α ( x )) ◦ r ( y ) − r ( α ( x )) ◦ l ( y ) + r ( α ( x )) ◦ r ( y ) − l ( α ( y )) ◦ l ( x ) + l ( α ( y )) ◦ r ( x ) + r ( α ( y )) ◦ l ( x ) − r ( α ( y )) ◦ r ( x )= { l ( α ( x )) ◦ l ( y ) − r ( α ( y )) ◦ r ( x ) } − l ( α ( x )) ◦ r ( y ) − r ( α ( x )) ◦ l ( y )+ { r ( α ( x )) ◦ r ( y ) − l ( α ( y )) ◦ l ( x ) } + r ( α ( y )) ◦ l ( x ) + l ( α ( y )) ◦ r ( x )= r ( α ( y )) ◦ l ( x ) − l ( α ( x )) ◦ r ( y ) + l ( α ( y )) ◦ r ( x ) − r ( α ( x )) ◦ l ( y )= r ( y · x ) ◦ ϕ − l ( y · x ) ◦ ϕ + l ( x · y ) ◦ ϕ − r ( x · y ) ◦ ϕ = ( l − r )([ x, y ]) ◦ ϕ. Therefore, (36) and (37) are satisfied.
Definition 7.8.
Let ( G , [ ., . ] G , α G ) and ( H , [ ., . ] H , α H ) be two Hom-Lie algebras. Let ρ H : H → End( G ) and µ G : G → End( H ) be two Hom-Lie algebra representations, and α G : G → G and α H : H → H two linear maps such that for all x, y ∈ G , a, b ∈ H ,µ G ( α G ( x )) [ a, b ] H = (cid:2) µ G ( x ) a, α H ( b ) (cid:3) H + (cid:2) α H ( a ) , µ G ( x ) b (cid:3) H − µ G ( ρ H ( a ) x )( α H ( b )) + µ G ( ρ H ( b ) x )( α H ( a )) , (38a) H ( α H ( a )) [ x, y ] G = (cid:2) ρ H ( a ) x, α G ( y ) (cid:3) G + (cid:2) α G ( x ) , ρ H ( a ) y (cid:3) G − ρ H ( µ G ( x ) a )( α G ( y )) + ρ H ( µ G ( y ) a )( α G ( x )) . (38b) Then, ( G , H , µ, ρ, α G , α H ) is called a matched pair of the Hom-Lie algebras G and H ,and denoted by H ⊲⊳ ρ H µ G G . In this case, ( G ⊕ H , [ ., . ] G ⊕ H , α G ⊕ α H ) defines a Hom-Liealgebra, where [( x + a ) , ( y + b )] G ⊕ H = [ x, y ] G + ρ H ( a ) y − ρ H ( b ) x + [ a, b ] H + µ G ( x ) b − µ G ( y ) a. (39) Theorem 7.9.
Let ( A, · , α A ) and ( B, ◦ , α B ) be two nearly Hom-associative algebras.Suppose there are linear maps l A , r A : A → End( B ) and l B , r B : B → End( A ) suchthat ( l A , r A , B, α B ) and ( l B , r B , A, α A ) are bimodules of the nearly Hom-associativealgebras ( A, · , α A ) and ( B, ◦ , α B ) , respectively and satisfying the following conditionsfor all x, y ∈ A and a, b ∈ B : α A ( x ) · ( r B ( a ) y ) + ( r B ( l A ( y ) a ) α A ( x ) − ( l B ( a ) x ) · α A ( y ) − l B ( r A ( x ) a ) α A ( y ) = 0 , (40a) α A ( x ) · ( l B ( a ) y ) + r B ( r A ( y ) a ) α A ( x ) − r B ( α B ( a ))( y · x ) = 0 , (40b) l B ( α B ( a ))( x · y ) − ( r B ( a ) y ) · α A ( x ) − l B ( l A ( y ) a ) α A ( x ) = 0 , (40c) α B ( a ) ◦ ( r A ( x ) b ) + r A ( l B ( b ) x ) α B ( a ) − ( l A ( x ) a ) ◦ α B ( b ) − l A ( r B ( a ) x ) α B ( b ) = 0 , (40d) α B ( a ) ◦ ( l A ( x ) b ) + r A ( r B ( b ) x ) α B ( a ) − r A ( α A ( x ))( b ◦ a ) = 0 , (40e) l A ( α A ( x ))( b ◦ a ) − ( r A ( x ) a ) ◦ α B ( b ) − l A ( l B ( a ) x ) α B ( b ) = 0 . (40f) Then, there is a bilinear product defined on A ⊕ B for all x, y ∈ A , and all a, b ∈ B , by ( x + a ) ∗ ( y + b ) = ( x · y + l B ( a ) y + r B ( b ) x ) + ( a ◦ b + l A ( x ) b + r A ( y ) a ) (41) such that ( A ⊕ B, ∗ , α A ⊕ α B ) is a nearly Hom-associative algebra.Proof. Let ( A, · , α A ), ( B, ◦ , α B ) be two nearly Hom-associative algebras, ( l A , r A , B, α B )a bimodule of ( A, · , α A ) and ( l B , r B , A, α A ) a bimodule of ( B, ◦ , α B ). For all x, y ∈ A and all a, b ∈ B ,( α A ( x ) + α B ( a )) ∗ (( y + b ) ∗ ( z + c ))= { ( α A ( x )) · ( l B ( b ) z ) + r B ( r A ( z ) b ) · ( α A ( x )) } + { ( α A ( x )) · ( r B ( c ) y ) + r B ( l A ( y ) c ) α A ( x ) } + ( α A ( x )) · ( y · z ) + l B ( α B ( a ))( y · z ) + l B ( α B ( a ))( l B ( b ) z )+ l B ( α B ( a ))( r B ( c ) y ) + r B ( b ◦ c )( α A ( x ))+ { ( α B ( a )) ◦ ( l A ( y ) c ) + r A ( r B ( c ) y ) α B ( a ) } + { ( α B ( a )) ◦ ( r A ( z ) b ) + r A ( l B ( b ) z ) α B ( a ) } + ( α B ( a )) ◦ ( b ◦ c ) + r A ( y · z ) α B ( a ) + l A ( α A ( x ))( b ◦ c )+ l A ( α A ( x ))( l A ( y ) c ) + l A ( α A ( x ))( r A ( z ) b );(( z + c ) ∗ ( x + a )) ∗ ( α A ( y ) + α B ( b ))= { ( l B ( c ) x ) · ( α A ( y )) + l B ( r A ( x ) c ) α A ( y ) } + { l B ( l A ( z ) a ) α A ( y ) + ( l B ( c ) x ) · α A ( y ) } + ( z · x ) · ( α A ( y )) + l B ( c ◦ a )( α A ( y )) + r B ( α B ( b )( z · x )+ r B ( α B ( b )( l B ( c ) x ) + r B ( α B ( b )( r B ( a ) z ) { ( l A ( z ) a ) ◦ ( α B ( b )) + l A ( r B ( a ) z ) α B } + { ( r A ( x ) c ) ◦ ( α B ( b )) + l A ( l B ( c ) x ) α B ( b ) } + ( c ◦ a ) ◦ ( α B ( b )) + l A ( z · x )( α B ( b )) + r A ( α A ( y ))( c ◦ a )+ r A ( α A ( y ))( l A ( z ) a ) + r A ( α A ( y ))( r A ( x ) c )Using (40a) - (40f) and the fact that ( l A , r A , B, α B ) and ( l B , r B , A, α A ) are bimodulesof the nearly Hom-associative algebras ( A, · , α A ) and ( B, ◦ , α B ), respectively, we obtainthat ( A ⊕ B, ∗ , α A ⊕ α B ) is a nearly associative algebra. Definition 7.10.
A matched pair of the nearly Hom-associative algebras ( A, · , α A ) and ( B, ◦ , α B ) is the high-tuple ( A, B, l A , r A , α B , l B , r B , α A ) , where l A , r A : A → End( B ) and l B , r B : B → End( A ) are linear maps such that ( l A , r A , B, α B ) and ( l B , r B , A, α A ) are bimodules of the nearly Hom-associative algebras ( A, · , α A ) and ( B, ◦ , α B ) , respec-tively, and satisfying (40a) - (40f) . Corollary 7.11.
Let ( A, B, l A , r A , α B , l B , r B , α A ) be a matched pair of the nearlyHom-associative algebras ( A, · , α A ) and ( B, ◦ , α B ) . Then, ( G ( A ) , G ( B ) , l A − r A , l B − r B , α A , α B ) is a matched pair of the underlying Hom-Lie algebras G ( A ) and G ( B ) ofthe nearly Hom-associative algebras ( A, · , α A ) and ( B, ◦ , α B ) .Proof. Let (
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