Operator forms of nonhomogeneous associative classical Yang-Baxter equation
aa r X i v : . [ m a t h . QA ] J u l OPERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVECLASSICAL YANG-BAXTER EQUATION
CHENGMING BAI, XING GAO, LI GUO, AND YI ZHANG
Abstract.
This paper studies operator forms of the nonhomogeneous associative clas-sical Yang-Baxter equation (nhacYBe), extending and generalizing such studies for theclassical Yang-Baxter equation and associative Yang-Baxter equation that can be trackedback to the works of Semonov-Tian-Shansky and Kupershmidt on Rota-Baxter Lie alge-bras and O -operators. In general, solutions of the nhacYBe are characterized in terms ofgeneralized O -operators. The characterization can be given by the classical O -operatorsprecisely when the solutions satisfy an invariant condition. When the invariant conditionis compatible with a Frobenius algebra, such solutions have close relationships with Rota-Baxter operators on the Frobenius algebra. In general, solutions of the nhacYBe can beproduced from Rota-Baxter operators, and then from O -operators when the solutions aretaken in semi-direct product algebras. In the other direction, Rota-Baxter operators canbe obtained from solutions of the nhacYBe in unitizations of algebras. Finally a classifica-tions of solutions of the nhacYBe satisfying the mentioned invariant condition in all unitalcomplex algebras of dimensions two and three are obtained. All these solutions are shownto come from Rota-Baxter operators. Contents
1. Introduction 21.1. CYBE, AYBE and their operator forms 21.2. Nonhomogeneous AYBE and its operator form 31.3. Outline of the paper 42. Characterizations of nhacYBe by generalized O -operators 52.1. O -operators and Rota-Baxter operators for bimodules 52.2. Operator forms of solutions of nhacYBe 72.3. Operator forms of solutions in a Frobenius algebra 102.4. Operator forms of symmetrized invariant solutions of nhacYBe 123. NhacYBe and Rota-Baxter operators 153.1. NhacYBe and Rota-Baxter operators on Frobenius algebras 153.2. From O -operators and dendriform algebras to nhacYBe on semi-direct productalgebras 163.3. From nhacYBe to Rota-Baxter operators on unitization algebras 204. Classification of symmetrized invariant solutions of nhacYBe in low dimensions 244.1. The classification in dimension two 244.2. The classification in dimension three 25References 28 Date : July 22, 2020.2010
Mathematics Subject Classification.
Key words and phrases. associative Yang-Baxter equation; classical Yang-Baxter equation; O -operator;Rota-Baxter operator; dendriform algebra. Introduction
The aim of this paper is to give operator forms of the nonhomogeneous associative clas-sical Yang-Baxter equation in terms of Rota-Baxter operators and the more general O -operators.1.1. CYBE, AYBE and their operator forms.
The classical Yang-Baxter equation(CYBE) was first given in the following tensor form[ r , r ] + [ r , r ] + [ r , r ] = 0 , where r ∈ g ⊗ g and g is a Lie algebra (see [15] for details). The CYBE arose from thestudy of inverse scattering theory in 1980s. Later it was recognized as the “semi-classicallimit” of the quantum Yang-Baxter equation which was encountered by C. N. Yang in thecomputation of the eigenfunctions of a one-dimensional fermion gas with delta functioninteractions [41] and by R. J. Baxter in the solution of the eight vertex model in statisticalmechanics [12]. The study of the CYBE is also related to classical integrable systems andquantum groups (see [15] and the references therein).An important approach in the study of the CYBE was through the interpretation of itstensor form in various operator forms which proved to be effective in providing solutions ofthe CYBE, in addition to the well-known work of Belavin and Drinfeld [13]. First Semonov-Tian-Shansky [39] showed that if there exists a nondegenerate symmetric invariant bilinearform on a Lie algebra g and if a solution r of the CYBE is skew-symmetric, then r can beequivalently expressed as a linear operator R : g → g satisfying the operator identity[ R ( x ) , R ( y )] = R ([ R ( x ) , y ]) + R ([ x, R ( y )]) , ∀ x, y ∈ g , (1)which is then regarded as an operator form of the CYBE. Note that Eq. (1) is exactlythe Rota-Baxter relation (of weight zero) in Eq. (4) for Lie algebras.In order for the approach to work more generally, Kupershmidt revisited operator formsof the CYBE in [27] and noted that, when r is skew-symmetric, the tensor form of theCYBE is equivalent to a linear map r : g ∗ → g satisfying[ r ( x ) , r ( y )] = r (ad ∗ r ( x )( y ) − ad ∗ r ( y )( x )) , ∀ x, y ∈ g ∗ , where g ∗ is the dual space of g and ad ∗ is the dual representation of the adjoint representa-tion (coadjoint representation) of the Lie algebra g . He further generalized the above ad ∗ toan arbitrary representation ρ : g → gl ( V ) of g , that is, a linear map T : V → g , satisfying[ T ( u ) , T ( v )] = T ( ρ ( T ( u )) v − ρ ( T ( v )) u ) , ∀ u, v ∈ V, which was regarded as a natural generalization of the CYBE. Such an operator is calledan O -operator associated to ρ . Note that the operator form (1) of the CYBE given bySemonov-Tian-Shansky is just an O -operator associated to the adjoint representation of g .Going in the other direction, any O -operator gives a skew-symmetric solution of theCYBE in a semi-direct product Lie algebra, completing the cycle from the tensor form to theoperator form and back to the tensor form of the CYBE. Moreover, there is a closely relatedalgebraic structure called the pre-Lie algebra. Any O -operator gives a pre-Lie algebra andconversely, any pre-Lie algebra naturally gives an O -operator of the commutator Lie algebra,and hence naturally gives rise to a solution of the CYBE [4]. PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 3
An analogue of the CYBE for associative algebras is the associative Yang-Baxterequation (AYBE) [2]: r r + r r − r r = 0 , for r ∈ A ⊗ A where A is an associative algebra (see Definition 2.6 for details). Its formwith spectral parameters was given in [36] in connection with the CYBE and the quantumYang-Baxter equation. The AYBE arose from the study of the (antisymmetric) infinitesimalbialgebras, a notion traced back to Joni and Rota in order to provide an algebraic frameworkfor the calculus of divided differences [23, 24] and, in the antisymmetric case, carryingthe same structures under the names of “associative D-algebra” in [45] and “balancedinfinitesimal bialgebra” in the sense of the opposite algebra in [2]. The AYBEs have foundapplications in various fields in mathematics and mathematical physics such as Poissonbrackets, integrable systems, quantum Yang-Baxter equation, and mirror symmetry [26,28, 32, 33, 38].Motivated by the operator approach to the CYBE and the Rota-Baxter operators withweights, O -operators with weights were introduced to give an operator approach to theAYBE [9], while a method of obtaining Rota-Baxter operators from solutions of the (op-posite) AYBE was obtained in [1]. Briefly speaking, under the skew-symmetric condition,a solution of the AYBE is an O -operator associated to the dual representation of the ad-joint representation, while an O -operator gives a skew-symmetric solution of the AYBEin a semi-direct product associative algebra. Furthermore, the dendriform algebra plays asimilar role as the pre-Lie algebra, that is, any O -operator induces a dendriform algebrastructure on the representation space and conversely, a dendriform algebra gives a natural O -operator and hence there is a construction of (skew-symmetric) solutions of the AYBEfrom dendriform algebras [7, 10].In turn, these studies of the AYBE by O -operators with weights led to the introduction ofsimilar O -operators to Lie algebras. These generalizations have found fruitful applicationsto the CYBE and further to Lax pairs, Lie bialgebras, and PostLie algebras [6, 8].1.2. Nonhomogeneous AYBE and its operator form.
The notion of a non-homo-geneous associative classical Yang-Baxter equation (nhacYBe) [34] is the equation(detailed in Definition 2.6) r r + r r − r r = µr , (2)where µ is a fixed constant. Its opposite form, given in Eq. (8), was called the associativeclassical Yang-Baxter equation of µ in [18]. Taking µ = 0 recovers the AYBE.The nhacYBe arose from the study of the quantum Yang-Baxter equation and Bezoutoperators. Another motivation for introducing the nhacYBe is the µ -infinitesimal bial-gebras , that is, a triple ( A, · , ∆) consisting of an algebra ( A, · ) and a coalgebra ( A, ∆)satisfying the compatibility condition∆( x · y ) = ( L ( x ) ⊗ id)∆( y ) + ∆( x )(id ⊗ R ( y )) − µx ⊗ y, ∀ x, y ∈ A, (3)where L ( x ) , R ( x ) are left and right multiplication operators of ( A, · ) respectively. When µ = 1, it was also called a unital infinitesimal bialgebra [31] and appeared in severaltopics such as rooted trees, operads and pre-Lie algebras [20, 21, 43, 44]. A solution of theopposite form of the nhacYBe in a unital algebra gives a µ -infinitesimal bialgebra [18, 34]. CHENGMING BAI, XING GAO, LI GUO, AND YI ZHANG
Note that while the AYBE has its origin from the CYBE for Lie algebras, when µ = 0,the nhacYBe does not have a counterpart for Lie algebras since r does not make sensefor a Lie algebra.As in the cases of the CYBE and the AYBE, it is important to study the nhacYBethrough its operator forms. This is the purpose of this paper. This approach gives furtherunderstanding on the nature of the equation, and provides constructions of its solutions.The O -operators and Rota-Baxter operators, in their newly generalized forms, continue toplay vital roles here, but in a different way from the homogeneous case (see Remark 3.14).1.3. Outline of the paper.
We next provide some details of our operator approach of thenhacYBe which also serve as an outline of the paper.In Section 2, we first generalize the notion of an O -operator whose weight is a scalarto one whose weight is a binary operation. We then interpret solutions of the nhacYBeequivalently in terms of generalized O -operators (Theorem 2.8) and, in the presence of asymmetric Frobenius algebra, in terms of generalized Rota-Baxter algebras (Theorem 2.16).On Frobenius algebras, such an interpretation also gives a correspondence between solu-tions of the AYBE and Rota-Baxter systems introduced in [14], rather than Rota-Baxteroperators by themselves (Corollary 2.18). In order to make a connection with the existingnotion of O -operators and Rota-Baxter operators, we explore the additional conditions forsolutions of the nhacYBe. As it turns out, a solution r of the nhacYBe can be interpreted interms of an O -operator precisely when the solution satisfies the symmetrized invariant condition that the extended symmetrizer r := r + σ ( r ) − µ ( ⊗ )of r is invariant, where σ is the flip map (Theorem 2.22). Note that the parameter µ appearsin both the nhacYBe and the invariant condition, especially as the scalar multiple of ⊗ for the latter. As a special case, the extended symmetrizer of a solution r is zero meansthat ( r, − σ ( r )) is an associative Yang-Baxter pair in the sense of [14] (Corollary 2.28).In Section 3, we present a close relationship between the nhacYBe and Rota-Baxteroperators including but exceeding the known relationships between the skew-symmetricsolutions of the AYBE and Rota-Baxter operators of weight zero on Frobenius algebrasgiven in [9]. In unital symmetric Frobenius algebras, when the extended symmetrizeris a multiple of the nondegenerate invariant tensor corresponding to the nondegeneratebilinear form defining the Frobenius algebra structure, that is, the extended symmetrizeris a nondegenerate invariant tensor or zero, there is a characterization of the solutions ofthe nhacYBe by Rota-Baxter operators (Theorem 3.1). Taking the matrix algebras givesthe correspondence in [34] and taking the trivial extended symmetrizer and µ = 0 yieldsthe correspondence in [9]. When the extended symmetrizer is degenerate, in one direction,there is a construction of solutions of the nhacYBe from Rota-Baxter operators satisfyingits own invariant conditions (Proposition 3.5). Based on such a construction, we obtainsymmetrized invariant solutions of the nhacYBe for µ = 0 in semi-direct product algebrasfrom O -operators of weight zero as well as from dendriform algebras of Loday [30]. Notethat these constructions are different from the construction of solutions of the AYBE from O -operators given in [9] due to the appearance of the new term µ ( ⊗ ) in the currentapproach (see Remark 3.14). In the other direction, Rota-Baxter operators can also beobtained from solutions of the nhacYBe in an augmented algebra, that is, the unitizationof an associative algebra (Theorem 3.17 and Corollary 3.19). PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 5
In Section 4, we give the classification of the symmetrized invariant solutions of thenhacYBe for µ = 0 in the unital complex algebras in dimensions two and three. Theseexamples indicate that the symmetrized invariant solutions of the nhacYBe only comprisea small part of all solutions of the nhacYBe. Moreover, we also find that all symmetrizedinvariant solutions of the nhacYBe for µ = 0 in the unital complex algebras in dimensionstwo and three are obtained from Rota-Baxter operators. Notations.
Throughout this paper, we fix a base field k . Unless otherwise specified, all thevector spaces and algebras are finite dimensional, although some results and notions remainvalid in the infinite-dimensional case. By a k -algebra, we mean an associative algebra over k not necessarily having a unit.2. Characterizations of nhacYBe by generalized O -operators We first recall some basic definitions and facts that will be used in this paper. We intro-duce the notion of generalized O -operators whose weight is a binary operation, especiallywhen the binary operations are obtained from A -bimodule k -algebras, we recover the notionof O -operators of weight λ . Then we give a general interpretation of the nhacYBe in termsof generalized O -operators, including a correspondence between solutions of the nhacYBewith µ = 0 and Rota-Baxter systems [14] on Frobenius algebras. Finally under the addi-tional invariant condition, this interpretation gives a correspondence between symmetrizedinvariant solutions of the nhacYBe and O -operators with weight λ .2.1. O -operators and Rota-Baxter operators for bimodules. We generalize the no-tions of O -operators and Rota-Baxter operators from those with scalar weights to the oneswith weights given by binary operations. We start with background that we refer the readerto [5, 9] for further details.Let ( A, · ) be a k -algebra. An A -bimodule is a k -module V , together with linear maps ℓ, r : A → End k ( V ) satisfying ℓ ( x · y ) v = ℓ ( x )( ℓ ( y ) v ) , vr ( x · y ) = ( vr ( x )) r ( y ) , ( ℓ ( x ) v ) r ( y ) = ℓ ( x )( vr ( y )) , ∀ x, y ∈ A, v ∈ V. If we want to be more precise, we also denote an A -bimodule V by the triple ( V, ℓ, r ).Given a k -algebra A = ( A, · ) and x ∈ A , define L ( x ) : A → A, L ( x ) y = xy ; R ( x ) : A → A, yR ( x ) = yx, ∀ y ∈ A to be the left and right actions on A . We further define L = L A : A → End k ( A ) , x L ( x ); R = R A : A → End k ( A ) , x R ( x ) , ∀ x ∈ A. Clearly, (
A, L, R ) is an A -bimodule, called the adjoint A -bimodule .There is a natural characterization of semi-direct product extensions of a k -algebra ( A, · )by an A -bimodule. Let ℓ, r : A → End k ( V ) be linear maps. Define a multiplication on A ⊕ V (still denoted by · ) by( a + u ) · ( b + v ) := a · b + ( ℓ ( a ) v + ur ( b )) , ∀ a, b ∈ A, u, v ∈ V. Then as is well-known, A ⊕ V is a k -algebra, denoted by A ⋉ ℓ,r V and called the semi-directproduct of A by V , if and only if ( V, ℓ, r ) is an A -bimodule.For a k -module V and its dual module V ∗ := Hom k ( V, k ), the usual pairing betweenthem is given by h , i : V ∗ × V → k , h u ∗ , v i = u ∗ ( v ) , ∀ u ∗ ∈ V ∗ , v ∈ V. CHENGMING BAI, XING GAO, LI GUO, AND YI ZHANG
Identifying V with ( V ∗ ) ∗ , we also use h v, u ∗ i = h u ∗ , v i .Let A be a k -algebra and let ( V, ℓ, r ) be an A -bimodule. Define linear maps ℓ ∗ , r ∗ : A → End k ( V ∗ ) by h u ∗ ℓ ∗ ( x ) , v i = h u ∗ , ℓ ( x ) v i , h r ∗ ( x ) u ∗ , v i = h u ∗ , vr ( x ) i , ∀ x ∈ A, u ∗ ∈ V ∗ , v ∈ V, respectively. Then ( V ∗ , r ∗ , ℓ ∗ ) is also an A -bimodule, called the dual A -bimodule of( V, ℓ, r ).To give an operator interpretation of solutions of the nhacYBe, we generalize the notion of O -operators with weights introduced in [9] by dropping the condition that the multiplication ◦ on R turns ( R, ◦ , ℓ, r ) into an A -bimodule k -algebra. Definition 2.1.
Let ( A, · ) be a k -algebra. Let ( R, ℓ, r ) be an A -bimodule and ◦ a binaryoperation on R . A linear map α : R → A is called an O -operator of weight ◦ associatedto ( R, ℓ, r ) or simply a generalized O -operator if α satisfies α ( u ) · α ( v ) = α ( ℓ ( α ( u )) v ) + α ( ur ( α ( v ))) + α ( u ◦ v ) , ∀ u, v ∈ R. In particular, if (
R, ℓ, r ) = (
A, L A , R A ) is the adjoint A -bimodule and ◦ is a binary operationon A , then an O -operator α : A → A of weight ◦ associated to the A -bimodule ( A, L A , R A )is called a Rota-Baxter operator of weight ◦ . In this case α satisfies α ( x ) · α ( y ) = α ( α ( x ) · y ) + α ( x · α ( y )) + α ( x ◦ y ) , ∀ x, y ∈ A. Example 2.2.
In the definition of Rota-Baxter operators with weight ◦ , when ◦ is given by x ◦ y := λx · y for a given λ ∈ k , we recover the usual Rota-Baxter operator of weight λ , with its defining operator identity P ( x ) · P ( y ) = P ( x · y ) + P ( P ( x ) · y ) + λP ( x · y ) , ∀ x, y ∈ A. (4)Here the notion is named after the mathematicians G.-C. Rota [37] and G. Baxter [11] fortheir early work motivated by fluctuation theory in probability and combinatorics, whichagain appeared in the work of Connes and Kreimer on renormalization of quantum fieldtheory [17] as a fundamental algebraic structure. See [22] for further details.We separately define a special case that will be important to us. Definition 2.3.
Let ( A, · ) be a k -algebra and ( R, ℓ, r ) be an A -bimodule. Let s : R → A be a linear map. A linear map α : R → A is called an O -operator right twisted by s associated to ( R, ℓ, r ) if α ( u ) · α ( v ) = α ( ℓ ( α ) u )) v ) + α ( ur ( α ( v ))) + α ( ur ( s ( v ))) , ∀ u, v ∈ R. Likewise α is called an O -operator left twisted by s associated to ( R, ℓ, r ) when thethird term in the above equation is replaced by α ( ℓ ( s ( u )) v ).When the A -bimodule is taken to be ( A, L A , R A ), the operator is called the Rota-Baxteroperator right twisted by s (resp. left twisted by s ).Obviously the operators in Definition 2.3 are the special cases of the operators in Defini-tion 2.1 when the binary operation ◦ are defined by u ◦ v := ur ( s ( v )) (resp. u ◦ v := ℓ ( s ( u )) v ) , ∀ u, v ∈ R. To recover the notion of O -operators with scalar weights introduced in [9], we recall aconcept combining A -bimodules with k -algebras [42]. PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 7
Definition 2.4.
Let ( A, · ) be a k -algebra with multiplication · and let ( R, ◦ ) be a k -algebrawith multiplication ◦ . Let ℓ, r : A → End k ( R ) be linear maps. We call R (or the quadruple( R, ◦ , ℓ, r )) an A -bimodule k-algebra if ( R, ℓ, r ) is an A -bimodule that is compatible withthe multiplication ◦ on R in the sense that ℓ ( x )( v ◦ w ) = ( ℓ ( x ) v ) ◦ w, ( v ◦ w ) r ( x ) = v ◦ ( wr ( x )) , ( vr ( x )) ◦ w = v ◦ ( ℓ ( x ) w ) , for all x, y ∈ A, v, w ∈ R .Obviously, ( A, · , L A , R A ) is an A -bimodule k -algebra.In Definition 2.1, when the A -bimodule ( R, ℓ, r ) with multiplication ∗ is assumed to bean A -bimodule k -algebra and when u ◦ v = λu ∗ v for λ ∈ k , we recover the following notionof an O -operator with weight λ in [9]: Definition 2.5.
Let ( A, · ) be a k -algebra and let ( R, ∗ , ℓ, r ) be an A -bimodule k -algebra.Let λ ∈ k . A linear map α : R → A is called an O -operator of weight λ associated to( R, ∗ , ℓ, r ) if α satisfies α ( u ) · α ( v ) = α ( ℓ ( α ( u )) v ) + α ( ur ( α ( v ))) + λα ( u ∗ v ) , ∀ u, v ∈ R. When ∗ = 0, then O is called an O -operator (of weight zero) associate to the A -bimodule( R, ℓ, r ).The new notion of an O -operator with weight ◦ in Definition 2.1 is more general in thatthe multiplication ◦ on R need not be compatible with A .When R is the A -bimodule k -algebra ( A, L A , R A ) with u ◦ v := λu · v for λ ∈ k and thedefault multiplication · of A , we recover the notion of a Rota-Baxter operator P of weight λ defined in Eq. (4).These structures can be summarized in the commutative diagram Rota-Baxter operatorsleft twisted by/right twisted by s (cid:31) (cid:127) / / _(cid:127) (cid:15) (cid:15) Rota-Baxter operatorsof weight ◦ _(cid:127) (cid:15) (cid:15) Rota-Baxter operatorsof weight λ ? _ o o _(cid:127) (cid:15) (cid:15) O -operatorsleft twisted by/right twisted by s (cid:31) (cid:127) / / O -operatorsof weight ◦ O -operatorsof weight λ ? _ o o Operator forms of solutions of nhacYBe.
We recall the notion of the nhacYBeand give an interpretation of solutions of the nhacYBe in terms of the generalized O -operators introduced in Definition 2.1.Let ( A, · , ) be a unital k -algebra whose multiplication · is often suppressed. For r = P i a i ⊗ b i ∈ A ⊗ A , denote r := X i a i ⊗ b i ⊗ , r := X i a i ⊗ ⊗ b i , r := X i ⊗ a i ⊗ b i . (5)Then r r , r r , r r are defined in the k -algebra A ⊗ A ⊗ A . Definition 2.6.
Let A be a unital k -algebra and let r ∈ A ⊗ A .(a) r is a solution of the associative Yang-Baxter equation (AYBE) r r + r r − r r = 0 (6)in A if the equation holds with the notation in Eq. (5). CHENGMING BAI, XING GAO, LI GUO, AND YI ZHANG (b) Fix a µ ∈ k . r is a solution of the µ -nonhomogeneous associative Yang-Baxterequation ( µ -nhacYBe) r r + r r − r r = µr (7)in A if the equation holds with the notation in Eq. (5).The opposite form of Eq. (7) is [18] r r + r r − r r = µr . (8) Definition 2.7.
Let A be a unital k -algebra and µ ∈ k . Let r ∈ A ⊗ A . Define the µ - extended symmetrizer of r to be r := r + σ ( r ) − µ ( ⊗ ) . (9)The prefix µ in Definitions 2.6 and 2.7 will be suppressed when its meaning is clear fromthe context.Let r ∈ A ⊗ A . Define linear maps r ♯ , r t♯ : A ∗ → A by the canonical bijections( ) ♯ : A ⊗ A ∼ = Hom k ( A ∗ , k ) ⊗ A ∼ = Hom k ( A ∗ , A ) , ( ) t♯ = ( ) ♯ σ : A ⊗ A → Hom k ( A ∗ , A ) . Explicitly, r ♯ and r t♯ are determined by h r ♯ ( a ∗ ) , b ∗ i = h r, a ∗ ⊗ b ∗ i , h r t♯ ( a ∗ ) , b ∗ i = h r, b ∗ ⊗ a ∗ i , ∀ a ∗ , b ∗ ∈ A ∗ . With these notations, r is called nondegenerate if the linear map r ♯ or r t♯ is a linearisomorphism. Otherwise, r is called degenerate . Furthermore, r is symmetric if and onlyif h r, a ∗ ⊗ b ∗ i = h r, b ∗ ⊗ a ∗ i , that is, h r ♯ ( a ∗ ) , b ∗ i = h r ♯ ( b ∗ ) , a ∗ i , ∀ a ∗ , b ∗ ∈ A ∗ . We now give an operator form of solutions of the nhacYBe in terms of the generalized O -operators with weights given by multiplications. Theorem 2.8.
Let ( A, · , ) be a unital k -algebra. For r ∈ A ⊗ A , let r be the extendedsymmetrizer of r and let r ♯ : A ∗ → A be the corresponding linear map. Then the followingstatements are equivalent. (a) The tensor r is a solution of the nhacYBe in A . (b) The following equation holds. r ♯ ( a ∗ ) · r ♯ ( b ∗ ) + r ♯ ( a ∗ L ∗ ( r t♯ ( b ∗ ))) − r ♯ ( R ∗ ( r ♯ ( a ∗ )) b ∗ ) − µr ♯ ( h , b ∗ i a ∗ ) = 0 , ∀ a ∗ , b ∗ ∈ A ∗ . (10)(c) The linear map r ♯ from r is an O -operator right twisted by − r ♯ associated to ( A ∗ , R ∗ , L ∗ ) . (d) The following equation holds. r t♯ ( a ∗ ) · r t♯ ( b ∗ ) − r t♯ ( a ∗ L ∗ ( r t♯ ( b ∗ )))+ r t♯ ( R ∗ ( r ♯ ( a ∗ )) b ∗ ) − µr t♯ ( h , a ∗ i b ∗ ) = 0 , ∀ a ∗ , b ∗ ∈ A ∗ . (11)(e) The linear map r t♯ from σ ( r ) is an O -operator left twisted by − r ♯ associated to ( A, R ∗ , L ∗ ) .Proof. Let r = P i a i ⊗ b i and a ∗ , b ∗ , c ∗ ∈ A ∗ .(a) ⇐⇒ (b). We have h r · r , a ∗ ⊗ b ∗ ⊗ c ∗ i = X i,j h a i · a j , a ∗ ih b i , b ∗ ih b j , c ∗ i = X j h r t♯ ( b ∗ ) · a j , a ∗ ih b j , c ∗ i = h r ♯ ( a ∗ L ∗ ( r t♯ ( b ∗ ))) , c ∗ i , PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 9 h r · r , a ∗ ⊗ b ∗ ⊗ c ∗ i = X i,j h a i , a ∗ ih a j , b ∗ ih b i · b j , c ∗ i = X j h a j , b ∗ ih r ♯ ( a ∗ ) · b j , c ∗ i = h r ♯ ( a ∗ ) · r ♯ ( b ∗ ) , c ∗ i , h− r · r , a ∗ ⊗ b ∗ ⊗ c ∗ i = − X i,j h a i , a ∗ ih a j · b i , b ∗ ih b j , c ∗ i = − X j h a j · r ♯ ( a ∗ ) , b ∗ ih b j , c ∗ i = h− r ♯ ( R ∗ ( r ♯ ( a ∗ )) b ∗ ) , c ∗ i , h− µr , a ∗ ⊗ b ∗ ⊗ c ∗ i = − µ X i h a i , a ∗ ih , b ∗ ih b i , c ∗ i = h− µr ♯ ( a ∗ ) , c ∗ ih , b ∗ i = h− µr ♯ ( h , b ∗ i a ∗ ) , c ∗ i . Hence r satisfies Eq. (7) if and only if Eq. (10) holds.(b) ⇐⇒ (c). From the definition of the extended symmetrizer of r : r = r + σ ( r ) − µ ( ⊗ ),we obtain r ♯ ( b ∗ ) = r ♯ ( b ∗ ) + r t♯ ( b ∗ ) − µ h , b ∗ i , ∀ b ∗ ∈ A ∗ and hence r t♯ ( b ∗ ) = − r ♯ ( b ∗ ) + r ♯ ( b ∗ ) + µ h , b ∗ i , ∀ b ∗ ∈ A ∗ . Further L ∗ ( ) is the identity map on A ∗ . Thus Eq. (10) is equivalent to r ♯ ( a ∗ ) · r ♯ ( b ∗ ) − r ♯ ( a ∗ L ∗ ( r ♯ ( b ∗ ))) − r ♯ ( R ∗ ( r ♯ ( a ∗ )) b ∗ ) + r ♯ ( a ∗ L ∗ ( r ♯ ( b ∗ ))) = 0 , ∀ a ∗ , b ∗ ∈ A ∗ , as needed.(a) ⇐⇒ (d). Similarly, we have h r · r , a ∗ ⊗ b ∗ ⊗ c ∗ i = X j h r t♯ ( b ∗ ) · a j , a ∗ ih b j , c ∗ i = h r t♯ ( b ∗ ) · r t♯ ( c ∗ ) , a ∗ i , h r · r , a ∗ ⊗ b ∗ ⊗ c ∗ i = X j h a i , a ∗ ih b i · r ♯ ( b ∗ ) , c ∗ i = h r t♯ ( R ∗ ( r ♯ ( b ∗ )) c ∗ ) , a ∗ i , h− r · r , a ∗ ⊗ b ∗ ⊗ c ∗ i = − X j h a i , a ∗ ih r t♯ ( c ∗ ) · b i , b ∗ i = −h r t♯ ( b ∗ L ∗ ( r t♯ ( c ∗ ))) , a ∗ i , h− µr , a ∗ ⊗ b ∗ ⊗ c ∗ i = h− µr t♯ ( c ∗ ) , a ∗ ih , b ∗ i = h− µr t♯ ( h , b ∗ i c ∗ ) , a ∗ i . Hence r satisfies Eq. (7) if and only if Eq. (11) holds.(d) ⇐⇒ (e). The proof is the same as for (b) ⇐⇒ (c). (cid:3) We now show that the oppositive nhacYBe in Eq. (8) also affords an operator form.
Lemma 2.9.
Let ( A, · , ) be a unital k -algebra. Let r ∈ A ⊗ A . Then r satisfies Eq. (7) ifand only if σ ( r ) satisfies Eq. (8) .Proof. Let r = P i a i ⊗ b i ∈ A ⊗ A . Then r satisfies Eq. (7) if and only if X i,j ( a i · a j ⊗ b i ⊗ b j + a i ⊗ a j ⊗ b i · b j − a j ⊗ a i · b j ⊗ b i − µa i ⊗ ⊗ b i ) = 0 . (12)On the other hand, σ ( r ) = P i b i ⊗ a i satisfies Eq. (8) if and only if X i,j ( b i · b j ⊗ a j ⊗ a i + b j ⊗ b i ⊗ a i · a j − b i ⊗ a i · b j ⊗ a j − µb i ⊗ ⊗ a i ) = 0 . (13) Let σ : A ⊗ A ⊗ A → A ⊗ A ⊗ A be the linear map defined by σ ( x ⊗ y ⊗ z ) = z ⊗ y ⊗ x forany x, y, z ∈ A . It is straightforward to check that the left hand side of Eq. (12) coincideswith the σ applied to the left hand side of Eq. (13). This completes the proof. (cid:3) Then we have
Corollary 2.10.
Let ( A, · , ) be a unital k -algebra. For r ∈ A ⊗ A , let r be the extendedsymmetrizer of r and let r ♯ : A ∗ → A be the corresponding linear map. Then r satisfiesEq. (8) if and only if the linear map r ♯ : A ∗ → A from r is an O -operator left twisted by − r ♯ associated to ( A ∗ , R ∗ , L ∗ ) .Proof. Since σ ( r ) ♯ = r t♯ , the conclusion follows from Theorem 2.8 and Lemma 2.9. (cid:3) Operator forms of solutions in a Frobenius algebra.
We now consider the so-lutions of the nhacYBe in a Frobenius algebra.
Definition 2.11.
Let ( A, · ) be a k -algebra. A tensor s ∈ A ⊗ A is called invariant if(id ⊗ L ( x ) − R ( x ) ⊗ id) s = 0 , ∀ x ∈ A. Lemma 2.12. ([9])
Let ( A, · ) be a k -algebra. Let s ∈ A ⊗ A be symmetric. Then thefollowing conditions are equivalent. (a) s is invariant. (b) s ♯ satisfies R ∗ ( s ♯ ( a ∗ )) b ∗ = a ∗ L ∗ ( s ♯ ( b ∗ )) , ∀ a ∗ , b ∗ ∈ A ∗ . (c) s ♯ satisfies s ♯ ( R ∗ ( x ) a ∗ ) = x · s ♯ ( a ∗ ) , s ♯ ( a ∗ L ∗ ( x )) = s ♯ ( a ∗ ) · x, ∀ x ∈ A, a ∗ ∈ A ∗ . Remark 2.13.
For a unital k -algebra ( A, ), it is obvious that ⊗ is not invariant whendim A ≥ Definition 2.14.
A bilinear form B := B ( , ) on a k -algebra ( A, · ) is called invariant if B ( a · b, c ) = B ( a, b · c ) , ∀ a, b, c ∈ A. A Frobenius algebra ( A, B ) is a k -algebra A with a nondegenerate invariant bilinearform B ( , ). A Frobenius algebra ( A, B ) is called symmetric if B ( , ) is symmetric.Let Iso k ( M, N ) denote the set of linear bijections between k -vector spaces M and N ofthe same dimension. Let NDHom( A ⊗ A, k ) and ND( A ⊗ A ) denote the set of nondegeneratebilinear forms on A and nondegenerate tensors in A ⊗ A respectively. Then by definition, thelinear bijection Hom k ( A ⊗ A, k ) ∼ = Hom k ( A, A ∗ ) restricts to a bijection NDHom k ( A ⊗ A, k ) ∼ =Iso k ( A, A ∗ ). Similarly, the linear bijection A ⊗ A ∼ = Hom k ( A ∗ , A ) restricts to a bijectionND( A ⊗ A ) ∼ = Iso k ( A ∗ , A ). Then thanks to the bijection Iso k ( A, A ∗ ) ∼ = Iso k ( A ∗ , A ) bytaking inverse, we obtain a bijectionNDHom k ( A ⊗ A, k ) ∼ = Iso k ( A, A ∗ ) ∼ = Iso k ( A ∗ , A ) ∼ = ND( A ⊗ A ) . (14)Explicitly, let B be a nondegenerate bilinear form. Let φ ♯ = φ ♯ B : A ∗ → A be the linearisomorphism defined by h φ ♯ − ( x ) , y i = B ( x, y ) , ∀ x, y ∈ A. (15)The corresponding tensor φ ∈ A ⊗ A is the one induced from the linear map φ ♯ . PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 11
Lemma 2.15.
Let ( A, · ) be a k -algebra. A nondegenerate bilinear form is symmetric andinvariant ( and hence gives a symmetric Frobenius algebra ( A, · , B )) if and only if the cor-responding φ ∈ A ⊗ A via Eq. (14) is symmetric and invariant.Proof. For any a ∗ , b ∗ ∈ A ∗ , let x = φ ♯ ( a ∗ ) and y = φ ♯ ( b ∗ ). Then from Eq. (15) we obtain B ( x, y ) = h ( φ ♯ ) − ( x ) , y i = h a ∗ , φ ♯ ( b ∗ ) i = h b ∗ ⊗ a ∗ , φ i . Thus B ( x, y ) − B ( y, x ) = h b ∗ ⊗ a ∗ − a ∗ ⊗ b ∗ , φ i which shows that B is symmetric if andonly if φ is symmetric.Then under the symmetric condition of B and hence of φ , for any z ∈ A , we have B ( y · z, x ) − B ( y, z · x ) = B ( φ ♯ ( b ∗ ) · z, φ ♯ ( a ∗ )) − B ( φ ♯ ( b ∗ ) , z · φ ♯ ( a ∗ ))= h a ∗ , φ ♯ ( b ∗ ) · z i − h b ∗ , z · φ ♯ ( a ∗ ) i = h a ∗ L ∗ ( φ ♯ ( b ∗ )) , z i − h R ∗ ( φ ♯ ( a ∗ )) b ∗ , z i = h a ∗ L ∗ ( φ ♯ ( b ∗ )) − h R ∗ ( φ ♯ ( a ∗ )) b ∗ , z i . By Lemma 2.12, this shows that B is symmetric and invariant if and only if φ is symmetricand invariant. (cid:3) Theorem 2.16.
Let ( A, · , , B ) be a unital symmetric Frobenius algebra. Let φ ♯ : A ∗ → A be the linear isomorphism defined by Eq. (15) . For r ∈ A ⊗ A , let the linear maps P r , P tr : A → A be defined respectively by P r ( x ) := r ♯ ( φ ♯ ) − ( x ) , P tr ( x ) := r t♯ ( φ ♯ ) − ( x ) , ∀ x ∈ A. (16) Let r ♯ ( a ∗ ) := r ♯ ( a ∗ ) + r t♯ ( a ∗ ) − µ h , a ∗ i , a ∗ ∈ A ∗ be defined by the extended symmetrizer r of r . Then the following statements are equivalent. (a) r is a solution of the nhacYBe in A . (b) The following equation holds. P r ( x ) · P r ( y ) = P r ( P r ( x ) · y ) − P r ( x · P tr ( y )) + µ B ( , y ) P r ( x ) , ∀ x, y ∈ A. (17)(c) The following equation holds. P tr ( x ) · P tr ( y ) = P tr ( − P r ( x ) · y ) + P tr ( x · P tr ( y )) + µ B ( , x ) P tr ( y ) , ∀ x, y ∈ A. (18)(d) The operator P r on A is a Rota-Baxter operator right twisted by − r ♯ ( φ ♯ ) − , that is, P r ( x ) · P r ( y ) = P r ( P r ( x ) · y ) + P r ( x · P r ( y )) − P r ( x · r ♯ ( φ ♯ ) − ( y )) , ∀ x, y ∈ A. (e) The operator P tr on A is a Rota-Baxter operator left twisted by − r ♯ ( φ ♯ ) − , that is, P tr ( x ) · P tr ( y ) = P tr ( P tr ( x ) · y ) + P tr ( x · P tr ( y )) − P tr ( r ♯ ( φ ♯ ) − ( x ) · y ) , ∀ x, y ∈ A. Proof.
For any x, y ∈ A , set a ∗ = φ ♯ − ( x ) , b ∗ = φ ♯ − ( y ), we have P r ( x ) · P r ( y ) = r ♯ ( a ∗ ) · r ♯ ( b ∗ ) ,P r ( P r ( x ) · y ) = r ♯ φ ♯ − ( r ♯ φ ♯ − ( x ) · φ ♯ ( b ∗ )) = r ♯ φ ♯ − ( r ♯ ( a ∗ ) · φ ♯ ( b ∗ )) = r ♯ ( R ∗ ( r ♯ ( a ∗ )) b ∗ ) ,P r ( x · P tr ( y )) = r ♯ φ ♯ − ( φ ♯ ( a ∗ ) · r t♯ φ ♯ − ( y )) = r ♯ φ ♯ − ( φ ♯ ( a ∗ ) · r t♯ ( b ∗ )) = r ♯ ( a ∗ L ∗ ( r t♯ ( b ∗ ))) , B ( , y ) P r ( x ) = P r φ ♯ ( a ∗ ) B ( , y ) = r ♯ ( h , b ∗ i a ∗ ) . Note that the invariance of φ given by Lemma 2.15 is used in deriving Eqs. (17) and (18).By Theorem 2.8, r satisfies Eq. (7) if and only if P r satisfies Eq. (17). Similarly, we show that r satisfies Eq. (7) if and only if P tr satisfies Eq. (18). Hence statements (a) – (c) areequivalent.Next for any x ∈ A and b ∗ ∈ A ∗ , we have h P r ( x ) + P tr ( x ) , b ∗ i = h r ♯ ( φ ♯ − ( x )) + r t♯ ( φ ♯ − ( x )) , b ∗ i = h r ♯ φ ♯ − ( x ) + µ h φ ♯ − ( x ) , i , b ∗ i = h r ♯ φ ♯ − ( x ) + µ B ( x, ) , b ∗ i . Hence P tr ( x ) = − P r ( x ) + r ♯ φ ♯ − ( x ) + µ B ( x, ) , ∀ x ∈ A. Then the equivalence of the statement (b) (resp. (c)) to the statement (d) (resp. (e))follows from applying this equation. (cid:3)
We give an application to Rota-Baxter systems introduced by Brzezi´nski [14].
Definition 2.17.
Let A be a k -algebra. Let P, S : A → A be two linear maps. The triple( A, P, S ) is called a
Rota-Baxter system if for any x, y ∈ A , the following equations hold P ( x ) P ( y ) = P ( P ( x ) y + xS ( y )) , S ( x ) S ( y ) = S ( P ( x ) y + xS ( y )) . Taking µ = 0 in the equivalent statements (a) – (c) in Theorem 2.16 gives Corollary 2.18.
Let ( A, · , , B ) be a unital symmetric Frobenius algebra. For r ∈ A ⊗ A ,let P r and P tr be defined as in Eq. (16) . Then r is a solution of the AYBE in Eq. (6) if andonly if ( A, P r , − P tr ) is a Rota-Baxter system. Operator forms of symmetrized invariant solutions of nhacYBe.
We nowshow that, under an invariant condition, solutions of the nhacYBe can be interpreted interms of the usual O -operators in Definition 2.5. Definition 2.19.
Let ( A, · ) be a k -algebra. A tensor r ∈ A ⊗ A is called symmetrizedinvariant if its extended symmetrizer r defined in Eq. (9) is invariant. Lemma 2.20. (a)
Let ( A, · , ) be a unital k -algebra. Let s ∈ A ⊗ A be symmetric andinvariant. Set a ∗ ◦ b ∗ := a ∗ L ∗ ( s ♯ ( b ∗ )) = R ∗ ( s ♯ ( a ∗ )) b ∗ , ∀ a ∗ , b ∗ ∈ A ∗ . (19) Then ( A ∗ , ◦ , R ∗ , L ∗ ) is an A -bimodule k -algebra. (b) Let ( A ∗ , ◦ , R ∗ , L ∗ ) be an A -bimodule k -algebra. Define a linear map s ♯ : A ∗ → A orequivalently s ∈ A ⊗ A by h s, a ∗ ⊗ b ∗ i := h s ♯ ( a ∗ ) , b ∗ i := h b ∗ ◦ a ∗ , i , ∀ a ∗ , b ∗ ∈ A ∗ . (20) Suppose h a ∗ ◦ b ∗ , i = h b ∗ ◦ a ∗ , i , ∀ a ∗ , b ∗ ∈ A ∗ , (21) and s ♯ satisfies h s ♯ ( a ∗ ) · x, b ∗ i = h b ∗ ◦ a ∗ , x i , ∀ x ∈ A, a ∗ , b ∗ ∈ A ∗ . (22) Then s is symmetric and invariant. PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 13
Proof. (a). Let a ∗ , b ∗ , c ∗ ∈ A ∗ and x, y ∈ A . Then we have( a ∗ ◦ b ∗ ) ◦ c ∗ = a ∗ L ∗ ( s ♯ ( b ∗ )) ◦ c ∗ = a ∗ L ∗ ( s ♯ ( b ∗ )) L ∗ ( s ♯ ( c ∗ )) ,a ∗ ◦ ( b ∗ ◦ c ∗ ) = a ∗ ◦ b ∗ L ∗ ( s ♯ ( c ∗ )) = a ∗ L ∗ ( s ♯ ( b ∗ L ∗ ( s ♯ ( c ∗ )))) = a ∗ L ∗ ( s ♯ ( b ∗ ) ∗ s ♯ ( c ∗ )) . Hence ( A ∗ , ◦ ) is a k -algebra. Moreover, h R ∗ ( x )( a ∗ ◦ b ∗ ) , y i = h a ∗ L ∗ ( s ♯ ( b ∗ )) , y · x i = h a ∗ , s ♯ ( b ∗ ) · y · x i , h ( R ∗ ( x ) a ∗ ) ◦ b ∗ , y i = h R ∗ ( x ) a ∗ , s ♯ ( b ∗ ) · y i = h a ∗ , s ♯ ( b ∗ ) · y · x i . Hence R ∗ ( x )( a ∗ ◦ b ∗ ) = ( R ∗ ( x ) a ∗ ) ◦ b ∗ . Similarly, we have( a ∗ ◦ b ∗ ) L ∗ ( x ) = a ∗ ◦ ( b ∗ L ∗ ( x )) , ( a ∗ L ∗ ( x )) ◦ b ∗ = a ∗ ◦ ( R ∗ ( x ) b ∗ ) . Therefore ( A ∗ , ◦ , R ∗ , L ∗ ) is an A -bimodule k -algebra.(b). Applying Eq. (21) gives h s, a ∗ ⊗ b ∗ i = h s ♯ ( a ∗ ) , b ∗ i = h b ∗ ◦ a ∗ , i = h a ∗ ◦ b ∗ , i = h s ♯ ( b ∗ ) , a ∗ i = h s, b ∗ ⊗ a ∗ i , ∀ a ∗ , b ∗ ∈ A ∗ . Hence s is symmetric. Since ( A ∗ , ◦ , R ∗ , L ∗ ) is an A -bimodule k -algebra, we have h x · s ♯ ( b ∗ ) , a ∗ i = h s ♯ ( b ∗ ) , a ∗ L ∗ ( x ) i = h ( a ∗ L ∗ ( x )) ◦ b ∗ , i = h a ∗ ◦ ( R ∗ ( x ) b ∗ ) , i = h s ♯ ( R ∗ ( x ) b ∗ ) , a ∗ i , h s ♯ ( b ∗ ) · x, a ∗ i = h s ♯ ( b ∗ ) , R ∗ ( x ) a ∗ i = h ( R ∗ ( x ) a ∗ ) ◦ b ∗ , i = h b ∗ ◦ ( R ∗ ( x ) a ∗ ) , i = h ( b ∗ L ∗ ( x )) ◦ a ∗ , i = h a ∗ ◦ ( b ∗ L ∗ ( x )) , i = h s ♯ ( b ∗ L ∗ ( x )) , a ∗ i , where x ∈ A, a ∗ , b ∗ ∈ A ∗ . Hence s is invariant. (cid:3) Remark 2.21.
In fact, under the same conditions as for Lemma 2.20, Eqs. (21) and (22)hold if and only if the following equation holds h s ♯ ( a ∗ ) · x, b ∗ i = h b ∗ ◦ a ∗ , x i = h x · s ♯ ( b ∗ ) , a ∗ i , ∀ x ∈ A, a ∗ , b ∗ ∈ A ∗ . Theorem 2.22.
Let ( A, · , ) be a unital k -algebra. Let r ∈ A ⊗ A whose extended sym-metrizer r is invariant. Let ◦ be the binary operation defined from r by Eq. (19) . Then thefollowing statements are equivalent. (a) The tensor r is a solution of the nhacYBe in Eq. (7) . (b) When r = 0 , the map r ♯ is an O -operator of weight zero associated to the A -bimodule ( A ∗ , R ∗ , L ∗ ) and when r = 0 , the map r ♯ is an O -operator of weight − associatedto the A -bimodule k -algebra ( A ∗ , ◦ , R ∗ , L ∗ ) . (c) When r = 0 , the map r t♯ is an O -operator of weight zero associated to the A -bimodule ( A ∗ , R ∗ , L ∗ ) and when r = 0 , the map r t♯ is an O -operator of weight − associatedto the A -bimodule k -algebra ( A ∗ , ◦ , R ∗ , L ∗ ) .Proof. ((a) ⇐⇒ (b)). Since a ∗ ◦ b ∗ := a ∗ L ∗ ( r ♯ ( b ∗ )) and by Lemma 2.20, ( A ∗ , ◦ , R ∗ , L ∗ ) is an A -bimodule k -algebra, the equivalence follows from Theorem 2.8.The proof of ((a) ⇐⇒ (c)) follows from the same argument. (cid:3) Corollary 2.23.
Let ( A, · , ) be a unital k -algebra. Let r ∈ A ⊗ A whose extended sym-metrizer is invariant. Then r is a solution of the nhacYBe if and only if r satisfies Eq. (8) . Proof.
By Theorem 2.22, the tensor r is a solution the nhacYBe if and only if σ ( r ) is asolution of the nhacYBe, which holds if and only if r is a solution of Eq. (8) by Lemma 2.9. (cid:3) Remark 2.24.
For a unital k -algebra ( A, ), it is obvious that µ ( ⊗ ) is a solution of thenhacYBe. However, if µ = 0 and dim A ≥
2, then the extended symmetrizer of µ ( ⊗ ) isnot invariant (see also Remark 2.13). Corollary 2.25.
Let ( A, · , ) be a unital k -algebra and ( A ∗ , ◦ , R ∗ , L ∗ ) be an A -bimodule k -algebra satisfying Eq. (21) . Let s ♯ : A ∗ → A be the linear map from ◦ defined by Eq. (20) satisfying Eq. (22) . Let P : A ∗ → A be a linear map satisfying P ( a ∗ ) + P ∗ ( a ∗ ) = s ♯ ( a ∗ ) + µ h a ∗ , i , ∀ a ∗ ∈ A ∗ , (23) where P ∗ : A ∗ → A ∗ is the dual map of P . Then the following statements are equivalent. (a) When s ♯ = 0 , P is an O -operator of weight 0 associated to ( A ∗ , R ∗ , L ∗ ) and when s ♯ = 0 , P is an O -operator of weight − associated to ( A ∗ , ◦ , R ∗ , L ∗ ) . (b) When s ♯ = 0 , P ∗ is an O -operator of weight zero associated to ( A ∗ , R ∗ , L ∗ ) and when s ♯ = 0 , P ∗ is an O -operator of weight − associated to ( A ∗ , ◦ , R ∗ , L ∗ ) . (c) The tensor r ∈ A ⊗ A defined by r ♯ = P is a symmetrized invariant solution of thenhacYBe. (d) The tensor r ∈ A ⊗ A defined by r t♯ = P is a symmetrized invariant solution of thenhacYBe.Proof. By Lemma 2.20, the tensor s from s ♯ is symmetric and invariant. Set P = r ♯ . Thenfor any a ∗ , b ∗ ∈ A ∗ , we have h P ( a ∗ ) + P ∗ ( a ∗ ) + s ♯ ( a ∗ ) − µ h a ∗ , i , b ∗ i = h r + σ ( r ) + s − µ ( ⊗ ) , a ∗ ⊗ b ∗ i . Hence P satisfies Eq. (23) if and only if the extended symmetrizer of r is symmetric andinvariant. By Theorem 2.22, statement (a) holds if and only if statement (c) holds. Notethat in this case, P ∗ = r t♯ . Therefore by Theorem 2.22, statement (b) holds if and only ifstatement (a) or statement (c) holds.Furthermore, by the symmetry of P and P ∗ , if we set P = r t♯ , then by the abovediscussion, we can directly show that statement (d) holds if and only if statement (b)holds. This proves that all the statements are equivalent. (cid:3) We end this subsection with displaying a relationship between solutions of the nhacYBewith trivial extended symmetrizers and associative Yang-Baxter pairs.
Definition 2.26. ([14]) Let A be a k -algebra. An associative Yang-Baxter pair is apair of elements r, s ∈ A ⊗ A satisfying r r − r r + r s = 0 , r s − s s + s s = 0 . Proposition 2.27. ([14])
Let ( A, ) be a unital k -algebra. Let r, s ∈ A ⊗ A . If r − s = ⊗ ,then the pair ( r, s ) is an associative Yang-Baxter pair if and only if r satisfies the nhacYBewith µ = 1 . Corollary 2.28.
Let ( A, ) be a unital k -algebra. Let r ∈ A ⊗ A . If r + σ ( r ) = µ ( ⊗ ) with µ = 0 , then r is a solution of the nhacYBe in Eq. (7) if and only if ( r, − σ ( r )) is anassociative Yang-Baxter pair. PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 15
Proof.
Let r ∈ A ⊗ A be a solution of the nhacYBe and r + σ ( r ) = µ ( ⊗ ) with µ = 0.Then r ′ = µ r is a solution of the nhacYBe with µ = 1 and r ′ + σ ( r ′ ) = ⊗ . ByProposition 2.27, ( r ′ , − σ ( r ′ )) is an associative Yang-Baxter pair. Hence ( r, − σ ( r )) is anassociative Yang-Baxter pair. Similarly, the converse also holds. (cid:3) NhacYBe and Rota-Baxter operators
In this section, we first give a correspondence between Rota-Baxter operators satisfyingadditional conditions and symmetrized invariant solutions of the nhacYBe with a spe-cific extended symmetrizer r in unital symmetric Frobenius algebras, when the extendedsymmetrizer r is a multiple of the nondegenerate invariant tensor corresponding to the non-degenerate bilinear form defining the Frobenius algebra structure. Then when the tensor r is degenerate, solutions of the nhacYBe in semi-direct product algebras can still be de-rived from Rota-Baxter operators, O -operators and dendriform algebras, while Rota-Baxteroperators can be derived from solutions of the nhacYBe in unitization algebras.3.1. NhacYBe and Rota-Baxter operators on Frobenius algebras.
Extending thecorrespondence between solutions of the AYBE and Rota-Baxter systems on Frobeniusalgebras given in Corollary 2.18 to the nhacYBe, we obtain
Theorem 3.1.
Let ( A, · , , B ) be a unital symmetric Frobenius algebra. Let φ ♯ : A ∗ → A bethe linear isomorphism from B defined by Eq. (15) and let φ ∈ A ⊗ A be the correspondinginvariant symmetric tensor. Suppose r ∈ A ⊗ A has its extended symmetrizer given by r := r + σ ( r ) − µ ( ⊗ ) = − λφ. (24) Define linear maps P r , P tr : A → A respectively by P r ( x ) := r ♯ φ ♯ − ( x ) , P tr ( x ) := r t♯ φ ♯ − ( x ) , ∀ x ∈ A. (25) Then the following conditions are equivalent. (a) r is a solution of the nhacYBe in A . (b) P r is a Rota-Baxter operator of weight λ , that is, Eq. (4) holds. (c) P tr is a Rota-Baxter operator of weight λ .Proof. It follows from Theorem 2.16 by taking r ♯ = − λφ ♯ . (cid:3) A different construction of Rota-Baxter operators from solutions of the opposite form ofthe nhacYBe in Eq. (8) can be found in [18].Taking λ = µ = 0 in Theorem 3.1, we obtain the following result. Note that in this case, P tr = − P r . Corollary 3.2. [9, Corollary 3.17]
A skew-symmetric r ∈ A ⊗ A is a solution of the AYBEin Eq. (6) if and only if the linear map P r defined by Eq. (25) is a Rota-Baxter operator ofweight zero. Example 3.3.
Let ( A, · ) = (End k ( V ) , · ) = ( M n ( k ) , · ) be the matrix algebra, where n =dim V . It is a Frobenius algebra with the invariant bilinear form being the trace form, thatis, B ( x, y ) := Tr( x · y ) , ∀ x, y ∈ A. (26) Take a basis { e , · · · , e n } of A such that B ( e i , e j ) = δ ij . Let φ = X i e i ⊗ e i . Therefore Eq. (15) holds. Moreover, since End k ( V ) ⊗ End k ( V ) ∼ = End k ( V ⊗ V ), it is knownthat φ is the flip map σ on V ⊗ V .Let r = P i a i ⊗ b i ∈ A ⊗ A . Then P r ( x ) = r ♯ φ ♯ − ( x ) = X i h φ ♯ − ( x ) , a i i b i = X i B ( x, a i ) b i = X i Tr( x · a i ) b i . Similarly, P tr ( x ) = P i Tr( x · b i ) a i . Suppose that r + σ ( r ) = − λσ + µ ( ⊗ ) = − λφ + µ ( ⊗ ) . If r satisfies Eq. (7), then both P r and P tr are Rota-Baxter operators of weight λ . This isexactly the example given in [34]. Example 3.4.
We can be more explicit with Example 3.3 when n = 2. Let E ij ∈ M ( k ),1 ≤ i, j ≤
2, be the matrix whose ( i, j )-entry is 1 and other entries are zero. Now thematrix algebra A = M ( C ) is a Frobenius algebra with the invariant bilinear form B givenby Eq. (26). An orthogonal basis with respect to the form is e = 1 √ E + E ) , e = 1 √ E − E ) , e = 1 √ E + E ) , e = 1 √− E − E ) . Hence the φ in Example 3.3 is φ = e ⊗ e + e ⊗ e + e ⊗ e + e ⊗ e = E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E . Note that the unit in M ( C ) is E + E . Then ⊗ = E ⊗ E + E ⊗ E + E ⊗ E + E ⊗ E . On the other hand, by a direct calculation, we find that r = E ⊗ E − E ⊗ E is asolution of the nhacYBe with µ = − M ( C ). Then we have r + σ ( r ) = E ⊗ E − E ⊗ E + E ⊗ E − E ⊗ E = φ − ⊗ . Hence by Theorem 3.1, we have a Rota-Baxter operator P r of weight − P r ( E ) = − E , P r ( E ) = E , P r ( E ) = P r ( E ) = 0 . From O -operators and dendriform algebras to nhacYBe on semi-direct prod-uct algebras. We now show that O -operators of weight zero and dendriform algebras cangive rise to solutions of the nhacYBe in some semidirect product algebras. We first gener-alize one direction of Theorem 3.1 by relaxing the condition that the extended symmetrizerof r is a multiple of a nondegenerate invariant tensor giving by a symmetric Frobeniusalgebra. Proposition 3.5.
Let ( A, · , ) be a unital k -algebra. Let s ∈ A ⊗ A be symmetric andinvariant. Let P : A → A be a linear map satisfying s ♯ P ∗ ( a ∗ ) + P s ♯ ( a ∗ ) = − λs ♯ ( a ∗ ) + µ h a ∗ , i , ∀ a ∗ ∈ A ∗ , PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 17 where P ∗ is the linear dual of P . Let r and r be defined by r ♯ = s ♯ P ∗ , r ♯ = P s ♯ . Explicitly,setting s = P i a i ⊗ b i , then r := X i P ( a i ) ⊗ b i , r := X i a i ⊗ P ( b i ) . (27) If P is a Rota-Baxter operator of weight λ , then r and r are symmetrized invariantsolutions of the nhacYBe in A .Conversely, suppose that s is nondegenerate. Let r ∈ A ⊗ A satisfy r + σ ( r ) = − λs + µ ( ⊗ ) . Let P r , P tr : A → A be the linear maps defined respectively by P r ( x ) := r ♯ s ♯ − ( x ) , P tr ( x ) := r t♯ s ♯ − ( x ) , ∀ x ∈ A. If r is a solution of the nhacYBe, then P r and P tr are Rota-Baxter operators of weight λ .Proof. In fact, we have r ♯ = r t ♯ since h r t ♯ ( a ∗ ) , b ∗ i = h s ♯ P ∗ ( b ∗ ) , a ∗ i = h s ♯ ( a ∗ ) , P ∗ ( b ∗ ) i = h P s ♯ ( a ∗ ) , b ∗ i = h r ♯ ( a ∗ ) , b ∗ i , ∀ a ∗ , b ∗ ∈ A ∗ . Hence r = σ ( r ). For any a ∗ , b ∗ ∈ A ∗ , we have h r + σ ( r ) + λs − µ ( ⊗ ) , a ∗ ⊗ b ∗ i = h s ♯ P ∗ ( a ∗ ) , b ∗ i + h s ♯ P ∗ ( b ∗ ) , a ∗ i + λ h s ♯ ( a ∗ ) , b ∗ i − µ h , a ∗ ih , b ∗ i = h s ♯ P ∗ ( a ∗ ) + P s ♯ ( a ∗ ) + λs ♯ ( a ∗ ) − µ h a ∗ , i , b ∗ i = 0 . Hence r + σ ( r ) + λs − µ ( ⊗ ) = 0. For any a ∗ , b ∗ , c ∗ ∈ A ∗ , we have h r ♯ ( a ∗ ) · r ♯ ( b ∗ ) , c ∗ i = h s ♯ P ∗ ( a ∗ ) · s ♯ P ∗ ( b ∗ ) , c ∗ i = h s ♯ P ∗ ( b ∗ ) , c ∗ L ∗ ( s ♯ P ∗ ( a ∗ )) i = h b ∗ , P ( s ♯ ( c ∗ ) · s ♯ P ∗ ( a ∗ )) i = h b ∗ , − P ( s ♯ ( c ∗ ) · P ( s ♯ ( a ∗ ))) i + h b ∗ , P ( − λs ♯ ( c ∗ ) · s ♯ ( a ∗ ) + µ h , a ∗ i s ♯ ( c ∗ )) i , h r ♯ ( a ∗ L ∗ ( r ♯ ( b ∗ ))) , c ∗ i = h s ♯ P ∗ ( a ∗ L ∗ ( s ♯ P ∗ ( b ∗ ))) , c ∗ i = h a ∗ , s ♯ P ∗ ( b ∗ ) · P ( s ♯ ( c ∗ )) i = h a ∗ , s ♯ ( P ∗ ( b ∗ ) L ∗ P ( s ♯ ( c ∗ ))) i = h b ∗ , P ( P ( s ♯ ( c ∗ )) · s ♯ ( a ∗ )) i , h r ♯ ( R ∗ ( r ♯ ( a ∗ )) b ∗ ) , c ∗ i = h s ♯ P ∗ ( R ∗ ( s ♯ P ∗ ( a ∗ )) b ∗ ) , c ∗ i = h R ∗ ( s ♯ P ∗ ( a ∗ )) b ∗ , P ( s ♯ ( c ∗ ) i = h b ∗ , P ( s ♯ ( c ∗ )) · s ♯ P ∗ ( a ∗ ) i = h b ∗ , − P ( s ♯ ( c ∗ )) · P ( s ♯ ( a ∗ )) i + h b ∗ , − λP ( s ♯ ( c ∗ )) · s ♯ ( a ∗ )+ µ h , a ∗ i P ( s ♯ ( c ∗ )) i , h λr ♯ ( a ∗ L · ( s ♯ ( b ∗ ))) , c ∗ i = h λs ♯ P ∗ ( a ∗ L ∗ ( s ♯ ( b ∗ ))) , c ∗ i = h a ∗ , λs ♯ ( b ∗ ) · P s ♯ ( c ∗ ) i = h a ∗ , λs ♯ ( b ∗ L ∗ ( P s ♯ ( c ∗ ))) i = h b ∗ , λP ( s ♯ ( c ∗ )) · s ♯ ( a ∗ ) i . Hence if P is a Rota-Baxter operator of weight λ , then r ♯ is an O -operator associatedto the A -bimodule k -algebra ( A ∗ , ◦ , R ∗ , L ∗ ), where ◦ is defined from − λs . Hence r is asolution of the nhacYBe by Theorem 2.22. By Theorem 2.22 again, r is also a solution ofthe nhacYBe since r ♯ = r t♯ = σ ( r ) ♯ .If s is nondegenerate, then from the above proof, it is obvious that the converse istrue. Alternatively, note that when s is nondegenerate, symmetric and invariant, then itcorresponds to a nondegenerate, symmetric and invariant bilinear form B by Lemma 2.15through Eq. (15) such that ( A, B ) is a Frobenius algebra. Then the conclusion follows fromTheorem 3.1. (cid:3) Remark 3.6.
When µ = 0, the tensor r in Eq. (27) recovers a construction in [18].In the rest of this subsection, we provide symmetrized invariant solutions of the nhacYBein semi-direct product algebras from O -operators of weight zero and dendriform algebrasby applying Proposition 3.5. We first supply more background.Let ( A, · ) be a k -algebra and ( V, l, r ) be an A -bimodule. Let ( V ∗ , r ∗ , l ∗ ) be the dual A -bimodule. Denote the semi-direct product algebras b A := A ⋉ l,r V, A := A ⋉ r ∗ ,l ∗ V ∗ . Identify a linear map β : V → A with an element in A ⊗ A by the injective mapHom k ( V, A ) ∼ = A ⊗ V ∗ ֒ → A ⊗ A . Proposition 3.7. ([7])
Let A be a k -algebra and ( V, ℓ, r ) be an A -bimodule. Let α : V → A be a linear map. Then α is an O -operator of weight zero if and only if the linear map b α ( x, u ) := ( α ( u ) , − λu ) , ∀ x ∈ A, u ∈ V, (28) is a Rota-Baxter operator of weight λ on the algebra b A . Lemma 3.8. ([9])
Let ( A, · ) be a k -algebra and ( V, l, r ) be an A -bimodule. Let β : V → A be a linear map. Then e β = β + σ ( β ) ∈ A ⊗ A is invariant if and only if β is a balanced A -bimodule homomorphism , that is, β ( l ( x ) u ) = x · β ( v ) , β ( ur ( x )) = β ( u ) · x, l ( β ( u )) v = ur ( β ( v )) , ∀ x ∈ A, u, v ∈ V. (29) Theorem 3.9.
Let ( A, · , ) be a unital k -algebra and ( V, ℓ, r ) be an A -bimodule. Assumethat α : V → A is an O -operator of weight zero and β : V ∗ → A is a balanced A -bimodulehomomorphism. Let b α be given by Eq. (28) and e β := β + σ ( β ) ∈ b A ⊗ b A . Let r , r ∈ b A ⊗ b A be defined by r ♯ := e β ♯ b α ∗ , r ♯ := b α e β ♯ . If α and β satisfy βα ∗ ( x ∗ ) + αβ ∗ ( x ∗ ) = µ h x ∗ , i , ∀ x ∗ ∈ A ∗ , then r and r are symmetrized invariant solutions of the nhacYBe in b A , with s = e β .Proof. By Proposition 3.7, b α is a Rota-Baxter operator of weight λ of b A . By Lemma 3.8, e β ∈ b A ⊗ b A is invariant. Moreover, we have b α ∗ ( x ∗ , u ∗ ) = (0 , α ∗ ( x ∗ ) − λu ∗ ) , e β ♯ ( x ∗ , u ∗ ) = ( β ( u ∗ ) , β ∗ ( x ∗ )) , ∀ x ∗ ∈ A ∗ , u ∗ ∈ V ∗ . Hence for any x ∗ ∈ A ∗ , u ∗ ∈ V , we have e β ♯ b α ∗ ( x ∗ , u ∗ ) + b α e β ♯ ( x ∗ , u ∗ ) + λ e β ♯ ( x ∗ , u ∗ ) − µ h ( x ∗ , u ∗ ) , ( , i ( , βα ∗ ( x ∗ ) − λβ ( u ∗ ) ,
0) + ( αβ ∗ ( x ∗ ) , − λβ ∗ ( x ∗ )) + λ ( β ( u ∗ ) , β ∗ ( x ∗ )) − ( µ h x ∗ , i , βα ∗ ( x ∗ ) + αβ ∗ ( x ∗ ) − µ h x ∗ , i ,
0) = 0 . By Proposition 3.5, the desired result follows. (cid:3)
Corollary 3.10.
Let ( A, ) be a unital k -algebra. Let s ∈ A ⊗ A be symmetric and invariant.Let P : A → A be a linear map satisfying s ♯ P ∗ ( a ∗ ) + P s ♯ ( a ∗ ) = µ h a ∗ , i , ∀ a ∗ ∈ A ∗ . Suppose that P is a Rota-Baxter operator of weight zero. PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 19 (a)
Let r , r ∈ A ⊗ A be defined by r ♯ := s ♯ P ∗ , r ♯ := P s ♯ . Then r and r are symmetrized invariant solutions of the nhacYBe in A whoseextended symmetrizers are zero. (b) Set b A := A ⋉ L,R A . Let b P be given by Eq. (28) with e s ♯ = s ♯ + σ ( s ♯ ) ∈ b A ⊗ b A . Let r , r ∈ b A ⊗ b A be defined by r ♯ := (cid:0) e s ♯ (cid:1) ♯ b P ∗ , r ♯ := b P (cid:0) e s ♯ (cid:1) ♯ . Then r and r are symmetrized invariant solutions of the nhacYBe in b A with s = e s ♯ .Proof. (a) follows from Proposition 3.5 with λ = 0.(b) follows from Theorem 3.9 where ( V, l, r ) = (
A, L, R ) and P = α , β = s ♯ . Note that inthis case, if s is invariant and symmetric, then s ♯ is a balanced A -module homomorphism,that is, s ♯ satisfies Eq. (29). (cid:3) Corollary 3.11.
Let ( A, · , ) be a unital k -algebra. Set b A := A ⋉ R ∗ ,L ∗ A ∗ . Assume that β : A → A is a linear map satisfying β ( x · y ) = β ( x ) · y = x · β ( y ) , ∀ x, y ∈ A. (30) Let α : A ∗ → A be an O -operator of weight zero associated to ( A ∗ , R ∗ , L ∗ ) . Let b α be givenby Eq. (28) and e β = β + σ ( β ) ∈ b A ⊗ b A . Let r, r ′ ∈ b A ⊗ b A be defined by r ♯ := e β ♯ b α ∗ , r ′ ♯ := b α e β ♯ . If α and β satisfy βα ∗ ( x ∗ ) + αβ ∗ ( x ∗ ) = µ h x ∗ , i , ∀ x ∗ ∈ A ∗ , then r and r ′ are symmetrized invariant solutions of the nhacYBe in b A , when taking s = e β .In particular, suppose that β = id . Then β satisfies Eq. (30) . Suppose that α ( x ∗ ) + α ∗ ( x ∗ ) = µ h x ∗ , i , ∀ x ∗ ∈ A ∗ . (a) Let r , r ∈ b A ⊗ b A be defined by r ♯ := e id ♯ b α ∗ , r ♯ := b α e id ♯ . Then r and r are symmetrized invariant solutions of the nhacYBe in b A with s = e id . (b) Let r , r ∈ A ⊗ A be defined by r ♯ := α, r ♯ := α ∗ . Then r and r are symmetrized invariant solutions of the nhacYBe in A .Proof. The first half part follows from Theorem 3.9 by taking (
V, l, r ) := ( A ∗ , R ∗ , L ∗ ). Notethat in this case, Eq. (29) is exactly Eq. (30).(a) follows from the above proof in the case when β = id.(b) follows from Corollary 2.25 in the case that the extended symmetrizer is zero. (cid:3) We finally provide solutions of the nhacYBe from dendriform algebras.
Definition 3.12. [30] Let A be a vector space with two bilinear products denoted by ≺ and ≻ . Then ( A, ≺ , ≻ ) is called a dendriform algebra if for all a, b, c ∈ A , ( a ≺ b ) ≺ c = a ≺ ( b ≺ c + b ≻ c ) , ( a ≻ b ) ≺ c = a ≻ ( b ≺ c ) , ( a ≺ b + a ≻ b ) ≻ c = a ≻ ( b ≻ c ) . Let ( A, ≺ , ≻ ) be a dendriform algebra. For any a ∈ A , let L ≺ ( a ), R ≺ ( a ) and L ≻ ( a ), R ≻ ( a ) denote the left and right multiplication operators on ( A, ≺ ) and ( A, ≻ ), respectively.Furthermore, define linear maps R ≺ , L ≻ : A → End k ( A ) , a R ≺ ( a ) , a L ≻ ( a ) , ∀ a ∈ A. As is well-known, for a dendriform algebra ( A, ≺ , ≻ ), the multiplication a ⋆ b := a ≺ b + a ≻ b, ∀ a, b ∈ A, defines a k -algebra ( A, ⋆ ), called the associated algebra of the dendriform algebra. More-over, (
A, L ≻ , R ≺ ) is a bimodule of the algebra ( A, ⋆ ) [5, 30].A unital dendriform algebra [19] is a k -module A := k1 ⊕ A + such that ( A + , ≺ , ≻ )is a dendriform algebra and the operations ≺ and ≻ are extended (partially) to A by x ≺ = ≻ x = x, x ≻ = ≺ x = 0 , ∀ x ∈ A + . Note that ≺ and ≻ are not defined. Then ( A, ⋆, ) is a unital k -algebra. Corollary 3.13.
Let ( A, ≺ , ≻ , ) be a unital dendriform algebra with the unit . Let ( A, ⋆ ) be the associated unital k -algebra with the unit . Suppose that there is a linear map β : A ∗ → A satisfying β ( R ∗≺ ( x ) y ∗ ) = x ⋆ β ( y ∗ ) , β ( y ∗ L ∗≻ ( x )) = β ( y ∗ ) ⋆ x, R ∗≺ ( β ( y ∗ )) z ∗ = y ∗ L ∗≻ ( β ( z ∗ )) , for any x ∈ A, y ∗ , z ∗ ∈ A ∗ . Set b A = A ⋉ L ≻ ,R ≺ A . Let b id be given by Eq. (28) , that is, b id( x, y ) = ( y, − λy ) , ∀ x, y ∈ A, and e β = β + σ ( β ) ∈ b A ⊗ b A . If in addition, β satisfies β ( x ∗ ) + β ∗ ( x ∗ ) = µ h x ∗ , i , ∀ x ∗ ∈ A ∗ , then r and r defined by r ♯ := e β ♯ b id ∗ , r ♯ := b id e β ♯ are symmetrized invariant solutions of the nhacYBe in b A , with s = e β .Proof. Note that the identity map id is an O -operator of the associated algebra ( A, ⋆ )associated to the bimodule (
A, L ≻ , R ≺ ). Hence the conclusion follows from Theorem 3.9. (cid:3) Remark 3.14.
The above constructions of symmetrized invariant solutions of the nhacYBeare different from the construction of solutions of the AYBE from O -operators given in [9],where the symmetric invariant tensors appearing in the symmetric parts of solutions in thesemi-direct product algebras can be “lifted” from linear maps from the bimodules to the k -algebras themselves as Lemma 3.8 illustrates. However, it is not true for the symmetrictensor ⊗ any more, that is, the approach in [9] does not apply here due to the appearanceof the new term µ ( ⊗ ).3.3. From nhacYBe to Rota-Baxter operators on unitization algebras.
We endthe section with constructions of Rota-Baxter operators from solutions of the nhacYBe inunitization algebras, or equivalently, augmented algebras.The unitization of a not necessarily unital k -algebra A ′ is the direct sum k -algebra A := k ⊕ A ′ . An augmentation map on a unital k -algebra ( A, · , ) is a k -algebra homomorphism ε : A → k . An augmented unital k-algebra is a unital k -algebra ( A, · , ) with anaugmentation map ε . PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 21
As is well-known [16, Theorem 5.1.1], augmented unital k -algebras are precisely theunitizations of (not necessarily unital) algebras given by k ⊕ A ′ ←→ ( A, ε ) , where A := k ⊕ A ′ , ε is the projection to k , while A ′ is ker ε . Remark 3.15.
For an augmented unital k -algebra ( A, · , , ε ) with augmentation map ε ,there is a basis { e , · · · , e n } of A such that e = and { e , · · · , e n } is a basis of ker ε = A ′ .Let { e ∗ , · · · , e ∗ n } be the dual basis. Then ε = e ∗ .The following conclusion is obvious. Lemma 3.16.
Let ( A, · , ) be a unital k -algebra and ε be an augmentation map. Then ε ( ) = 1 k , and ε ( x · y · z ) = ε ( y · z · x ) = ε ( z · x · y ) = ε ( x ) ε ( y ) ε ( z ) , ∀ x, y, z ∈ A. (31)Let ( A, · , , ε ) be an augmented unital k -algebra. Define linear maps ε l : A ⊗ A → k ⊗ A, ε r : A ⊗ A → A ⊗ k respectively by ε l := ε ⊗ id , ε r := id ⊗ ε. Similarly, define linear maps ε : A ⊗ A ⊗ A → k ⊗ k ⊗ A, ε : A ⊗ A ⊗ A → A ⊗ k ⊗ k , ε : A ⊗ A ⊗ A → k ⊗ A ⊗ k respectively by ε := ε ⊗ ε ⊗ id , ε := id ⊗ ε ⊗ ε, ε := ε ⊗ id ⊗ ε. Denote the natural isomorphisms of algebras [22] β ℓ : k ⊗ A → A, k ⊗ a a ; β r : A ⊗ k → A, x ⊗ k x, ∀ x ∈ A. Similarly, define natural isomorphisms of algebras β : k ⊗ k ⊗ A → A, k ⊗ k ⊗ x x,β : A ⊗ k ⊗ k → A, x ⊗ k ⊗ k x,β : k ⊗ A ⊗ k → A, k ⊗ x ⊗ k x, ∀ x ∈ A. For any x ∈ A , set x ( l ) := x ⊗ ∈ A ⊗ A, x ( r ) := ⊗ x ∈ A ⊗ A,x (1) := x ⊗ ⊗ ∈ A ⊗ A ⊗ A, x (2) := ⊗ x ⊗ ∈ A ⊗ A ⊗ A, x (3) := ⊗ ⊗ x ∈ A ⊗ A ⊗ A. Theorem 3.17.
Let ( A, · , , ε ) be an augmented unital k -algebra. Let r = P i a i ⊗ b i ∈ A ⊗ A be a solution of the nhacYBe and r be the extended symmetrizer of r . Define linear maps P, P ′ : A → A by P ( x ) := X i ε ( a i · x ) b i , P ′ ( x ) := X i ε ( b i · x ) a i , ∀ x ∈ A. (32)(a) If r is nonzero and satisfies β l ( ε l ( r · x ( l ) )) = x, ∀ x ∈ A, (33) then P and P ′ are Rota-Baxter operators of weight − . (b) If r = 0 , then P and P ′ are Rota-Baxter operators of weight zero. Proof. (a). Let x, y ∈ A . By definition, we have P ( x ) = β l ε l ( r · x ( l ) ) = β ( ε ( r · x (1) )) = β ( ε ( r · x (1) )) = β ( ε ( r · x (2) )) , (34) P ′ ( x ) = β r ε r ( r · x ( r ) ) = β l ε l ( σ ( r ) · x ( l ) )= β ( ε ( r · x (2) )) = β ( ε ( r · x (3) )) = β ( ε ( r · x (3) )) . (35)Since r satisfies Eq. (7), we have r · r · x (1) · y (2) + r · r · x (1) · y (2) − r · r · x (1) · y (2) = µr · x (1) · y (2) . Applying β ε : A ⊗ A ⊗ A → A to both sides of the above equation, we get β ε (cid:0) r · r · x (1) · y (2) + r · r · x (1) · y (2) − r · r · x (1) · y (2) (cid:1) = µβ (cid:0) ε ( r · x (1) · y (2) ) (cid:1) . (36)Furthermore, we have β (cid:0) ε ( r · r · x (1) · y (2) ) (cid:1) = β ( ε ( X i,j ( a i · a j · x ) ⊗ ( b i · y ) ⊗ b j ))= β ( X i,j ε ( a i · a j · x ) ⊗ ε ( b i · y ) ⊗ b j )= X i,j ε ( a i · a j · x ) ε ( b i · y ) b j (32) = X j ε ( P ′ ( y ) · a j · x ) b j (31) = X j ε ( a j · x · P ′ ( y )) b j (32) = P ( x · P ′ ( y )) . (37)Similarly, we have β (cid:0) ε ( r · r · x (1) · y (2) ) (cid:1) = P ( x ) · P ( y ) , (38) β (cid:0) ε ( r · r · x (1) · y (2) ) (cid:1) = P ( P ( x ) · y ) , (39) β (cid:0) ε ( r · x (1) · y (2) ) (cid:1) = ε ( y ) P ( x ) . (40)Substituting Eqs. (37)-(40) into Eq. (36) gives P ( x ) · P ( y ) + P ( x · P ′ ( y )) − P ( P ( x ) · y ) = µε ( y ) P ( x ) . (41)Since the extended symmetrizer r of r is nonzero, we have β l ε l (( r + σ ( r )) · x ( l ) − µx ( l ) ) = β l ε l ( r · x ( l ) ) . By Eqs. (34), (35) and Eq. (33), we obtain P ′ ( x ) = x + µε ( x ) − P ( x ) . (42)Substituting Eq. (42) into Eq. (41) yields P ( x ) · P ( y ) + P (cid:16) x · (cid:0) y + µε ( y ) − P ( y ) (cid:1)(cid:17) − P ( P ( x ) · y )= P ( x ) · P ( y ) + P ( x · y ) + µε ( y ) P ( x ) − P ( x · P ( y )) − P ( P ( x ) · y )= µε ( y ) P ( x ) , PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 23 that is, P ( x ) · P ( y ) = P ( P ( x ) · y ) + P ( x · P ( y )) − P ( x · y ) , as required. Similarly, we prove that P ′ is also a Rota-Baxter operator of weight − P ( x ) · P ( y ) + P ( x · P ′ ( y )) − P ( P ( x ) · y ) = µε ( y ) P ( x ) . (43)Since the extended symmetrizer of r is zero, we obtain r + σ ( r ) − µ ( ⊗ ) = 0 , and so β l ε l (( r + σ ( r )) · x ( l ) − µx ( l ) ) = 0 . By Eqs. (34)-(35), we have P ′ ( x ) = µε ( x ) − P ( x ) . (44)Substituting Eq. (44) into Eq. (43) shows that P is a Rota-Baxter operator of weight zero.A similar argument proves that P ′ is a Rota-Baxter operator of weight zero. (cid:3) Corollary 3.18.
Let ( A, · , , ε ) be an augmented unital k -algebra. Let r ∈ A ⊗ A be anti-symmetric ( i.e. r + σ ( r ) = 0) . If r satisfies the AYBE, then the operator P defined byEq. (32) is a Rota-Baxter operator of weight zero.Proof. It follows from Theorem 3.17 (b) by taking µ = 0. (cid:3) Corollary 3.19.
With the conditions in Theorem 3.17, suppose that r ∈ A ⊗ A is nonzeroand invariant, that is, r · x ( l ) = x ( r ) · r , ∀ x ∈ A. As in Remark 3.15, let { e = , e , · · · , e n } be a basis of A and { e ∗ , e ∗ , · · · , e ∗ n } be the dual basis such that ε = e ∗ . Moreover, suppose r = ⊗ + X i,j> s ij e i ⊗ e j . Then linear maps P and P ′ defined by Eq. (32) are Rota-Baxter operators of weight − .Proof. For all x ∈ A , we have β l ε l ( r · x ( l ) ) = β l ε l ( x ( r ) · r ) = β l ( ε ( ) ⊗ x ) + X i,j> β l ( s ij ε ( e i ) ⊗ ( x · e j )) = x, that is, r satisfies Eq. (33). Hence the conclusion follows from Theorem 3.17. (cid:3) Proposition 3.20.
Let ( A, · , ) be a unital k -algebra. If ε : A → k is an augmentationmap, then the bilinear form B on A defined by B ( x, y ) := ε ( x ) ε ( y ) , ∀ x, y ∈ A, (45) is symmetric and invariant. Moreover, B satisfies B ( x · y, z ) = B ( y · x, z ) , ∀ x, y, z ∈ A. In particular, if B is nondegenerate, then A is commutative. Conversely, if B is a sym-metric invariant bilinear form satisfying B ( x, y ) = B ( x · y,
1) = B ( x, B ( y, , ∀ x, y ∈ A, then the linear map ε : A → k defined by ε ( x ) := B ( x, , ∀ x ∈ A, is an augmentation map. Proof.
All the statements can be verified directly from the definitions. (cid:3)
Example 3.21.
Let ( A, · , , ε ) be an augmented unital commutative k -algebra. Let B bethe bilinear form defined by Eq. (45). Suppose that B is nondegenerate. Then ( A, · , B )is a symmetric Frobenius algebra. Let φ ♯ : A ∗ → A be the linear isomorphism defined byEq. (15). Let { e = , e , · · · , e n } be a basis of A satisfying B ( e i , e j ) = δ ij , ∀ i, j = 1 , · · · , n. Then φ ∈ A ⊗ A is invariant and φ = n X i =1 e i ⊗ e i = ⊗ + n X i =2 e i ⊗ e i . By Theorem 3.17 and Corollary 3.19, we show that if r satisfies Eqs. (7) and (24), thenthe linear maps P and P ′ defined by Eq. (32) are Rota-Baxter operators of weight λ . Notethat this conclusion also follows form Theorem 3.1, since in this case, P = P r and P ′ = P tr ,where P r and P tr are defined by Eq. (25).4. Classification of symmetrized invariant solutions of nhacYBe in lowdimensions
In this section, we classify symmetrized invariant solutions of the nhacYBe for µ = 0in the unital complex algebras in dimensions two and three and find that all of them areobtained from Rota-Baxter operators through Theorem 3.1. It would be interesting to seewhat happens for algebras in higher dimensions.4.1. The classification in dimension two.
The set of symmetric invariant tensors of a k -algebra A is a subspace of A ⊗ A and is denoted by Inv( A ).There are two two-dimensional unital C -algebras whose nonzero products with respectto a basis { e , e } are given by [35]( A
1) : e e = e , e e = e e = e ;( A
2) : e e = e , e e = e . By [29], for the algebra ( A r = µe ⊗ e of the nhacYBeEq. (7). By Remark 2.24, this solution is not symmetrized invariant.For the algebra ( A r = µ ( e ⊗ e + e ⊗ e + e ⊗ e ) , r = µ ( e ⊗ e + e ⊗ e + e ⊗ e ) ,r = µe ⊗ e , r = µe ⊗ e ,r = µ ( e ⊗ e + e ⊗ e ) , r = µ ( e ⊗ e + e ⊗ e ) ,r = µ ( e ⊗ e + e ⊗ e ) , r = µ ( e ⊗ e + e ⊗ e ) . Moreover, all of these solutions are obtained from Rota-Baxter operators through Theo-rem 3.1.To see this, note that r = σ ( r ) , r = σ ( r ) , r = σ ( r ) , r = σ ( r ) , PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 25 and the unit of the algebra ( A
2) is e + e . It is straightforward to show that Inv( A
2) =span { e ⊗ e , e ⊗ e } . Let B and B be the bilinear forms on ( A
2) defined respectivelyby B ( e , e ) = B ( e , e ) = 1 , B ( e , e ) = B ( e , e ) = 0; B ( e , e ) = 1 , B ( e , e ) = − , B ( e , e ) = B ( e , e ) = 0 . Then both B and B are symmetric, nondegenerate and invariant. Their correspondingsymmetric, invariant tensors from Lemma 2.15 are φ = e ⊗ e + e ⊗ e , φ = e ⊗ e − e ⊗ e , so that B i ( x, y ) = h φ ♯i − ( x ) , y i for any x, y ∈ ( A
2) and i = 1 ,
2. Now the 8 symmetrizedinvariant solutions of the nhacYBe satisfy r + σ ( r ) = r + σ ( r ) = r + r = µφ + µ ( e + e ) ⊗ ( e + e ); r + σ ( r ) = r + σ ( r ) = r + r = − µφ + µ ( e + e ) ⊗ ( e + e ); r + σ ( r ) = r + σ ( r ) = r + r = µφ + µ ( e + e ) ⊗ ( e + e ); r + σ ( r ) = r + σ ( r ) = r + r = − µφ + µ ( e + e ) ⊗ ( e + e ) . Thus by Theorem 3.1, the corresponding linear operator P r , P r , P r , P r are Rota-Baxteroperators of weight − µ and P r , P r , P r , P r are Rota-Baxter operators of weight µ . Ex-plicitly, the operators are defined by P r ( e ) = e + e , P r ( e ) = e ; P r ( e ) = e , P r ( e ) = e + e ; P r ( e ) = e , P r ( e ) = 0; P r ( e ) = 0 , P r ( e ) = e ; P r ( e ) = e + e , P r ( e ) = 0; P r ( e ) = e , P r ( e ) = − e ; P r ( e ) = e , P r ( e ) = − e ; P r ( e ) = 0 , P r ( e ) = − e − e . The classification in dimension three.
Any three-dimensional unital C -algebrais isomorphic to one of the following five [25, 40], with their nonzero products on a basis { e , e , e } given by( B
1) : e e = e , e e = e , e e = e ;( B
2) : e e = e , e e = e , e e = e e = e ;( B
3) : e e = e , e e = e e = e , e e = e e = e , e e = e ;( B
4) : e e = e , e e = e e = e , e e = e e = e , e e = e , e e = e ;( B
5) : e e = e , e e = e e = e , e e = e e = e . Solutions of the nhacYBe in these algebras were classified in [29] as follows. For thealgebras ( B
3) and ( B r = µe ⊗ e and it is notsymmetrized invariant.For the algebra ( B B
4) = 0. Hence in this case,by the classification of solutions of the nhacYBe given in [29], none of the nonzero solutionsis symmetrized invariant.For the algebra ( B e + e is the unit. Moreover, the vector subspace S spanned by e , e is a unital subalgebra of ( B A
2) in Section 4.1. As discussed there,there are 8 symmetrized invariant solutions r i , 1 ≤ i ≤
8, of the nhacYBe in S , togetherwith the corresponding Rota-Baxter operators P r i , ≤ i ≤ A B Rota-Baxter operators on ( B
2) are extended from P r i , i = 1 , · · · , P r i ( e ) = 0,as shown in [3].For the algebra ( B is e + e + e andInv( B
1) = span { e ⊗ e , e ⊗ e , e ⊗ e } . Set φ := e ⊗ e + e ⊗ e + e ⊗ e , φ := e ⊗ e + e ⊗ e − e ⊗ e ,φ := e ⊗ e − e ⊗ e + e ⊗ e , φ := − e ⊗ e + e ⊗ e + e ⊗ e . According to their extended symmetrizers r := r + σ ( r ) − µ ( ⊗ ) , these 48 solutions and their corresponding Rota-Baxter operators are grouped together asfollows. r = µ ( e ⊗ e + e ⊗ e + e ⊗ e ) , P r ( e ) = 0 , P r ( e ) = e , P r ( e ) = e + e ; r = µ ( e ⊗ e + e ⊗ e + e ⊗ e ) , P r ( e ) = e + e , P r ( e ) = e , P r ( e ) = 0; r = µ ( e ⊗ e + e ⊗ e + e ⊗ e ) , P r ( e ) = 0 , P r ( e ) = e + e , P r ( e ) = e ; r = µ ( e ⊗ e + e ⊗ e + e ⊗ e ) , P r ( e ) = e + e , P r ( e ) = 0 , P r ( e ) = e ; r = µ ( e ⊗ e + e ⊗ e + e ⊗ e ) , P r ( e ) = e , P r ( e ) = e + e , P r ( e ) = 0; r = µ ( e ⊗ e + e ⊗ e + e ⊗ e ) , P r ( e ) = e , P r ( e ) = 0 , P r ( e ) = e + e , for which r = − µφ and the Rota-Baxter operators are of weight µ . r = r + µφ , P r ( e ) = e , P r ( e ) = e + e , P r ( e ) = e + e + e ; r = r + µφ , P r ( e ) = e + e + e , P r ( e ) = e + e , P r ( e ) = e ; r = r + µφ , P r ( e ) = e , P r ( e ) = e + e + e , P r ( e ) = e + e ; r = r + µφ , P r ( e ) = e + e + e , P r ( e ) = e , P r ( e ) = e + e ; r = r + µφ , P r ( e ) = e + e , P r ( e ) = e + e + e , P r ( e ) = e ; r = r + µφ , P r ( e ) = e + e , P r ( e ) = e , P r ( e ) = e + e + e , for which r = µφ and the Rota-Baxter operators are of weight − µ . r = r + µ ( e ⊗ e ) , P r ( e ) = 0 , P r ( e ) = e , P r ( e ) = − e − e − e ; r = r + µ ( e ⊗ e ) , P r ( e ) = e + e , P r ( e ) = e , P r ( e ) = − e ; r = r + µ ( e ⊗ e ) , P r ( e ) = 0 , P r ( e ) = e + e , P r ( e ) = − e − e ; r = r + µ ( e ⊗ e ) , P r ( e ) = e + e , P r ( e ) = 0 , P r ( e ) = − e − e ; r = r + µ ( e ⊗ e ) , P r ( e ) = e , P r ( e ) = e + e , P r ( e ) = − e ; r = r + µ ( e ⊗ e ) , P r ( e ) = e , P r ( e ) = 0 , P r ( e ) = − e − e − e , for which r = − µφ and the Rota-Baxter operators are of weight µ . r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e , P r ( e ) = e + e , P r ( e ) = − e − e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e + e + e , P r ( e ) = e + e , P r ( e ) = 0; PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 27 r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e , P r ( e ) = e + e + e , P r ( e ) = − e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e + e + e , P r ( e ) = e , P r ( e ) = − e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e + e , P r ( e ) = e + e + e , P r ( e ) = 0; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e + e , P r ( e ) = e , P r ( e ) = − e − e , for which r = µφ and the Rota-Baxter operators are of weight − µ . r = r + µ ( e ⊗ e ) , P r ( e ) = 0 , P r ( e ) = − e − e , P r ( e ) = e + e ; r = r + µ ( e ⊗ e ) , P r ( e ) = e + e , P r ( e ) = − e − e , P r ( e ) = 0; r = r + µ ( e ⊗ e ) , P r ( e ) = 0 , P r ( e ) = − e − e − e , P r ( e ) = e ; r = r + µ ( e ⊗ e ) , P r ( e ) = e + e , P r ( e ) = − e , P r ( e ) = e ; r = r + µ ( e ⊗ e ) , P r ( e ) = e , P r ( e ) = − e − e − e , P r ( e ) = 0; r = r + µ ( e ⊗ e ) , P r ( e ) = e , P r ( e ) = − e , P r ( e ) = e + e , for which r = − µφ and the Rota-Baxter operators are of weight µ . r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e , P r ( e ) = − e , P r ( e ) = e + e + e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e + e + e , P r ( e ) = − e , P r ( e ) = e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e , P r ( e ) = − e − e , P r ( e ) = e + e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e + e + e , P r ( e ) = 0 , P r ( e ) = e + e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e + e , P r ( e ) = − e − e , P r ( e ) = e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = e + e , P r ( e ) = 0 , P r ( e ) = e + e + e , for which r = µφ and the Rota-Baxter operators are of weight − µ . r = r + µ ( e ⊗ e ) , P r ( e ) = − e , P r ( e ) = e , P r ( e ) = e + e ; r = r + µ ( e ⊗ e ) , P r ( e ) = − e − e − e , P r ( e ) = e , P r ( e ) = 0; r = r + µ ( e ⊗ e ) , P r ( e ) = − e , P r ( e ) = e + e , P r ( e ) = e ; r = r + µ ( e ⊗ e ) , P r ( e ) = − e − e − e , P r ( e ) = 0 , P r ( e ) = e ; r = r + µ ( e ⊗ e ) , P r ( e ) = − e − e , P r ( e ) = e + e , P r ( e ) = 0; r = r + µ ( e ⊗ e ) , P r ( e ) = − e − e , P r ( e ) = 0 , P r ( e ) = e + e , for which r = − µφ and the Rota-Baxter operators are of weight µ . r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = 0 , P r ( e ) = e + e , P r ( e ) = e + e + e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = − e − e , P r ( e ) = e + e , P r ( e ) = e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = 0 , P r ( e ) = e + e + e , P r ( e ) = e + e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = − e − e , P r ( e ) = e , P r ( e ) = e + e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = − e , P r ( e ) = e + e + e , P r ( e ) = e ; r = r + µ ( e ⊗ e + e ⊗ e ) , P r ( e ) = − e , P r ( e ) = e , P r ( e ) = e + e + e , for which r = µφ and the Rota-Baxter operators are of weight − µ . Acknowledgements.
This work is supported by National Natural Science Foundation ofChina (Grant Nos. 11931009, 11771190). C. Bai is also supported by the FundamentalResearch Funds for the Central Universities and Nankai ZhiDe Foundation. Y. Zhang is supported by China Scholarship Council to visit University of Southern California and hewould like to thank Prof. Susan Montgomery for hospitality during his visit.
References [1] M. Aguiar, Pre-Poisson algebras, Lett. Math. Phys. (2000), 263-277. 3[2] M. Aguiar, On the associative analog of Lie bialgebras, J. Algebra (2001), 492-532. 3[3] H. An and C. Bai, From Rota-Baxter algebras to pre-Lie algebras,
J. Phys. A: Math. Theor. (2008),015201. 26[4] C. Bai, A unified algebraic approach to the classical Yang-Baxter equation, J. Phys. A: Math. Thero (2007), 11073-11082. 2[5] C. Bai, Double constructions of Frobenius algebras, Connes cocycles and their duality, J. Noncommut.Geom. (2010), 475-530. 5, 20[6] C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation andPostLie algebras, Comm. Math. Phys. (2010), 553-596. 3[7] C. Bai, L. Guo and X. Ni, O -operators on associative algebras, associative Yang-Baxter equations anddendriform algebras, in “Quantized Algebra and Physics”, Nankai Series in Pure, Applied Mathematicsand Theoretical Physics , World Scientific, Singapore (2011), 10-51. 3, 18[8] C. Bai, L. Guo and X. Ni, Generalizations of the classical Yang-Baxter equation and O -operators, J.Math. Phys. (2011), 063515. 3[9] C. Bai, L. Guo and X. Ni, O -operators on associative algebras and associative Yang-Baxter equations, Pacific J. Math. (2012), 257-289. 3, 4, 5, 6, 7, 10, 15, 18, 20[10] C. Bai, L. Guo and X. Ni, Relative Rota-Baxter operators and tridendriform algebras,
J. Algebra Appl. (2013), 1350027, 18 pp. 3[11] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J.Math. (1960), 731-742. 6[12] R.J. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys. (1972), 193-228. 2[13] A. A. Belavin and V. G. Drinfeld, Solutions of the classical Yang-Baxter equation for simple Liealgebras, Funct. Anal. Appl. (1982), 159-180. 2[14] T. Brzezi´nski, Rota-Baxter systems, dendriform algebras and covariant bialgebras, J. Algebra (2016), 1-25. 4, 5, 12, 14[15] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge(1994). 2[16] P. M. Cohn, Basic Algebra: Groups, Rings and Fields, Springer, 2003. 21[17] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert prob-lem. I.
Comm. Math. Phys. (2000), 249-273. 6[18] K. Ebrahimi-Fard, Rota-Baxter algebras and the Hopf algebra of renormalization, Ph.D. Dissertation,University of Bonn, 2006. 3, 8, 15, 18[19] L. Foissy, Bidendriform bialgebras, trees, and free quasi-symmetric functions,
J. Pure Appl. Algebra (2007), 439-459. 20[20] L. Foissy, The infinitesimal Hopf algebra and the poset of planar forests,
J. Algebr. Comb. (2009),277-309. 3[21] L. Foissy, The infinitesimal Hopf algebra and the operads of planar forests, Int. Math. Res. Not. IMRN (2010), 395-435. 3[22] L. Guo, An Introduction to Rota-Baxter Algebra, Surveys of Modern Mathematics , InternationalPress, Somerville, MA; Higher Education Press, Beijing, 2012. 6, 21[23] P. S. Hirschhorn and L. A. Raphael, Coalgebraic foundations of the method of divided differences, Adv. Math. (1992), 75-135. 3[24] S.A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Studies in Appl. Math. (1979), 93-139. 3[25] Y. Kobayashi, K. Shirayanagi, S. -E. Takahasi and M. Tsukada, A complete classification of three-dimensional algebras over R and C , arXiv:1903.01623. 25[26] T. Krasnov and A. Zotov, Trigonometric integrable tops from solutions of associative Yang-Baxterequation, Ann. Henri Poincar´e (2019), 2671-2697. 3 PERATOR FORMS OF NONHOMOGENEOUS ASSOCIATIVE YANG-BAXTER EQUATION 29 [27] B. A. Kupershmidt, What a classical r -matrix really is, J. Nonlinear Math. Phys. (1999), 448-488. 2[28] Y. Lekili and A. Polishchuk, Associative Yang-Baxter equation and Fukaya categories of square-tiledsurfaces, Adv. Math. (2019), 273-315. 3[29] X. Liu, Solutions to non-homogenous associative classical Yang-Baxter equation in low dimensions,Thesis for Bachelor Degree, Nankai University (2017). 24, 25, 26[30] J.-L. Loday, Dialgebras. In “Dialgebras and Related Operads”,
Lecture Notes in Math. , Springer,Berlin (2001), 7-66. 4, 19, 20[31] J.-L. Loday and M. O. Ronco, On the structure of cofree Hopf algebras,
J. Reine Angew. Math. (2006), 123-155. 3[32] A. V. Odesskii, V. Rubtsov and V. Sokolov, Parameter-dependent associative Yang-Baxter equationsand Poisson brackets,
Inter. J. Geom. Methods Modern Phys. (2014), 1460036. 3[33] A. V. Odesskii and V. V. Sokolov, Pairs of compatible associative algebras, classical Yang-Baxterequation and quiver representations, Comm. Math. Phys. (2008) 83-99. 3[34] O. Ogievetsky and T. Popov, R -matrices in rime, Adv. Theor. Math. Phys. (2010), 439-505. 3, 4,16[35] B. Peirce, Linear associative algebra, Amer. J. Math. (1881), 97-229. 24[36] A. Polishchuk, Classical Yang-Baxter equation and the A ∞ -constraint, Adv. Math. (2002), 56-95.3[37] G.-C. Rota, Baxter operators, an introduction, In: Gian-Carlo Rota on Combinatorics, Joseph P.S.Kung, Editor, Birkh¨auser, Boston, 1995, 504-512. 6[38] T. Schedler, Trigonometric solutions of the associative Yang-Baxter equation,
Math. Res. Lett. ,301-321. 3[39] M.A. Semenov-Tian-Shansky, What is a classical R-matrix? Funct. Anal. Appl. (1983), 259-272. 2[40] E. Study, ¨Uber systeme complexer zahlen und ihre anwendung in der theorie der transformationsgrup-pen, Monatsh. Math. u. Phisik (1890), 283-354. 25[41] C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. (1967), 1312-1314. 2[42] Y. Zhang, C. Bai and L. Guo, Totally compatible associative and Lie dialgebras, tridendriform algebrasand PostLie algebras, Sci. China Math. (2014), 259-273. 6[43] Y. Zhang, D. Chen, X. Gao and Y. F. Luo, Weighted infinitesimal unitary bialgebras on rooted forestsand weighted cocycles, Pacific J. Math. (2019), 741-766. 3[44] Y. Zhang, X. Gao and Y. F. Luo, Weighted infinitesimal unitary bialgebras on rooted forests, symmetriccocycles and pre-Lie algebras,
J Algebr. Comb. (2020), Doi: 10.1007/s10801-020-00942-7. 3[45] V. N. Zhelyabin, Jordan bialgebras and their connection with Lie bialgebras,
Algebra and Logic (1997), 1-15. 3 Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, China
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