aa r X i v : . [ m a t h . R A ] M a y ANTI-FLEXIBLE BIALGEBRAS
MAFOYA LANDRY DASSOUNDO ⋆ , CHENGMING BAI † , AND MAHOUTON NORBERT HOUNKONNOU ‡ Abstract.
We establish a bialgebra theory for anti-flexible algebras in this paper. We intro-duce the notion of an anti-flexible bialgebra which is equivalent to a Manin triple of anti-flexiblealgebras. The study of a special case of anti-flexible bialgebras leads to the introduction of anti-flexible Yang-Baxter equation in an anti-flexible algebra which is an analogue of the classicalYang-Baxter equation in a Lie algebra or the associative Yang-Baxter equation in an associativealgebra. It is a unexpected consequence that both the anti-flexible Yang-Baxter equation and theassociative Yang-Baxter equation have the same form. A skew-symmetric solution of anti-flexibleYang-Baxter equation gives an anti-flexible bialgebra. Finally the notions of an O -operator of ananti-flexible algebra and a pre-anti-flexible algebra are introduced to construct skew-symmetricsolutions of anti-flexible Yang-Baxter equation. Contents
1. Introduction 12. Bimodules and matched pairs of anti-flexible algebras 33. Manin triples of anti-flexible algebras and anti-flexible bialgebras 54. A special class of anti-flexible bialgebras 85. O -operators of anti-flexible algebras and pre-anti-flexible algebras 13References 161. Introduction
At first, we recall the definition of a flexible algebra.
Definition 1.1.
Let A be a vector space over a field F equipped with a bilinear product ( x, y ) → xy . Set the associator as ( x, y, z ) = ( xy ) z − x ( yz ) , ∀ x, y, z ∈ A. (1.1) A is called a flexible algebra if the following identity is satisfied( x, y, x ) = 0 , or equivalently , ( xy ) x = x ( yx ) , ∀ x, y ∈ A. (1.2)As a natural generalization of associative algebras, flexible algebras were studied widely. Forexample, using the solvability and reducibility of the radicals of their underlying Lie algebras,finite-dimensional flexible Lie-admissible algebras were characterized in [5]; any simple strictlypower-associative algebra of characteristic prime to 6 of degree greater than 2 is a flexible algebra([14]). Note that the “linearization” of the identity (1.2) gives the following equivalent identityby substituting x + z for x in Eq. (1.2):( x, y, z ) + ( z, y, x ) = 0 , ∀ x, y, z ∈ A. (1.3) Mathematics Subject Classification.
Key words and phrases. anti-flexible algebra, anti-flexible bialgebra, anti-flexible Yang-Baxter equation, O -operator. It is also natural to consider certain generalization of flexible algebras which leads to the intro-duction of several classes of nonassociative algebras [17]. In particular, the so-called anti-flexiblealgebras were introduced as follows.
Definition 1.2.
Let A be a vector space equipped with a bilinear product ( x, y ) → xy . A iscalled an anti-flexible algebra if the following identity is satisfied( x, y, z ) = ( z, y, x ) , or equivalently , ( xy ) z − x ( yz ) = ( zy ) x − z ( yx ) , ∀ x, y, z ∈ A. (1.4)Note that the identity (1.4) means that the associator (1.1) is symmetric in x, z and thusan anti-flexible algebra is also called a center-symmetric algebra in [12] (it is also calleda G -associative algebra in [11]). The study of anti-flexible algebras is fruitful, too. Forexample, simplicity and semi-simplicity of anti-flexible algebras were investigated in [18]; thesimple, semisimple (totally) anti-flexible algebras over splitting fields of characteristic different to2 and 3 were studied and classified in [6, 19, 20]; the primitive structures and prime anti-flexiblerings were investigated in [7]; furthermore, it were shown that a simple nearly anti-flexible algebraof characteristic prime to 30 satisfying the identity ( x, x, x ) = 0 in which its commutator givesnon-nilpotent structure possesses a unity element ([8]).On the other hand, a bialgebra structure on a given algebraic structure is obtained as a coal-gebra structure together which gives the same algebraic structure on the dual space with a setof compatibility conditions between the multiplications and comultiplications. One of the mostfamous examples of bialgebras is the Lie bialgebra ([9]) and more importantly there have been alot of bialgebra theories for other algebra structures that essentially follow the approach of Liebialgebras such as antisymmetric infinitesimal bialgebras ([1, 3]), left-symmetric bialgebras ([2]),alternative D-bialgebras ([10]) and Jordan bialgebras ([21]).In this paper, we give a bialgebra theory for anti-flexible algebras. We still take a similarapproach as of the study on Lie bialgebras, that is, the compatibility condition is still decided byan analogue of Manin triple of Lie algebras, which we call a Manin triple of anti-flexible algebras.The notion of anti-flexible bialgebra is thus introduced as an equivalent structure of a Manin tripleof anti-flexible algebras, which is interpreted in terms of matched pairs of anti-flexible algebras.Here the dual bimodule of a bimodule of an anti-flexible algebra plays an important role. Wewould like to point out that both anti-flexible and associative algebras have the same forms ofdual bimodules, which is quite different from other generalizations of associative algebras such asleft-symmetric algebras ([2]) or other G -associative algebras in [11].Although to our knowledge, a well-constructed cohomology theory for anti-flexible algebrasis unknown yet, we still consider a special case of anti-flexible bialgebras following the studyof coboundary Lie bialgebras for Lie algebras ([9]) or coboundary antisymmetric infinitesimalbialgebras for associative algebras ([3]). The study of such a class of anti-flexible bialgebrasalso leads to the introduction of anti-flexible Yang-Baxter equation in an anti-flexible equationwhich is an analogue of the classical Yang-Baxter equation in a Lie algebra or the associative Yang-Baxter equation in an associative algebra. A skew-symmetric solution of anti-flexible Yang-Baxterequation gives an anti-flexible bialgebra.There is an unexpected consequence that both the anti-flexible Yang-Baxter equation and theassociative Yang-Baxter equation have the same form. It is partly due to the fact that bothanti-flexible and associative algebras have the same forms of dual bimodules. Therefore someproperties of anti-flexible Yang-Baxter equation can be obtained directly from the correspondingones of associative Yang-Baxter equation. NTI-FLEXIBLE BIALGEBRAS 3
In particular, as for the study on the associative Yang-Baxter equation, in order to obtainskew-symmetric solutions of anti-flexible Yang-Baxter equation, we introduce the notions of an O -operator of an anti-flexible algebra which is an analogue of an O -operator of a Lie algebra intro-duced by Kupershmidt in [13] as a natural generalization of the classical Yang-Baxter equation in aLie algebra, and a pre-anti-flexible algebra. The former gives a construction of skew-symmetric so-lutions of anti-flexible Yang-Baxter equation in a semi-direct product anti-flexible algebra, whereasthe latter as a generalization of a dendriform algebra ([15]) gives a bimodule of the associatedanti-flexible algebra such that the identity is a natural O -operator associated to it. Therefore aconstruction of skew-symmetric solutions of anti-flexible Yang-Baxter equation and hence anti-flexible bialgebras from pre-anti-flexible algebras is given. Note that from the point of view ofoperads, pre-anti-flexible algebras are the splitting of anti-flexible algebras ([4, 16]).The paper is organized as follows. In Section 2, we study bimodules and matched pairs ofanti-flexible algebras. In particular, we give the dual bimodule of a bimodule of an anti-flexiblealgebra. In Section 3, we give the notion of a Manin triple of anti-flexible algebras and theninterpret it in terms of matched pairs of anti-flexible algebras. The notion of an anti-flexiblebialgebra is thus introduced as an equivalent structure of a Manin triple of anti-flexible algebras.In Section 4, we consider the special class of anti-flexible bialgebras which lead to the introductionof anti-flexible Yang-Baxter equation. A skew-symmetric solution of anti-flexible Yang-Baxterequation gives such a anti-flexible bialgebra. In Section 5, we introduce the notions of an O -operator of an anti-flexible algebra and a pre-anti-flexible algebra. The relationships betweenthem and the anti-flexible Yang-Baxter equation are given. In particular, we give constructions ofskew-symmetric solutions of anti-flexible Yang-Baxter equation from O -operators of anti-flexiblealgebras and pre-anti-flexible algebras.Throughout this paper, all vector spaces are finite-dimensional over a base field F whose char-acteristic is not 2, although many results still hold in the infinite dimension.2. Bimodules and matched pairs of anti-flexible algebras
In this section, we first introduce the notion of a bimodule of an anti-flexible algebra. Then westudy the dual bimodule of a bimodule of an anti-flexible algebra. We also give the notion of amatched pair of anti-flexible algebras.
Definition 2.1.
Let ( A, · ) be an anti-flexible algebra and V be a vector space. Let l, r : A → End( V ) be two linear maps. If for any x, y ∈ A , l ( x · y ) − l ( x ) l ( y ) = r ( x ) r ( y ) − r ( y · x ) , (2.1) l ( x ) r ( y ) − r ( y ) l ( x ) = l ( y ) r ( x ) − r ( x ) l ( y ) , (2.2)then it is called a bimodule of ( A, · ), denoted by ( l, r, V ). Two bimodules ( l , r , V ) and ( l , r , V )of an anti-flexible algebra A is called equivalent if there exists a linear isomorphism ϕ : V → V satisfying ϕl ( x ) = l ( x ) ϕ, ϕr ( x ) = r ( x ) ϕ, ∀ x ∈ A. (2.3) Remark 2.2.
Note that if both sides of Eqs. (2.1) and (2.2) are zero, then they exactly give thedefinition of a bimodule of an associative algebra.Let ( A, · ) be an anti-flexible algebra. For any x, y ∈ A , let L x and R x denote the left and rightmultiplication operators respectively, that is, L x ( y ) = xy and R x ( y ) = yx . Let L, R : A → End( A )be two linear maps with x → L x and x → R x for any x ∈ A respectively. MAFOYA LANDRY DASSOUNDO, CHENGMING BAI, AND MAHOUTON NORBERT HOUNKONNOU
Example 2.3.
Let ( A, · ) be an anti-flexible algebra. Then ( L, R, A ) is a bimodule of ( A, · ), whichis called the regular bimodule of ( A, · ). Proposition 2.4.
Let ( A, · ) be an anti-flexible algebra and V be a vector space. Let l, r : A → End( V ) be two linear maps. Then ( l, r, V ) is a bimodule of ( A, · ) if and only if the direct sum A ⊕ V of vector spaces is turned into an anti-flexible algebra by defining the multiplication in A ⊕ V by ( x + u ) ∗ ( y + v ) = x · y + l ( x ) v + r ( y ) u, ∀ x, y ∈ A, u, v ∈ V. (2.4) We call it semi-direct product and denote it by A ⋉ l,r V or simply A ⋉ V. Proof.
It is straightforward or follows from Theorem 2.9 as a direct consequence. (cid:3)
It is known that an anti-flexible algebra is a Lie-admissible algebra ([11]).
Proposition 2.5.
Let ( A, · ) be an anti-flexible algebra. Define the commutator by [ x, y ] = x · y − y · x, ∀ x, y ∈ A. (2.5) Then it is a Lie algebra and we denote it by ( g ( A ) , [ , ]) or simply g ( A ) , which is called theassociated Lie algebra of ( A, · ) . Corollary 2.6.
Let ( l, r, V ) be a bimodule of an anti-flexible algebra ( A, · ) . Then ( l − r, V ) is arepresentation of the associated Lie algebra ( g ( A ) , [ , ]) .Proof. For any x, y ∈ A , we have[( l − r )( x ) , ( l − r )( y )] = [ l ( x ) , l ( y )] + [ r ( x ) , r ( y )] − [ l ( x ) , r ( y )] − [ r ( x ) , l ( y )]= [ l ( x ) , l ( y )] + [ r ( x ) , r ( y )] = l ( x · y − y · x ) − r ( x · y − y · x )= ( l − r )([ x, y ]) . Hence ( l − r, V ) is a representation of ( g ( A ) , [ , ]). (cid:3) Let ( A, · ) be an anti-flexible algebra. Let V be a vector space and α : A → End( V ) be a linearmap. Define a linear map α ∗ : A → End( V ∗ ) as h α ∗ ( x ) u ∗ , v i = h u ∗ , α ( x ) v i , ∀ x ∈ A, v ∈ V, u ∗ ∈ V ∗ , (2.6)where h , i is the usual pairing between V and the dual space V ∗ . Proposition 2.7.
Let ( l, r, V ) be a bimodule of an anti-flexible algebra ( A, · ) . Then ( r ∗ , l ∗ , V ∗ ) isbimodule of ( A, · ) .Proof. For all x, y ∈ A, u ∗ ∈ V ∗ , v ∈ V , we have h ( r ∗ ( x · y ) − r ∗ ( x ) r ∗ ( y )) u ∗ , v i = h u ∗ , ( r ( x · y ) − r ( y ) r ( x ))( v ) i = h u ∗ , ( l ( y ) l ( x ) − l ( y · x ))( v ) i = h ( l ∗ ( x ) l ∗ ( y ) − l ∗ ( y · x )) u ∗ , v i ; h ( l ∗ ( x ) r ∗ ( y ) − r ∗ ( y ) l ∗ ( x )) u ∗ , v i = h u ∗ , ( r ( y ) l ( x ) − l ( x ) r ( y ))( v ) i = h u ∗ , ( r ( x ) l ( y ) − l ( y ) r ( x ))( v ) i = h ( l ∗ ( y ) r ∗ ( x ) − r ∗ ( x ) l ∗ ( y )) u ∗ , v i . Hence ( r ∗ , l ∗ , V ∗ ) is bimodule of ( A, · ). (cid:3) Remark 2.8.
Note that for a bimodule ( l, r, V ) of an associative algebra, ( r ∗ , l ∗ , V ∗ ) is also abimodule. Therefore, for both associative and anti-flexible algebras, the “dual bimodules” in theabove sense have the same form. NTI-FLEXIBLE BIALGEBRAS 5
Theorem 2.9. ([12])
Let ( A, · ) and ( B, ◦ ) be two anti-flexible algebras. Suppose that there are fourlinear maps l A , r A : A → End( B ) and l B , r B : B → End( A ) such that ( l A , r A , B ) and ( l B , r B , A ) are bimodules of ( A, · ) and ( B, ◦ ) respectively, obeying the following relations: l B ( a )( x · y ) + r B ( a )( y · x ) − r B ( l A ( x ) a ) y − y · ( r B ( a ) x ) − l B ( r A ( x ) a ) y − ( l B ( a ) x ) · y = 0 , (2.7) l A ( x )( a ◦ b ) + r A ( x )( b ◦ a ) − r A ( l B ( a ) x ) b − b ◦ ( r A ( x ) a ) + l A ( r B ( a ) x ) b − ( l A ( x ) a ) ◦ b = 0 , (2.8) y · ( l B ( a ) x ) + ( r B ( a ) x ) · y − ( r B ( a ) y ) · x − l B ( l A ( y ) a ) x + r B ( r A ( x ) a ) y + l B ( l A ( x ) a ) y − x · ( l B ( a ) y ) − r B ( r A ( y ) a ) x = 0 , (2.9) b ◦ ( l A ( x ) a ) + ( r A ( x ) a ) ◦ b − ( r A ( x ) b ) ◦ a − l A ( l B ( b ) x ) a + r A ( r B ( a ) x ) b + l A ( l B ( a ) x ) b − a ◦ ( l A ( x ) b ) − r A ( r B ( b ) x ) a = 0 , (2.10) for any x, y ∈ A, a, b ∈ B . Then there is an anti-flexible algebra structure on A ⊕ B given 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 ) , ∀ x, y ∈ A, a, b ∈ B. (2.11) Conversely, every anti-flexible algebra which is a direct sum of the underlying vector spaces of twosubalgebras can be obtained from the above way.
Definition 2.10.
Let ( A, · ) and ( B, ◦ ) be two anti-flexible algebras. Suppose that there arefour linear maps l A , r A : A → End( B ) and l B , r B : B → End( A ) such that ( l A , r A , B ) and( l B , r B , A ) are bimodules of ( A, · ) and ( B, ◦ ) and Eqs. (2.7)-(2.10) hold. Then we call the six-tuple( A, B, l A , r A , l B , r B ) a matched pair of anti-flexible algebras . We also denote the anti-flexiblealgebra defined by Eq. (2.11) by A ⊲⊳ l A ,r A l B ,r B B or simply by A ⊲⊳ B .3.
Manin triples of anti-flexible algebras and anti-flexible bialgebras
In this section, we introduce the notions of a Manin triple of anti-flexible algebras and ananti-flexible bialgebra. The equivalence between them is interpreted in terms of matched pairs ofanti-flexible algebras.
Definition 3.1.
A bilinear form B on an anti-flexible algebra ( A, · ) is called invariant if B ( x · y, z ) = B ( x, y · z ) , ∀ x, y, z ∈ A. (3.1) Proposition 3.2.
Let ( A, · ) be an anti-flexible algebra. If there is a nondegenerate symmetricinvariant bilinear form B on A , then as bimodules of the anti-flexible algebra ( A, · ) , ( L, R, A ) and ( R ∗ , L ∗ , A ∗ ) are equivalent. Conversely, if as bimodules of an anti-flexible algebra ( A, · ) , ( L, R, A ) and ( R ∗ , L ∗ , A ∗ ) are equivalent, then there exists a nondegenerate invariant bilinear form B on A .Proof. Since B is nondegenerate, there exists a linear isomorphism ϕ : A → A ∗ defined by h ϕ ( x ) , y i = B ( x, y ) , ∀ x, y ∈ A. Hence for any x, y, z ∈ A , we have h ϕL ( x ) y, z i = B ( x · y, z ) = B ( z, x · y ) = B ( z · x, y ) = h ϕ ( y ) , z · x i = h R ∗ ( x ) ϕ ( y ) , z i ; h ϕR ( x ) y, z i = B ( y · x, z ) = B ( y, x · z ) = h ϕ ( y ) , x · z i = h L ∗ ( x ) ϕ ( y ) , z i . Hence (
L, R, A ) and ( R ∗ , L ∗ , A ∗ ) are equivalent. Conversely, by a similar way, we can get theconclusion. (cid:3) MAFOYA LANDRY DASSOUNDO, CHENGMING BAI, AND MAHOUTON NORBERT HOUNKONNOU
Definition 3.3. A Manin triple of anti-flexible algebras is a triple of anti-flexible algebras(
A, A + , A − ) together with a nondegenerate symmetric invariant bilinear form B on A such thatthe following conditions are satisfied.(a) A + and A − are anti-flexible subalgebras of A ;(b) A = A + ⊕ A − as vector spaces;(c) A + and A − are isotropic with respect to B , that is, B ( x + , y + ) = B ( x − , y − ) = 0 for any x + , y + ∈ A + , x − , y − ∈ A − .A isomorphism between two Manin triples ( A, A + , A − ) and ( B, B + , B − ) of anti-flexible algebrasis an isomorphism ϕ : A → B of anti-flexible algebras such that ϕ ( A + ) = B + , ϕ ( A − ) = B − , B A ( x, y ) = B B ( ϕ ( x ) , ϕ ( y )) , ∀ x, y ∈ A. (3.2) Definition 3.4.
Let ( A, · ) be an anti-flexible algebra. Suppose that “ ◦ ” is an anti-flexible algebrastructure on the dual space A ∗ of A and there is an anti-flexible algebra structure on the directsum A ⊕ A ∗ of the underlying vector spaces of A and A ∗ such that ( A, · ) and ( A ∗ , ◦ ) are subalgebrasand the natural symmetric bilinear form on A ⊕ A ∗ given by B d ( x + a ∗ , y + b ∗ ) := h a ∗ , y i + h x, b ∗ i , ∀ x, y ∈ A ; a ∗ , b ∗ ∈ A ∗ , (3.3)is invariant, then ( A ⊕ A ∗ , A, A ∗ ) is called a standard Manin triple of anti-flexible algebrasassociated to B d . Obviously, a standard Manin triple of anti-flexible algebras is a Manin triple of anti-flexiblealgebras. Conversely, we have
Proposition 3.5.
Every Manin triple of anti-flexible algebras is isomorphic to a standard one.Proof.
Since in this case A − and ( A + ) ∗ are identified by the nondegenerate invariant bilinear form,the anti-flexible algebra structure on A − is transferred to ( A + ) ∗ . Hence the anti-flexible algebrastructure on A + ⊕ A − is transferred to A + ⊕ ( A + ) ∗ . Then the conclusion holds. (cid:3) Proposition 3.6.
Let ( A, · ) be an anti-flexible algebra. Suppose that there is an anti-flexiblealgebra structure “ ◦ ” on the dual space A ∗ . Then there exists an anti-flexible algebra structureon the vector space A ⊕ A ∗ such that ( A ⊕ A ∗ , A, A ∗ ) is a standard Manin triple of anti-flexiblealgebras associated to B d defined by Eq. (3.3) if and only if ( A, A ∗ , R ∗· , L ∗· , R ∗◦ , L ∗◦ ) is a matchedpair of anti-flexible algebras.Proof. It follows from the same proof of [3, Theorem 2.2.1]. (cid:3)
Proposition 3.7.
Let ( A, · ) be an anti-flexible algebra. Suppose that there exists an anti-flexiblealgebra structure “ ◦ ” on the dual space A ∗ . Then ( A, A ∗ , R ∗· , L ∗· , R ∗◦ , L ∗◦ ) is a matched pair ofanti-flexible algebras if and only if for any x, y ∈ A, a ∈ A ∗ , − R ∗◦ ( a )( x · y ) − L ∗◦ ( a )( y · x ) + L ∗◦ ( R ∗· ( x ) a ) y + y · ( L ∗◦ ( a ) x ) + R ∗◦ ( L ∗· ( x ) a ) y + ( R ∗◦ ( a ) x ) · y = 0 , (3.4) y · ( R ∗◦ ( a ) x ) − x · ( R ∗◦ ( a ) y ) + ( L ∗◦ ( a ) x ) · y − ( L ∗◦ ( a ) y ) · x + L ∗◦ ( L ∗· ( x ) a ) y − R ∗◦ ( R ∗· ( y ) a ) x + R ∗◦ ( R ∗· ( x ) a ) y − L ∗◦ ( L ∗· ( y ) a ) x = 0 . (3.5) Proof.
Obviously, Eq. (3.4) is exactly Eq. (2.8) and Eq. (3.5) is exactly Eq. (2.10) in the case l A = R ∗· , r A = L ∗· , l B = l A ∗ = R ∗◦ , r B = r A ∗ = L ∗◦ . For any x, y ∈ A, a, b ∈ A ∗ , we have: h R ∗◦ ( a )( x · y ) , b i = h x · y, R ◦ ( a ) b i = h x · y, b ◦ a i = h L · ( x ) y, b ◦ a i = h y, L ∗· ( x )( b ◦ a ) i ; h L ∗◦ ( a )( y · x ) , b i = h y · x, L ◦ ( a ) b i = h y · x, a ◦ b i = h R · ( x ) y, a ◦ b i = h y, R ∗· ( x )( a ◦ b ) i ; h L ∗◦ ( R ∗· ( x ) a ) y, b i = h y, L ◦ ( R ∗· ( x ) a ) b i = h y, ( R ∗· ( x ) a ) ◦ b i ; NTI-FLEXIBLE BIALGEBRAS 7 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 ∗◦ ( L ∗· ( x ) a ) y, b i = h y, R ◦ ( L ∗· ( x ) a ) b i = h y, b ◦ ( L ∗· ( x ) a ) i ; h ( R ∗◦ ( a ) x ) · y, b i = h L · ( R ∗◦ ( a ) x ) y, b i = h y, L ∗· ( R ∗◦ ( a ) x ) b i . Then Eq. (2.7) holds if and only if Eq. (2.8) holds. Similarly, Eq. (2.9) holds if and only ifEq. (2.10) holds. Therefore the conclusion holds. (cid:3)
Let V be a vector space. Let σ : V ⊗ V → V ⊗ V be the flip defined as σ ( x ⊗ y ) = y ⊗ x, ∀ x, y ∈ V. (3.6) Theorem 3.8.
Let ( A, · ) be an anti-flexible algebra. Suppose there is an anti-flexible alge-bra structure “ ◦ ” on its dual space A ∗ given by a linear map ∆ ∗ : A ∗ ⊗ A ∗ → A ∗ . Then ( A, A ∗ , R ∗· , L ∗· , R ∗◦ , L ∗◦ ) is a matched pair of anti-flexible algebras if and only if ∆ : A → A ⊗ A satisfies the following two conditions: ∆( x · y ) + σ ∆( y · x ) = ( σ (id ⊗ L · ( y )) + R · ( y ) ⊗ id)∆( x ) + ( σ ( R · ( x ) ⊗ id) + id ⊗ L · ( x ))∆( y ) , (3.7)( σ (id ⊗ R · ( y )) − id ⊗ R · ( y ) − σ ( L · ( y ) ⊗ id) + L · ( y ) ⊗ id)∆( x ) =( σ (id ⊗ R · ( x )) − id ⊗ R · ( x ) − σ ( L · ( x ) ⊗ id) + L · ( x ) ⊗ id)∆( y ) , (3.8) for any x, y ∈ A .Proof. For any x, y ∈ A and any a, b ∈ A ∗ , we have h ∆( x · y ) , a ⊗ b i = h x · y, a · b i , = h L ∗◦ ( a )( x · y ) , b i , h σ ∆( y · x ) , a ⊗ b i = h y · x, b ◦ a i = h R ∗◦ ( a )( y · 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 , h ( R · ( y ) ⊗ id)∆( x ) , a ⊗ b i = h x, ( R ∗· ( y ) a ) ◦ b i = h L ∗◦ ( R ∗· ( y ) a ) x, b i , h σ ( R · ( x ) ⊗ id)∆( y ) , a ⊗ b i = h y, ( R ∗· ( x ) b ) ◦ a i = h ( R ∗◦ ( a ) y ) · x, b i , h (id ⊗ L · ( x ))∆( y ) , a ⊗ b i = h y, a ◦ ( L ∗· ( x ) b ) i = h x · ( L ∗◦ ( a ) y ) , b i . Then Eq. (3.4) is equivalent to Eq. (3.7). Moreover, we have h σ (id ⊗ R · ( y ))∆( x ) , a ⊗ b i = h x, b ◦ ( R ∗· ( y ) a ) i = h R ∗◦ ( R ∗· ( y ) a ) x, b i , h (id ⊗ R · ( y ))∆( x ) , a ⊗ b i = h x, a ◦ ( R ∗· ( y ) b ) i = h ( L ∗◦ ( a ) x ) · 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 ( L · ( y ) ⊗ id)∆( x ) , a ⊗ b i = h x, ( L ∗· ( y ) a ) ◦ b i = h L ∗◦ ( L ∗· ( y ) a ) x, b i . Then Eq. (3.5) is equivalent to Eq. (3.8). Hence the conclusion holds. (cid:3)
Remark 3.9.
From the symmetry of the anti-flexible algebras ( A, · ) and ( A ∗ , ◦ ) in the standardManin triple of anti-flexible algebras associated to B d , we also can consider a linear map γ : A ∗ → A ∗ ⊗ A ∗ such that γ ∗ : A ⊗ A → A gives the anti-flexible algebra structure “ · ” on A . It isstraightforward to show that ∆ satisfies Eqs. (3.7) and (3.8) if and only if γ satisfies γ ( a ◦ b ) + σγ ( b ◦ a ) = ( σ (id ⊗ L ◦ ( b )) + R ◦ ( b ) ⊗ id) γ ( a ) + ( σ ( R ◦ ( a ) ⊗ id) + id ⊗ L ◦ ( a )) γ ( b ) , (3.9)( σ (id ⊗ R ◦ ( b )) − id ⊗ R ◦ ( b ) − σ ( L ◦ ( b ) ⊗ id) + ( L ◦ ( b ) ⊗ id)) γ ( a ) =(( L ◦ ( a ) ⊗ id) − σ ( L ◦ ( a ) ⊗ id) + σ (id ⊗ R ◦ ( a )) − (id ⊗ R ◦ ( a ))) γ ( b ) , (3.10)for any a, b ∈ A ∗ . Definition 3.10.
Let ( A, · ) be an anti-flexible algebra. An anti-flexible bialgebra structure on A is a linear map ∆ : A → A ⊗ A such that(a) ∆ ∗ : A ∗ ⊗ A ∗ → A ∗ defines an anti-flexible algebra structure on A ∗ ;(b) ∆ satisfies Eqs. (3.7) and (3.8). MAFOYA LANDRY DASSOUNDO, CHENGMING BAI, AND MAHOUTON NORBERT HOUNKONNOU
We denote it by ( A, ∆) or ( A, A ∗ ). Example 3.11.
Let ( A, ∆) be an anti-flexible bialgebra on an anti-flexible algebra A . Then ( A ∗ , γ )is an anti-flexible bialgebra on the anti-flexible algebra A ∗ , where γ is given in Remark 3.9.Combining Proposition 3.6 and Theorem 3.8 together, we have the following conclusion. Theorem 3.12.
Let ( A, · ) be an anti-flexible algebra. Suppose that there is an anti-flexible algebrastructure on its dual space A ∗ denoted “ ◦ ” which is defined by a linear map ∆ : A → A ⊗ A . Thenthe following conditions are equivalent. (a) ( A ⊕ A ∗ , A, A ∗ ) is a standard Manin triple of anti-flexible algebras associated to B d definedby Eq. (3.3) . (b) ( A, A ∗ , R ∗· , L ∗· , R ∗◦ , L ∗◦ ) is a matched pair of anti-flexible algebras. (c) ( A, ∆) is an anti-flexible bialgebra. Recall a Lie bialgebra structure on a Lie algebra g is a linear map δ : g → g ⊗ g such that δ ∗ : g ∗ ⊗ g ∗ → g ∗ defines a Lie algebra structure on g ∗ and δ satisfies δ [ x, y ] = (ad( x ) ⊗ id + id ⊗ ad( x )) δ ( y ) − (ad( y ) ⊗ id + id ⊗ ad( y )) δ ( x ) , ∀ x, y ∈ g , (3.11)where ad( x )( y ) = [ x, y ] for any x, y ∈ g . We denoted it by ( g , δ ). Proposition 3.13.
Let ( A, ∆) be an anti-flexible bialgebra. Then ( g ( A ) , δ ) is a Lie bialgebra,where δ = ∆ − σ ∆ .Proof. It is straightforward. (cid:3) A special class of anti-flexible bialgebras
In this section, we consider a special class of anti-flexible bialgebras, that is, the anti-flexiblebialgebra ( A, ∆) on an anti-flexible algebra ( A, · ), with the linear map ∆ defined by∆( x ) = (id ⊗ L · ( x ))r + ( R · ( x ) ⊗ id) σ r , ∀ x ∈ A, (4.1)where r ∈ A ⊗ A . Lemma 4.1.
Let ( A, · ) be an anti-flexible algebra and r ∈ A ⊗ A . Let ∆ : A → A ⊗ A be a linearmap defined by Eq. (4.1) . Then σ ∆( x ) = ( L · ( x ) ⊗ id) σ r + (id ⊗ R · ( x ))r , ∀ x ∈ A. (4.2) Proof.
It is straightforward. (cid:3)
Proposition 4.2.
Let ( A, · ) be an anti-flexible algebra and r ∈ A ⊗ A . Let ∆ : A → A ⊗ A be alinear map defined by Eq. (4.1) . (a) Eq. (3.7) holds if and only if ( L · ( x ) ⊗ R · ( y ) + R · ( x ) ⊗ L · ( y ))(r + σ r) = 0 , ∀ x, y ∈ A. (4.3)(b) Eq. (3.8) holds if and only if ( R · ( x ) ⊗ R · ( y ) − R · ( y ) ⊗ R · ( x ) + L · ( x ) ⊗ L · ( y ) − L · ( y ) ⊗ L · ( x ))(r + σ r) = 0 , ∀ x, y ∈ A. (4.4) Proof. (a) Let x, y ∈ A . By Lemma 4.1, we have∆( x · y ) + σ ∆( y · x ) = (id ⊗ ( L · ( x · y ) + R · ( y · x )))r + (( R · ( x · y ) + L · ( y · x )) ⊗ id) σ r . By the definition of an anti-flexible algebra, we have∆( x · y ) + σ ∆( y · x ) = (id ⊗ ( L · ( x ) L · ( y ) + R · ( x ) R · ( y )))r + (( R · ( y ) R · ( x ) + L · ( y ) L · ( x )) ⊗ id) σ r . NTI-FLEXIBLE BIALGEBRAS 9
Moreover, we have σ (id ⊗ L · ( y ))∆( x ) = σ (id ⊗ L · ( y ))(id ⊗ L · ( x ))r + σ (id ⊗ L · ( y ))( R · ( x ) ⊗ id) σ r= ( L · ( y ) L · ( x ) ⊗ id) σ r + ( L · ( y ) ⊗ id)(id ⊗ R · ( x ))r , ( R · ( y ) ⊗ id)∆( x ) = ( R · ( y ) ⊗ id)(id ⊗ L · ( x ))r + ( R · ( y ) ⊗ id)( R · ( x ) ⊗ id) σ r= ( R · ( y ) ⊗ id)(id ⊗ L · ( x ))r + ( R · ( y ) R · ( x ) ⊗ id) σ r ,σ ( R · ( x ) ⊗ id)∆( y ) = σ ( R · ( x ) ⊗ id)(id ⊗ L · ( y ))r + σ ( R · ( x ) ⊗ id)( R · ( y ) ⊗ id) σ r= (id ⊗ R · ( x ))( L · ( y ) ⊗ id) σ r + (id ⊗ R · ( x ) R · ( y ))r , (id ⊗ L · ( x ))∆( y ) = (id ⊗ L · ( x ))(id ⊗ L · ( y ))r + (id ⊗ L · ( x ))( R · ( y ) ⊗ id) σ r= (id ⊗ L · ( x ) L · ( y ))r + (id ⊗ L · ( x ))( R · ( y ) ⊗ id) σ r . Hence we have( R · ( y ) ⊗ id + σ (id ⊗ L · ( y )))∆( x ) + (id ⊗ L · ( x ) + σ ( R · ( x ) ⊗ id))∆( y ) − ∆( x · y ) − σ ∆( y · x ) = ( R · ( y ) ⊗ L · ( x ) + L · ( y ) ⊗ R · ( x ))(r + σ r) . Therefore Eq. (3.7) hold if and only if Eq. (4.3) holds.(b) Let x, y ∈ A . Then we have σ (id ⊗ R · ( y ))∆( x ) = σ (id ⊗ R · ( y ))(id ⊗ L · ( x ))r + σ (id ⊗ R · ( y ))( R · ( x ) ⊗ id) σ r= ( R · ( y ) L · ( x ) ⊗ id) σ r + ( R · ( y ) ⊗ id)(id ⊗ R · ( x ))r , (id ⊗ R · ( y ))∆( x ) = (id ⊗ R · ( y ))(id ⊗ L · ( x ))r + (id ⊗ R · ( y ))( R · ( x ) ⊗ id) σ r= (id ⊗ R · ( y ) L · ( x ))r + (id ⊗ R · ( y ))( R · ( x ) ⊗ id) σ r ,σ ( L · ( y ) ⊗ id)∆( x ) = σ ( L · ( y ) ⊗ id)(id ⊗ L · ( x ))r + σ ( L · ( y ) ⊗ id)( R · ( x ) ⊗ id) σ r= (id ⊗ L · ( y ))( L · ( x ) ⊗ id) σ r + (id ⊗ L · ( y ) R · ( x ))r , ( L · ( y ) ⊗ id)∆( x ) = ( L · ( y ) ⊗ id)(id ⊗ L · ( x ))r + ( L · ( y ) ⊗ id)( R · ( x ) ⊗ id) σ r= ( L · ( y ) ⊗ id)(id ⊗ L · ( x ))r + ( L · ( y ) R · ( x ) ⊗ id) σ r . Therefore we have( σ (id ⊗ R · ( y )) + ( L · ( y ) ⊗ id) − (id ⊗ R · ( y )) − σ ( L · ( y ) ⊗ id))∆( x ) − ( σ (id ⊗ R · ( x )) + ( L · ( x ) ⊗ id) − (id ⊗ R · ( x )) − σ ( L · ( x ) ⊗ id))∆( y )= ( R · ( y ) ⊗ R · ( x ) + L · ( y ) ⊗ L · ( x ) − R · ( x ) ⊗ R · ( y ) − L · ( x ) ⊗ L · ( y ))(r + σ r)+(([ L · ( y ) , R · ( x )] − [ L · ( x ) , R · ( y )]) ⊗ id) σ r + (id ⊗ ([ L · ( x ) , R · ( y )] − [ L · ( y ) , R · ( x )]))r= ( R · ( y ) ⊗ R · ( x ) + L · ( y ) ⊗ L · ( x ) − R · ( x ) ⊗ R · ( y ) − L · ( x ) ⊗ L · ( y ))(r + σ r) . Note that the last equal sign is due to the definition of an anti-flexible algebra. Hence Eq. (3.8)hold if and only if Eq. (4.4) holds. (cid:3)
Lemma 4.3.
Let A be a vector space and ∆ : A → A ⊗ A be a linear map. Then the dual map ∆ ∗ : A ∗ ⊗ A ∗ → A ∗ defines an anti-flexible algebra structure on A ∗ if and only if E ∆ = 0 , where E ∆ = (∆ ⊗ id)∆ − (id ⊗ ∆)∆ + (( σ ∆) ⊗ id)( σ ∆) − (id ⊗ ( σ ∆))( σ ∆) . (4.5) Proof.
Denote by “ ◦ ” the product on A ∗ defined by ∆ ∗ , that is, h a ◦ b, x i = h ∆ ∗ ( a ⊗ b ) , x i = h a ⊗ b, ∆( x ) i ∀ x ∈ A, a, b ∈ A ∗ . Therefore, for all a, b, c ∈ A ∗ and x ∈ A , we have h ( a, b, c ) , x i = h ( a ◦ b ) ◦ c − a ◦ ( b ◦ c ) , x i = h (∆ ∗ (∆ ∗ ⊗ id) − ∆ ∗ (id ⊗ ∆ ∗ )) ( a ⊗ b ⊗ c ) , x i = h ((∆ ⊗ id)∆ − (id ⊗ ∆)∆) ( x ) , a ⊗ b ⊗ c i ; h ( c, b, a ) , x i = h ( c ◦ b ) ◦ a − c ◦ ( b ◦ a ) , x i = h (∆ ∗ (∆ ∗ ⊗ id) − ∆ ∗ (id ⊗ ∆ ∗ )) ( c ⊗ b ⊗ a ) , x i = h ((∆ ∗ σ ∗ )((∆ ∗ σ ∗ ) ⊗ id) − (∆ ∗ σ ∗ )(id ⊗ (∆ ∗ σ ∗ ))) ( a ⊗ b ⊗ c ) , x i = h ((( σ ∆) ⊗ id)( σ ∆) − (id ⊗ ( σ ∆))( σ ∆)) ( x ) , a ⊗ b ⊗ c i . Therefore, ( A ∗ , ◦ ) is an anti-flexible algebra if and only if E ∆ = 0. (cid:3) Let ( A, · ) be an anti-flexible algebra and r = X i a i ⊗ b i ∈ A ⊗ A . Setr = X i a i ⊗ b i ⊗ , r = X i a i ⊗ ⊗ b i , r = X i ⊗ a i ⊗ b i , (4.6)r = X i b i ⊗ a i ⊗ , r = X i b i ⊗ ⊗ a i , r = X i ⊗ b i ⊗ a i , (4.7)where 1 is the unit if ( A, · ) has a unit, otherwise is a symbol playing a similar role of the unit.Then the operation between two rs is in an obvious way. For example,r r = X i,j a i · a j ⊗ b i ⊗ b j , r r = X i,j a i ⊗ a j ⊗ b i · b j , r r = X i,j a j ⊗ a i · b j ⊗ b i , (4.8)and so on. Theorem 4.4.
Let ( A, · ) be an anti-flexible algebra and r ∈ A ⊗ A . Let ∆ : A → A ⊗ A be a linearmap defined by Eq. (4.1) . Then ∆ ∗ defines an anti-flexible algebra structure on A ∗ if and only iffor any x ∈ A , (id ⊗ id ⊗ L · ( x ))( M (r)) + (id ⊗ id ⊗ R · ( x ))( P (r))+( L · ( x ) ⊗ id ⊗ id)( N (r)) + ( R · ( x ) ⊗ id ⊗ id)( Q (r)) = 0 , (4.9) where M (r) = r r + r r − r r , N (r) = r r − r r − r r ,P (r) = r r + r r − r r , Q (r) = r r − r r − r r . Proof.
Set r = X i a i ⊗ b i . Let x ∈ A . Then we have(∆ ⊗ id)∆( x ) = X i,j { a j ⊗ ( a i · b j ) ⊗ ( x · b i ) + ( b j · a i ) ⊗ a j ⊗ ( x · b i )+ a j ⊗ (( b i · x ) b j ) ⊗ a i + ( b j · ( b i · x )) ⊗ a j ⊗ a i } = X i,j { a j ⊗ (( b i · x ) · b j ) ⊗ a i + ( b j · ( b i · x )) ⊗ a j ⊗ a i } + (id ⊗ id ⊗ L · ( x ))(r r + r r ) , (( σ ∆) ⊗ id)( σ ∆)( x ) = X i,j { (( x · b i ) · b j ) ⊗ a j ⊗ a i + a j ⊗ ( b j · ( x · b i )) ⊗ a i + ( a i · b j ) ⊗ a j ⊗ ( b i · x ) + a j ⊗ ( b j · a i ) ⊗ ( b i · x ) } = X i,j { (( x · b i ) · b j ) ⊗ a j ⊗ a i + a j ⊗ ( b j · ( x · b i )) ⊗ a i } + (id ⊗ id ⊗ R · ( x ))(r r + r r ) , (id ⊗ ∆)∆( x ) = X i,j { a i ⊗ a j ⊗ (( x · b i ) · b j ) + a i ⊗ ( b j · ( x · b i )) ⊗ a j + ( b i · x ) ⊗ a j ⊗ ( a i · b j ) + ( b i · x ) ⊗ ( b j · a i ) ⊗ a j } = X i,j { a i ⊗ a j ⊗ (( x · b i ) · b j ) + a i ⊗ ( b j · ( x · b i )) ⊗ a j } NTI-FLEXIBLE BIALGEBRAS 11 + ( R · ( x ) ⊗ id ⊗ id)(r r + r r ) , (id ⊗ ( σ ∆))( σ ∆)( x ) = X i,j { ( x · b i ) ⊗ ( a i · b j ) ⊗ a j + ( x · b i ) ⊗ a j ⊗ ( b j · a i )+ a i ⊗ (( b i · x ) · b j ) ⊗ a j + a i ⊗ a j ⊗ ( b j · ( b i · x )) } = X i,j { a i ⊗ (( b i · x ) · b j ) ⊗ a j + a i ⊗ a j ⊗ ( b j · ( b i · x )) } + ( L · ( x ) ⊗ id ⊗ id)(r r + r r ) . Thus E ∆ ( x ) = ( A
1) + ( A
2) + ( A A
1) = (id ⊗ id ⊗ L · ( x ))(r r + r r ) + (id ⊗ id ⊗ R · ( x ))(r r + r r ) − ( R · ( x ) ⊗ id ⊗ id)(r r + r r ) − ( L · ( x ) ⊗ id ⊗ id)(r r + r r ) , ( A
2) = X i,j { a j ⊗ (( b i · x ) · b j + b j · ( x · b i )) ⊗ a i − a i ⊗ ( b j · ( x · b i ) + ( b i · x ) · b j ) ⊗ a j } , ( A
3) = X i,j { (( x · b i ) · b j + b j · ( b i · x )) ⊗ a j ⊗ a i − a i ⊗ a j ⊗ (( x · b i ) · b j + b j · ( b i · x )) } . By exchanging the indices i and j , we have ( A
2) = 0.By the definition of an anti-flexible algebra, we have( A
3) = X i,j { ( x · ( b i · b j ) + ( b j · b i ) · x ) ⊗ a j ⊗ a i − a i ⊗ a j ⊗ ( x · ( b i · b j ) + ( b j · b i ) · x ) } = ( L · ( x ) ⊗ id ⊗ id)(r r ) + ( R · ( x ) ⊗ id ⊗ id)(r r ) − (id ⊗ id ⊗ R · ( x ))(r r ) − (id ⊗ id ⊗ L · ( x ))(r r ) . Then we have E ∆ ( x ) = (id ⊗ id ⊗ L · ( x ))(r r + r r − r r ) + (id ⊗ id ⊗ R · ( x ))(r r + r r − r r ) − ( R · ( x ) ⊗ id ⊗ id)(r r + r r − r r ) − ( L · ( x ) ⊗ id ⊗ id)(r r + r r − r r ) . Hence the conclusion follows. (cid:3)
Remark 4.5.
In fact, for any r ∈ A ⊗ A , we have N (r) = − σ M (r) , P (r) = σ M (r) , Q (r) = − σ σ M (r) , where σ ( x ⊗ y ⊗ z ) = y ⊗ x ⊗ z , σ ( x ⊗ y ⊗ z ) = z ⊗ y ⊗ x , for any x, y, z ∈ A .Combing Proposition 4.2, Theorem 4.4 and Remark 4.5 together, we have the following conclu-sion. Theorem 4.6.
Let ( A, · ) be an anti-flexible algebra and r ∈ A ⊗ A . Let ∆ : A → A ⊗ A be alinear map defined by Eq. (4.1) . Then ( A, ∆) is an anti-flexible bialgebra if and only if r satisfiesEqs. (4.3) , (4.4) and ((id ⊗ id ⊗ L · ( x )) − ( R · ( x ) ⊗ id ⊗ id) σ σ + ((id ⊗ id ⊗ R · ( x )) σ − ( L · ( x ) ⊗ id ⊗ id) σ ) M (r) = 0 , (4.10) where M (r) = r r + r r − r r . As a direct consequence of Theorem 4.6, we have the following result.
Corollary 4.7.
Let ( A, · ) be an anti-flexible algebra and r ∈ A ⊗ A . Let ∆ : A → A ⊗ A be alinear map defined by Eq. (4.1) . If in addition, r is skew-symmetric and r satisfies r r − r r + r r = 0 , (4.11) then ( A, ∆) is an anti-flexible bialgebra. Remark 4.8.
In fact, there is certain ”freedom degree” for the construction of ∆ : A → A ⊗ A defined by Eq. (4.1). Explicitly, assume∆ ′ ( x ) = ∆( x ) + Π( x )(r + σ (r)) = (id ⊗ L · ( x ) + Π( x ))r + ( R · ( x ) ⊗ id + Π( x )) σ (r) , ∀ x ∈ A, (4.12)where Π( x ) is an operator depending on x acting on A ⊗ A . Then by a direct and similar proofas of Theorem 4.6 or by Theorem 4.6 through the relationship between ∆ ′ and ∆, one can showthat ( A, ∆ ′ ) is an anti-flexible bialgebra if and only if the following equations hold:LHS of Eq . (4 .
3) + A ( x )(r + σ (r)) = 0 , LHS of Eq . (4 .
4) + B ( x )(r + σ (r)) = 0 , LHS of Eq . (4 .
10) + C ( x )(r + r ) + C ( x )(r + r ) + C ( x )(r + r ) = 0 , where A ( x ) , B ( x ) are operators depending on x acting on A ⊗ A , C ( x ) , C ( x ) , C ( x ) are oper-ators depending on x acting on A ⊗ A ⊗ A (it is the component itself when the component acts on1), and all of them are related to Π( x ). Hence by this conclusion (which is independent of Theo-rem 4.6) directly, we still show that ( A, ∆ ′ ) is an anti-flexible bialgebra when r is skew-symmetricand r satisfies Eq. (4.11). That is, this ∆ ′ defined by Eq. (4.12), also leads to the introduction ofEq. (4.11). Note that when r is skew-symmetric, ∆ ′ = ∆. Definition 4.9.
Let ( A, · ) be an anti-flexible algebra and r ∈ A ⊗ A . Eq. (4.11) is called the anti-flexible Yang-Baxter equation (AFYBE) in ( A, · ). Remark 4.10.
The notion of anti-flexible Yang-Baxter equation in an anti-flexible algebra is dueto the fact that it is an analogue of the classical Yang-Baxter equation in a Lie algebra ([9]) orthe associative Yang-Baxter equation in an associative algebra ([3]).It is a remarkable observation and an unexpected consequence that both the anti-flexible Yang-Baxter equation in an anti-flexible algebra and the associative Yang-Baxter equation ([3]) in anassociative algebra have the same form Eq. (4.11). Hence both these two equations have somecommon properties. At the end of this section, we give two properties of anti-flexible Yang-Baxterequation whose proofs are omitted since the proofs are the same as in the case of associativeYang-Baxter equation.Let A be a vector space. For any r ∈ A ⊗ A , r can be regarded as a linear map from A ∗ to A in the following way: h r , u ∗ ⊗ v ∗ i = h r( u ∗ ) , v ∗ i , ∀ u ∗ , v ∗ ∈ A ∗ . (4.13) Proposition 4.11.
Let ( A, · ) be an anti-flexible algebra and r ∈ A ⊗ A be skew-symmetric. Then r is solution of anti-flexible Yang-Baxter equation if and only if r satisfies r( a ) · r( b ) = r( R ∗· (r( a )) b + L ∗· (r( b )) a ) , ∀ a, b ∈ A ∗ . (4.14) Remark 4.12.
Since the dual bimodules of both anti-flexible and associative algebras have thesame form (see Remark 2.8), the interpretation of anti-flexible Yang-Baxter equation in terms ofoperator form (4.14) in the above Proposition 4.11 explains partly why the anti-flexible Yang-Baxter equation has the same form as of the associative Yang-Baxter equation.
Theorem 4.13.
Let ( A, · ) be an anti-flexible algebra and r ∈ A ⊗ A . Suppose that r is antisym-metric and nondegenerate. Then r is a solution of anti-flexible Yang-Baxter equation in ( A, · ) ifand only if the inverse of the isomorphism A ∗ → A induced by r , regarded as a bilinear form ω on A (that is, ω ( x, y ) = h r − x, y i for any x, y ∈ A ), satisfies ω ( x · y, z ) + ω ( y · z, x ) + ω ( z · x, y ) = 0 , ∀ x, y, z ∈ A. (4.15) NTI-FLEXIBLE BIALGEBRAS 13 O -operators of anti-flexible algebras and pre-anti-flexible algebras In this section, we introduce the notions of O -operators of anti-flexible algebras and pre-anti-flexible algebras to construct skew-symmetric solutions of anti-flexible Yang-Baxter equation andhence to construct anti-flexible bialgebras. Definition 5.1.
Let ( l, r, V ) be a bimodule of an anti-flexible algebra ( A, · ). A linear map T : V → A is called an O -operator associated to ( l, r, V ) if T satisfies T ( u ) · T ( v ) = T ( l ( T ( u )) v + r ( T ( v )) u ) , ∀ u, v ∈ V. (5.1) Example 5.2.
Let ( A, · ) be an anti-flexible algebra. An O -operator R B associated to the regularbimodule ( L, R, A ) is called a
Rota-Baxter operator of weight zero , that is, R B satisfies R B ( x ) · R B ( y ) = R B ( R B ( x ) · y + x · R B ( y )) , ∀ x, y ∈ A. (5.2) Example 5.3.
Let ( A, · ) be an anti-flexible algebra and r ∈ A ⊗ A . If r is skew-symmetric, thenby Proposition 4.11, r is a solution of anti-flexible Yang-Baxter equation if and only if r regardedas a linear map from A ∗ to A is an O -operator associated to the bimodule ( R ∗· , L ∗· , A ∗ ).There is the following construction of (skew-symmetric) solutions of anti-flexible Yang-Baxterequation in a semi-direct product anti-flexible algebra from an O -operator of an anti-flexiblealgebra which is similar as for associative algebras ([3, Theorem 2.5.5], hence the proof is omitted). Theorem 5.4.
Let ( l, r, V ) be a bimodule of an anti-flexible algebra ( A, · ) . Let T : V → A be alinear map which is identified as an element in ( A ⋉ r ∗ ,l ∗ V ∗ ) ⊕ ( A ⋉ r ∗ ,l ∗ V ∗ ) . Then r = T − σ ( T ) is a skew-symmetric solution of anti-flexible Yang-Baxter equation in A ⋉ r ∗ ,l ∗ V ∗ if only if T isan O -operator associated to the bimodule ( l, r, V ) . Definition 5.5.
Let A be a vector space with two bilinear products ≺ , ≻ : A ⊗ A → A . We call ita pre-anti-flexible algebra denoted by ( A, ≺ , ≻ ) if for any x, y, z ∈ A , the following equationsare satisfied ( x, y, z ) m = ( z, y, x ) m , (5.3)( x, y, z ) l = ( z, y, x ) r , (5.4)where ( x, y, z ) m := ( x ≻ y ) ≺ z − x ≻ ( y ≺ z ) , (5.5)( x, y, z ) l := ( x ∗ y ) ≻ z − x ≻ ( y ≻ z ) , (5.6)( x, y, z ) r := ( x ≺ y ) ≺ z − x ≺ ( y ∗ z ) , (5.7)here x ∗ y = x ≺ y + x ≻ y . Remark 5.6.
Note that if both hand sides in Eqs. (5.5), (5.6) and (5.7) are zero, that is,( x, y, z ) m = 0 , ( z, y, x ) l = 0 , ( x, y, z ) r = 0 , (5.8)then it exactly gives the definition of a dendriform algebra which was introduced by Loday in[15]. Hence any dendriform algebra is a pre-anti-flexible algebra, that is, pre-anti-flexible algebrascan be regarded as a natural generalization of dendriform algebras. On the other hand, fromthe point of view of operads, like dendriform algebras being the splitting of associative algebras,pre-anti-flexible algebras are the splitting of anti-flexible algebras ([4, 16]). Proposition 5.7.
Let ( A, ≺ , ≻ ) be a pre-anti-flexible algebra. Define a bilinear product ∗ : A ⊗ A → A by x ∗ y = x ≺ y + x ≻ y, ∀ x, y ∈ A. (5.9) Then ( A, ∗ ) is an anti-flexible algebra, which is called the associated anti-flexible algebra of ( A, ≺ , ≻ ) .Proof. Set ( x, y, z ) ∗ = ( x ∗ y ) ∗ z − x ∗ ( y ∗ z ) , ∀ x, y, z ∈ A. Then for any x, y, z ∈ A , we have( x, y, z ) ∗ = ( x, y, z ) m + ( x, y, z ) l + ( x, y, z ) r = ( z, y, x ) m + ( z, y, x ) l + ( z, y, x ) r = ( z, y, x ) ∗ . Hence ( A, ∗ ) is an anti-flexible algebra. (cid:3) Let ( A, ≺ , ≻ ) be a pre-anti-flexible algebra. For any x ∈ A , let L ≻ ( x ) , R ≺ ( x ) denote the leftmultiplication operator of ( A, ≺ ) and the right multiplication operator of ( A, ≻ ) respectively, thatis, L ≻ ( x )( y ) = x ≻ y, R ≺ ( x )( y ) = y ≺ x, ∀ x, y ∈ A . Moreover, let L ≻ , R ≺ : A → gl ( A ) be twolinear maps with x → L ≻ ( x ) and x → R ≺ ( x ) respectively. Proposition 5.8.
Let ( A, ≺ , ≻ ) be a pre-anti-flexible algebra. Then ( L ≻ , R ≺ , A ) is a bimodule ofthe associated anti-flexible algebra ( A, ∗ ) , where ∗ is defined by Eq. (5.9) .Proof. For any x, y, z ∈ A , we have( L ≻ ( x ∗ y ) − L ≻ ( x ) L ≻ ( y ))( z ) = ( x ∗ y ) ≻ z − x ≻ ( y ≻ z ) = ( x, y, z ) l , ( R ≺ ( x ) R ≺ ( y ) − R ≺ ( y ∗ x ))( z )( z ≺ y ) ≺ x − z ≺ ( y ∗ x ) = ( z, y, x ) r , ( L ≻ ( x ) R ≺ ( y ) − R ≺ ( y ) L ≻ ( x ))( z ) = x ≻ ( z ≺ y ) − ( x ≻ z ) ≺ y = ( x, z, y ) m , ( L ≻ ( y ) R ≺ ( x ) − R ≺ ( x ) L ≻ ( y ))( z ) = y ≻ ( z ≺ x ) − ( y ≻ z ) ≺ x = ( y, z, x ) m . Hence ( L ≻ , R ≺ , A ) is a bimodule of ( A, ∗ ). (cid:3) A direct consequence is given as follows.
Corollary 5.9.
Let ( A, ≺ , ≻ ) be a pre-anti-flexible algebra. Then the identity map id is an O -operator of the associated anti-flexible algebra ( A, ∗ ) associated to the bimodule ( L ≻ , R ≺ , A ) . Theorem 5.10.
Let ( l, r, V ) be a bimodule of an anti-flexible algebra ( A, · ) . Let T : V → A bean O -operator associated to ( l, r, V ) . Then there exists a pre-anti-flexible algebra structure on V given by u ≻ v = l ( T ( u )) v, u ≺ v = r ( T ( v )) u, ∀ u, v ∈ V. (5.10) So there is an associated anti-flexible algebra structure on V given by Eq. (5.9) and T is a ho-momorphism of anti-flexible algebras. Moreover, T ( V ) = { T ( v ) | v ∈ V } ⊂ A is an anti-flexiblesubalgebra of ( A, · ) and there is an induced pre-anti-flexible algebra structure on T ( V ) given by T ( u ) ≻ T ( v ) = T ( u ≻ v ) , T ( u ) ≺ T ( v ) = T ( u ≺ v ) , ∀ u, v ∈ V. (5.11) Its corresponding associated anti-flexible algebra structure on T ( V ) given by Eq. (5.9) is just theanti-flexible subalgebra structure of ( A, · ) and T is a homomorphism of pre-anti-flexible algebras.Proof. For all u, v, w ∈ V , we have ( u, v, w ) m = ( u ≻ v ) ≺ w − u ≻ ( v ≺ w ) = r ( T ( w )) l ( T ( u )) v − l ( T ( u )) r ( T ( w )) v = r ( T ( u )) l ( T ( w )) v − l ( T ( u )) r ( T ( w )) v = ( w, v, u ) m , ( u, v, w ) l = ( u ≻ v + u ≺ v ) ≻ w − u ≻ ( v ≻ w ) = ( l ( T ( l ( T ( u )) v + r ( T ( v )) u )) − l ( T ( u )) l ( T ( v ))) w = ( l ( T ( u ) · T ( v )) − l ( T ( u )) l ( T ( v ))) w = ( r ( T ( u )) r ( T ( v )) − r ( T ( v ) · T ( u )) w NTI-FLEXIBLE BIALGEBRAS 15 = ( r ( T ( u )) r ( T ( v )) − r ( T ( u ≻ v + u ≺ v ))) w = ( w ≺ v ) ≺ u − w ≺ ( u ≻ v + u ≺ v )= ( w, v, u ) r Therefore, ( V, ≺ , ≻ ) is a pre-anti-flexible algebra. For T ( V ), we have T ( u ) ∗ T ( v ) = T ( u ≻ v + u ≺ v ) = T ( u ∗ v ) = T ( u ) · T ( v ) , ∀ u, v ∈ V. The rest is straightforward. (cid:3)
Corollary 5.11.
Let ( A, · ) be an anti-flexible algebra. Then there exists a pre-anti-flexible algebrasstructure on A such that its associated anti-flexible algebra is ( A, · ) if and only if there exists aninvertible O -operator.Proof. Suppose that there exists an invertible O -operator T : V → A associated to a bimodule( l, r, V ). Then the products “ ≻ , ≺ ” given by Eq. (5.10) defines a pre-anti-flexible algebra structureon V . Moreover, there is a pre-anti-flexible algebra structure on T ( V ) = A given by Eq. (5.11),that is, x ≻ y = T ( l ( x ) T − ( y )) , x ≺ y = T ( r ( y ) T − ( x )) , ∀ x, y ∈ A. Moreover, for any x, y ∈ A , we have x ≻ y + x ≺ y = T ( l ( x ) T − ( y ) + r ( y ) T − ( x )) = T ( T − ( x )) · T ( T − ( y )) = x · y. Hence the associated anti-flexible algebra of ( A, ≻ , ≺ ) is ( A, · ).Conversely, let ( A, ≻ , ≺ ) be pre-anti-flexible algebra such that its associated anti-flexible is( A, · ). Then by Corollary 5.9, the identity map id is an O -operator of ( A, · ) associated to thebimodule ( L ≻ , R ≺ , A ). (cid:3) Corollary 5.12.
Let ( A, · ) be an anti-flexible algebra and ω be a nondegenerate skew-symmetricbilinear form satisfying Eq. (4.15) . Then there exists a pre-anti-flexible algebra structure ≻ , ≺ on A given by ω ( x ≻ y, z ) = ω ( y, z · x ) , ω ( x ≺ y, z ) = ω ( x, y · z ) , ∀ x, y, z ∈ A, (5.12) such that the associated anti-flexible algebra is ( A, · ) .Proof. Define a linear map T : A → A ∗ by h T ( x ) , y i = ω ( x, y ) , ∀ x, y ∈ A. Then T is invertible and T − is an O -operator of the anti-flexible algebra ( A, · ) associated to thebimodule ( R ∗· , L ∗· , A ∗ ). By Corollary 5.11, there is a pre-anti-flexible algebra structure ≻ , ≺ on( A, ∗ ) given by x ≻ y = T − R ∗ ( x ) T ( y ) , x ≺ y = T − L ∗ ( y ) T ( x ) , ∀ x, y ∈ A, which gives exactly Eq. (5.12) such that the associated anti-flexible algebra is ( A, · ). (cid:3) Finally we give the following construction of skew-symmetric solutions of anti-flexible Yang-Baxter equation (hence anti-flexible bialgebras) from a pre-anti-flexible algebra.
Proposition 5.13.
Let ( A, ≻ , ≺ ) be a pre-anti-flexible algebra. Then r = n X i ( e i ⊗ e ∗ i − e ∗ i ⊗ e i ) (5.13) is a solution of anti-flexible Yang-Baxter equation in A ⋉ R ∗≺ ,L ∗≻ A ∗ , where { e , · · · , e n } is a basisof A and { e ∗ , · · · , e ∗ n } is its dual basis. Proof.
Note that the identity map id = n P i =1 e i ⊗ e ∗ i . Hence the conclusion follows from Theorem 5.4and Corollary 5.9. (cid:3) Acknowledgements.
This work is supported by NSFC (11931009). C. Bai is also supported bythe Fundamental Research Funds for the Central Universities and Nankai ZhiDe Foundation.
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