aa r X i v : . [ m a t h . QA ] A p r QUADRI-BIALGEBRAS
XIANG NI AND CHENGMING BAI
Abstract.
We introduce a notion of quadri-bialgebra, which gives a bialgebra theory builtupon a quadri-algebra introduced by Aguiar and Loday. A quadri-bialgebra is equivalent to aManin triple of dendriform dialgebras associated to a nondegenerate 2-cocycle or a Manin tripleof quadri-algebras associated to a nondegenerate invariant bilinear form. Moreover, quadri-bialgebras fit into a framework of construction of certain linear operators on the double spaces. Introduction
At the beginning of 1990s, in order to study periodicity phenomena in algebraic K-theory,J.-L. Loday introduced the notion of dendriform dialgebra ([Lo1]). It has been attracting a greatinterest because of its connections with various fields in mathematics and physics (see [EMP]and the references therein).There is a remarkable fact that a Rota-Baxter operator (of weight zero), which first arose inprobability theory ([Bax]) and later became a subject in combinatorics ([R1, R2]), on an asso-ciative algebra naturally gives a dendriform dialgebra structure on the underlying vector spaceof the associative algebra ([Ag1, Ag2, E1]). Such unexpected relationships between dendriformdialgebras in the field of operads and algebraic topology and Rota-Baxter operators in the fieldof combinatorics and probability theory attracted many mathematicians’ attentions right away.Later, in 2003, in order to determine the algebraic structures behind a pair of commuting Rota-Baxter operators (on an associative algebra), which appear, for example, in the space of thelinear endomorphisms of an infinitesimal bialgebra, Aguiar and Loday introduced the notionof quadri-algebra ([AL]), which is a vector space equipped with four operations satisfying nineaxioms. Moreover, quadri-algebras have deep relationships with combinatorics and the theory ofHopf algebras ([AL, EG1]). A quadri-algebra is also regarded as the underlying algebra structureof a dendriform algebra with a nondegenerate 2-cocycle ([Bai3]).In this paper, we construct a bialgebra theory of quadri-algebras. We are mainly motivatedby the theory of Lie bialgebras ([CP, D]). Explicitly, in the finite-dimensional case, we consideran analogue of Manin triple of Lie algebras which is equivalent to a Lie bialgebra, namely, aManin triple of dendriform dialgebras associated to a nondegenerate 2-cocycle. We find that itis in fact equivalent to certain bialgebra structure of the underlying quadri-algebra, which leads
Mathematics Subject Classification.
Key words and phrases.
Quadri-algebra, dendriform dialgebra, classical Yang-Baxter equation, Rota-Baxteroperator, Nijenhuis operator. to the notion of quadri-bialgebra. Furthermore, it is interesting to show that such a structureis also equivalent to another Manin triple at the level of quadri-algebras, that is, a Manin tripleof quadri-algebras with a nondegenerate invariant bilinear form. Quadri-bialgebras have certainsimilar properties of Lie bialgebras. For example, there are the so-called coboundary quadri-bialgebras which lead to a construction from an analogue of the classical Yang-Baxter equationand there also exists a “Drinfeld double” construction for a finite-dimensional quadri-bialgebra.Moreover, we find that quadri-bialgebras fit into a framework of construction of certain linearoperators, such as Rota-Baxter operators and Nijenhuis operators in combinatorics ([Bax, E1,E2, R1, R2]), renormalization of perturbative quantum field theory (pQFT) ([CK, EG2, EGK1,EGK2]) and quantum physics ([CGM]), on the “double spaces”. We would like to point out thatquadri-bialgebras might be put into the framework of the so-called generalized bialgebras in thesense of Loday ([Lo2]), which will be considered elsewhere.The paper is organized as follows. In Section 2, we recall some basic facts on dendriform dial-gebras and quadri-algebras. In Section 3, we introduce the notion of Manin triple of dendriformdialgebras associated to a nondegenerate 2-cocycle and then interpret it in terms of matchedpairs of dendriform dialgebras. In Section 4, we introduce the notion of Manin triple of quadri-algebras associated to a nondegenerate invariant bilinear form and then interpret it in termsof matched pairs of quadri-algebras. We also show the equivalence between these two Manintriples. In Section 5, we introduce the notion of quadri-bialgebra as an equivalent bialgebrastructure corresponding to the aforementioned Manin triples by assuming that there is a quadri-algebra structure on the dual space. In Section 6, we study the coboundary cases which leadto a construction from certain algebraic equations, which could be regarded as analogues of theclassical Yang-Baxter equation. In Section 7, we recall some results in [Bai3] which reduce theaforementioned algebraic equations in a simple form, namely, Q -equation referring to a set of twoequations. We list some properties of Q -equation including the ones given in [Bai3] from anotherpoint of view. In Section 8, we construct families of Nijenhuis operators and Rota-Baxter oper-ators on certain “double spaces” of quadri-algebras, including the Drinfeld Q -doubles obtainedfrom quadri-bialgebras.Throughout this paper, all algebras and vector spaces are finite-dimensional over a fixed basefield F . We give some notations as follows.(1) Let V be a vector space. Let B : V ⊗ V → F be a symmetric or skew-symmetric bilinearform on a vector space V . A subspace W is called isotropic if W ⊂ W ⊥ , where(1.1) W ⊥ = { x ∈ V | B ( x, y ) = 0 , ∀ y ∈ W } . (2) Let ( A, ⋄ ) be a vector space with a bilinear operation ⋄ : A ⊗ A → A . Let L ⋄ ( x ) and R ⋄ ( x )denote the left and right multiplication operator respectively, that is, L ⋄ ( x ) y = R ⋄ ( y ) x = x ⋄ y UADRI-BIALGEBRAS 3 for any x, y ∈ A . We also simply denote them by L ( x ) and R ( x ) respectively without confusion.Moreover, let L ⋄ , R ⋄ : A → gl ( A ) be two linear maps with x → L ⋄ ( x ) and x → R ⋄ ( x ) respectively.(3) Let V be a vector space and let r = P i a i ⊗ b i ∈ V ⊗ V . Set(1.2) r = X i a i ⊗ b i ⊗ , r = X i a i ⊗ ⊗ b i , r = X i ⊗ a i ⊗ b i , where 1 is a symbol playing a similar role of unit. If in addition, there exists a bilinear operation ⋄ : V ⊗ V → V on V , then the operation between two r s is in an obvious way. For example,(1.3) 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 . and so on. Note Eq. (1.3) is independent of the existence of the unit.(4) Let V be a vector space. Any r ∈ V ⊗ V can be identified as a linear map T r : V ∗ → V inthe following way:(1.4) h u ∗ ⊗ v ∗ , r i = h u ∗ , T r ( v ∗ ) i , ∀ u ∗ , v ∗ ∈ V ∗ , where h , i is the canonical paring between V and V ∗ . r ∈ V ⊗ V is called nondegenerate if theabove induced linear map T r is invertible.(5) Let V , V be two vector spaces and let T : V → V be a linear map. Denote the dual(linear) map by T ∗ : V ∗ → V ∗ defined by(1.5) h v , T ∗ ( v ∗ ) i = h T ( v ) , v ∗ i , ∀ v ∈ V , v ∗ ∈ V ∗ . (6) Let A be an algebra and let V be a vector space. For any linear map ρ : A → gl ( V ), definea linear map ρ ∗ : A → gl ( V ∗ ) by(1.6) h ρ ∗ ( x ) v ∗ , u i = h v ∗ , ρ ( x ) u i , ∀ x ∈ A, u ∈ V, v ∗ ∈ V ∗ . Note that in this case, ρ ∗ is different from the one given by Eq. (1.5) which regards gl ( V ) as avector space, too.(7) Let V be a vector space, we sometimes use 1 to denote the identity transformation of V .2. Dendriform dialgebras and Quadri-algebras
Definition 2.1. ([Lo1]) A dendriform dialgebra ( A, ≺ , ≻ ) is a vector space A together with twobilinear operations ≺ , ≻ : A ⊗ A → A such that (for any x, y, z ∈ A )(2.1) ( x ≺ y ) ≺ z = x ≺ ( y ⋆ z ) , ( x ≻ y ) ≺ z = x ≻ ( y ≺ z ) , ( x ⋆ y ) ≻ z = x ≻ ( y ≻ z ) , where x ⋆ y = x ≺ y + x ≻ y . Moreover, a homomorphism between two dendriform dialgebras isdefined as a linear map (between the two dendriform dialgebras) which preserves the operationsrespectively. Remark 2.2. ([Lo1]) For a dendriform dialgebra ( A, ≺ , ≻ ), the bilinear operation given by(2.2) x ⋆ y := x ≺ y + x ≻ y, ∀ x, y ∈ A, XIANG NI AND CHENGMING BAI defines an associative algebra, which is denoted by ( As ( A ) , ⋆ ). Definition 2.3. ([Ag3, Bai2]) Let ( A, ≺ , ≻ ) be a dendriform dialgebra and let V be a vectorspace. Let l ≺ , l ≻ , r ≺ , r ≻ : A → gl ( V ) be four linear maps. V or ( V, l ≺ , r ≺ , l ≻ , r ≻ ) is called a bimodule of A if and only if the following equations holds (for any x, y ∈ A ):(2.3) r ≺ ( y ) r ≺ ( x ) = r ≺ ( x ⋆ y ) , r ≺ ( y ) l ≺ ( x ) = l ≺ ( x ) r ⋆ ( y ) , l ≺ ( x ≺ y ) = l ≺ ( x ) l ⋆ ( y ) , (2.4) r ≺ ( y ) r ≻ ( x ) = r ≻ ( x ≺ y ) , r ≺ ( y ) l ≻ ( x ) = l ≻ ( x ) r ≺ ( y ) , l ≺ ( x ≻ y ) = l ≻ ( x ) l ≺ ( y ) , (2.5) r ≻ ( y ) r ⋆ ( x ) = r ≻ ( x ≻ y ) , r ≻ ( y ) l ⋆ ( x ) = l ≻ ( x ) r ≻ ( y ) , l ≻ ( x ⋆ y ) = l ≻ ( x ) l ≻ ( y ) . Proposition 2.4. ([Bai2])
Let ( V, l ≺ , r ≺ , l ≻ , r ≻ ) be a bimodule of a dendriform dialgebra ( A, ≺ , ≻ ) . Then ( V ∗ , − r ∗≻ , l ∗≻ + l ∗≺ , r ∗≻ + r ∗≺ , − l ∗≺ ) is a bimodule of ( A, ≺ , ≻ ) . Definition 2.5. ([Bai2, K]) Let (
V, l ≺ , r ≺ , l ≻ , r ≻ ) be a bimodule of a dendriform dialgebra ( A, ≺ , ≻ ). A linear map T : V → A is called an O -operator associated to ( V, l ≺ , r ≺ , l ≻ , r ≻ ) if T satisfies(2.6) T ( u ) ≺ T ( v ) = T ( l ≺ ( T ( u )) v + r ≺ ( T ( v )) u ) , T ( u ) ≻ T ( u ) = T ( l ≻ ( T ( u )) v + r ≻ ( T ( v )) u ) , ∀ u, v ∈ V. Definition 2.6. ([AL]) A quadri-algebra ( A, տ , ր , ւ , ց ) is a vector space A together with fourbilinear operations տ , ր , ւ and ց : A ⊗ A → A satisfying the axioms below (for any x, y, z ∈ A )(2.7) ( x տ y ) տ z = x տ ( y ⋆ z ) , ( x ր y ) տ z = x ր ( y ≺ z ) , ( x ∧ y ) ր z = x ր ( y ≻ z ) , (2.8) ( x ւ y ) տ z = x ւ ( y ∧ z ) , ( x ց y ) տ z = x ց ( y տ z ) , ( x ∨ y ) ր z = x ց ( y ր z ) , (2.9) ( x ≺ y ) ւ z = x ւ ( y ∨ z ) , ( x ≻ y ) ւ z = x ց ( y ւ z ) , ( x ⋆ y ) ց z = x ց ( y ց z ) , where(2.10) x ≻ y := x ր y + x ց y, x ≺ y := x տ y + x ւ y, (2.11) x ∨ y := x ւ y + x ց y, x ∧ y := x տ y + x ր y, (2.12) x ⋆ y := x ց y + x ր y + x տ y + x ւ y = x ≻ y + x ≺ y = x ∨ y + x ∧ y. A homomorphism between two quadri-algebras is defined as a linear map (between the twoquadri-algebras) which preserves the operations respectively. Proposition 2.7. ([AL])
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra. (1) The bilinear operation given by Eq. (2.10) defines a dendriform dialgebra A h := ( A, ≺ , ≻ ) ,which is called the associated horizontal dendriform dialgebra. (2) The bilinear operation given by Eq. (2.11) defines a dendriform dialgebra A v := ( A, ∧ , ∨ ) ,which is called the associated vertical dendriform dialgebra. UADRI-BIALGEBRAS 5
Proposition 2.8. ([Bai3])
Let A be a vector space with four bilinear operations denoted by տ , ր , ւ and ց : A ⊗ A → A . Then the following conditions are equivalent: (1) ( A, տ , ր , ւ , ց ) is a quadri-algebra; (2) ( A, ≺ , ≻ ) defined by Eq. (2.10) is a dendriform dialgebra and ( A, L ւ , R տ , L ց , R ր ) is abimodule. (3) ( A, ∧ , ∨ ) defined by Eq. (2.11) is a dendriform dialgebra and ( A, L ր , R տ , L ց , R ւ ) is abimodule. Corollary 2.9. ([Bai3])
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra. Then ( A ∗ , − R ∗ր , L ∗∨ , R ∗∧ , − L ∗ւ ) is a bimodule of the associated horizontal dendriform dialgebra ( A, ≺ , ≻ ) and ( A ∗ , − R ∗ւ , L ∗≻ , R ∗≺ , − L ∗ր ) is a bimodule of the associated vertical dendriform dialgebra ( A, ∧ , ∨ ) . For brevity, we pay our main attention to the study of the associated vertical dendriformdialgebras of quadri-algebras in this paper. The corresponding study on the associated horizontaldendriform dialgebras is completely similar.
Definition 2.10. ([Bai2]) Let ( A, ∧ , ∨ ) be a dendriform dialgebra and let ( As ( A ) , ⋆ ) be theassociated associative algebra. Suppose that B : A ⊗ A → F is a symmetric bilinear form. B iscalled a 2 -cocycle of A if B satisfies(2.13) B ( x ⋆ y, z ) = B ( y, z ∧ x ) + B ( x, y ∨ z ) , ∀ x, y, z ∈ A. Proposition 2.11. ([Bai3])
Let ( A, ∧ , ∨ ) be a dendriform dialgebra equipped with a nondegen-erate -cocycle B . Define four bilinear operations տ , ր , ւ , ց : A ⊗ A → A by (2.14) B ( x տ y, z ) = B ( x, y ⋆ z ) , B ( x ր y, z ) = − B ( y, z ∨ x ) , (2.15) B ( x ւ y, z ) = − B ( x, y ∧ z ) , B ( x ց y, z ) = B ( y, z ⋆ x ) , ∀ x, y, z ∈ A. Then ( A, տ , ր , ւ , ց ) is a quadri-algebra such that ( A, ∧ , ∨ ) is the associated vertical dendriformdialgebra. Manin triples of dendriform dialgebras associated to a nondegenerate2-cocycle and matched pairs of dendriform dialgebras
Definition 3.1. A Manin triple of dendriform dialgebras associated to a nondegenerate 2-cocycle is a triple of dendriform dialgebras (
A, A + , A − ) together with a nondegenerate 2-cocycle B on A , such that:(1) A + and A − are sub-dialgebras of A ;(2) A = A + ⊕ A − as vector spaces;(3) A + and A − are isotropic with respect to B . XIANG NI AND CHENGMING BAI
It is denoted by (
A, A + , A − , B ). A homomorphism between two Manin triples of dendriformdialgebras associated to a nondegenerate 2-cocycle ( A, A + , A − , B A ) and ( B, B + , B − , B B ) is ahomomorphism of dendriform dialgebras ϕ : A → B such that(3.1) ϕ ( A + ) ⊂ B + , ϕ ( A − ) ⊂ B − , B A ( x, y ) = B B ( ϕ ( x ) , ϕ ( y )) , ∀ x, y ∈ A. Definition 3.2.
Let ( A, ∧ , ∨ ) be a dendriform dialgebra. If there is a dendriform dialgebrastructure on the direct sum of the underlying vector spaces of A and the dual space A ∗ such that A and A ∗ are sub-dialgebras and the natural symmetric bilinear form on A ⊕ A ∗ given by(3.2) B S ( x + a ∗ , y + b ∗ ) := h a ∗ , y i + h x, b ∗ i , ∀ x, y ∈ A ; a ∗ , b ∗ ∈ A ∗ , is a 2-cocycle, then ( A ⊕ A ∗ , A, A ∗ , B S ) is called a standard Manin triple of dendriform dialgebrasassociated to B S . Obviously, a standard Manin triple of dendriform dialgebras is a Manin triple of dendriformdialgebras. Conversely, we have
Proposition 3.3.
Every Manin triple of dendriform dialgebras is isomorphic to a standard one.Proof.
Since in this case A − and ( A + ) ∗ are identified by the nondegenerate 2-cocycle, the dendri-form dialgebra structure on A − is transferred to ( A + ) ∗ . Hence the dendriform dialgebra structureon A + ⊕ A − is transferred to A + ⊕ ( A + ) ∗ . Then the conclusion holds. (cid:3) Proposition 3.4. ([Bai2])
Let ( A, ∧ A , ∨ A ) and ( B, ∧ B , ∨ B ) be two dendriform dialgebras. Sup-pose that there are linear maps l ∧ A , r ∧ A , l ∨ A , r ∨ A : A → gl ( B ) and l ∧ B , r ∧ B , l ∨ B , r ∨ B : B → gl ( A ) such that ( l ∧ A , r ∧ A , l ∨ A , r ∨ A ) is a bimodule of A and ( l ∧ B , r ∧ B , l ∨ B , r ∨ B ) is a bimodule of B , andthey satisfy the following conditions: (3.3) ( l ∧ B ( a ) x ) ∧ A y + l ∧ B ( r ∧ A ( x ) a ) y = l ∧ B ( a )( x ⋆ A y ) , (3.4) l ∧ B ( l ∧ A ( x ) a ) y + ( r ∧ B ( a ) x ) ∧ A y = x ∧ A ( l ⋆ B ( a ) y ) + r ∧ B ( r ⋆ A ( y ) a ) x, (3.5) r ∧ B ( a )( x ∧ A y ) = r ∧ B ( l ⋆ A ( y ) a ) x + x ∧ A ( r ⋆ B ( a ) y ) , (3.6) ( l ∨ B ( a ) x ) ∧ A y + l ∧ B ( r ∨ A ( x ) a ) y = l ∨ B ( a )( x ∧ A y ) , (3.7) l ∧ B ( l ∨ A ( x ) a ) y + ( r ∨ B ( a ) x ) ∧ A y = x ∨ A ( l ∧ B ( a ) y ) + r ∨ B ( r ∧ A ( y ) a ) x, (3.8) r ∧ B ( a )( x ∨ A y ) = r ∨ B ( l ∧ A ( y ) a ) x + x ∨ A ( r ∧ B ( a ) y ) , (3.9) ( l ⋆ B ( a ) x ) ∨ A y + l ∨ B ( r ⋆ A ( x ) a ) y = l ∨ B ( a )( x ∨ A y ) , (3.10) l ∨ B ( l ⋆ A ( x ) a ) y + ( r ⋆ B ( a ) x ) ∨ A y = x ∨ A ( l ∨ B ( a ) y ) + r ∨ B ( r ∨ A ( y ) a ) x, (3.11) r ∨ B ( a )( x ⋆ A y ) = r ∨ B ( l ∨ A ( y ) a ) x + x ∨ A ( r ∨ B ( a ) y ) , UADRI-BIALGEBRAS 7 (3.12) ( l ∧ A ( x ) a ) ∧ B b + l ∧ A ( r ∧ B ( a ) x ) b = l ∧ A ( x )( a ⋆ B b ) , (3.13) l ∧ A ( l ∧ B ( a ) x ) b + ( r ∧ A ( x ) a ) ∧ B b = a ∧ B ( l ⋆ A ( x ) b ) + r ∧ A ( r ⋆ B ( b ) x ) a, (3.14) r ∧ A ( x )( a ∧ B b ) = r ∧ A ( l ⋆ B ( b ) x ) a + a ∧ B ( r ⋆ A ( x ) b ) , (3.15) ( l ∨ A ( x ) a ) ∧ B b + l ∧ A ( r ∨ B ( a ) x ) b = l ∨ A ( x )( a ∧ B b ) , (3.16) l ∧ A ( l ∨ B ( a ) x ) b + ( r ∨ A ( x ) a ) ∧ B b = a ∨ B ( l ∧ A ( x ) b ) + r ∨ A ( r ∧ B ( b ) x ) a, (3.17) r ∧ A ( x )( a ∨ B b ) = r ∨ A ( l ∧ B ( b ) x ) a + a ∨ B ( r ∧ A ( x ) b ) , (3.18) ( l ⋆ A ( x ) a ) ∨ B b + l ∨ A ( r ⋆ B ( a ) x ) b = l ∨ A ( x )( a ∨ B b ) , (3.19) l ∨ A ( l ⋆ B ( a ) x ) b + ( r ⋆ A ( x ) a ) ∨ B b = a ∨ B ( l ∨ A ( x ) b ) + r ∨ A ( r ∨ B ( b ) x ) a, (3.20) r ∨ A ( x )( a ⋆ B b ) = r ∨ A ( l ∨ B ( b ) x ) a + a ∨ B ( r ∨ A ( x ) b ) , for all x, y ∈ A, a, b ∈ B . Then there is a dendriform dialgebra structure on the vector space A ⊕ B which is given by (3.21) ( x + a ) ∧ ( y + b ) := x ∧ A y + l ∧ B ( a ) y + r ∧ B ( b ) x + a ∧ B b + l ∧ A ( x ) b + r ∧ A ( y ) a, (3.22) ( x + a ) ∨ ( y + b ) := x ∨ A y + l ∨ B ( a ) y + r ∨ B ( b ) x + a ∨ B b + l ∨ A ( x ) b + r ∨ A ( y ) a, for all x, y ∈ A, a, b ∈ B. We denote this dendriform dialgebra by
A ⊲⊳ l ∧ B ,r ∧ B ,l ∨ B ,r ∨ B l ∧ A ,r ∧ A ,l ∨ A ,r ∨ A B orsimply A ⊲⊳ B . Moreover ( A, B, l ∧ A , r ∧ A , l ∨ A , r ∨ A , l ∧ B , r ∧ B , l ∨ B , r ∨ B ) is called a matched pair ofdendriform dialgebras. On the other hand, every dendriform dialgebra which is the direct sum ofthe underlying vector spaces of two sub-dialgebras can be obtained from the above way. Proposition 3.5.
Let ( A, տ A , ր A , ւ A , ց A ) be a quadri-algebra. Suppose that there is a quadri-algebra structure տ B , ր B , ւ B , ց B on the dual space A ∗ . Then there exists a dendriform dial-gebra structure on the vector space A ⊕ A ∗ such that A v and ( A ∗ ) v are isotropic sub-dialgebrasassociated to the -cocycle (3 . , that is, ( A v ⊕ ( A ∗ ) v , A v , ( A ∗ ) v , B S ) is a Manin triple of dendri-form dialgebras, if and only if ( A v , ( A ∗ ) v , − R ∗ւ A , L ∗≻ A , R ∗≺ A , − L ∗ր A , − R ∗ւ B , L ∗≻ B , R ∗≺ B , − L ∗ր B ) isa matched pair of dendriform dialgebras.Proof. If ( A v , ( A ∗ ) v , − R ∗ւ A , L ∗≻ A , R ∗≺ A , − L ∗ր A , − R ∗ւ B , L ∗≻ B , R ∗≺ B , − L ∗ր B ) is a matched pair ofdendriform dialgebras, then it is straightforward to check that the bilinear form given by Eq.(3.2) on A v ⊲⊳ − R ∗ւ B ,L ∗≻ B ,R ∗≺ B , − L ∗ր B − R ∗ւ A ,L ∗≻ A ,R ∗≺ A , − L ∗ր A ( A ∗ ) v is a 2-cocycle. So ( A v ⊕ ( A v ) ∗ , A v , ( A ∗ ) v , B S ) is aManin triple of dendriform dialgebras.Conversely, if ( A v ⊕ ( A v ) ∗ , A v , ( A v ) ∗ , B S ) is a Manin triple of dendriform dialgebras, then forall x, y ∈ A, a ∗ , b ∗ ∈ A ∗ , we have h x ∧ a ∗ , y i = B S ( y, x ∧ a ∗ ) = − B S ( y ւ A x, a ∗ ) = h y, − R ∗ւ A ( x ) a ∗ i , XIANG NI AND CHENGMING BAI h x ⋆ a ∗ , b ∗ i = B S ( b ∗ , x ⋆ a ∗ ) = B S ( a ∗ ց B b ∗ , x ) = h L ∗ց B ( a ∗ ) x, b ∗ i , h x ∨ a ∗ , b ∗ i = B S ( b ∗ , x ∨ a ∗ ) = − B S ( a ∗ ր B b ∗ , x ) = h− L ∗ր B ( a ∗ ) x, b ∗ i . So x ∧ a ∗ = − R ∗ւ A ( x ) a ∗ + L ∗≻ B ( a ∗ ) x . Similarly, we show that x ∨ a ∗ = R ∗≺ A ( x ) a ∗ − L ∗ր B ( a ∗ ) x, a ∗ ∧ x = − R ∗ւ B ( a ∗ ) x + L ∗≻ A ( x ) a ∗ , a ∗ ∨ x = R ∗≺ B ( a ∗ ) x − L ∗ր A ( x ) a ∗ , for any x ∈ A, a ∗ ∈ A ∗ . Hence ( A v , ( A ∗ ) v , − R ∗ւ A , L ∗≻ A , R ∗≺ A , − L ∗ր A , − R ∗ւ B , L ∗≻ B , R ∗≺ B , − L ∗ր B )is a matched pair of dendriform dialgebras. (cid:3) Bimodules and matched pairs of quadri-algebras
Definition 4.1. ([Bai3]) Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let V be a vector space.Let l ◦ , r ◦ : A → gl ( V ) be eight linear maps, where ◦ ∈ {տ , ր , ւ , ց} . V or ( V, l տ , r տ , l ր , r ր , l ւ , r ւ , l ց , r ց )) is called a bimodule of A if for all x, y ∈ A ,(4.1) r տ ( y ) r տ ( x ) = r տ ( x ⋆ y ) , r տ ( y ) l տ ( x ) = l տ ( x ) r ⋆ ( y ) , l տ ( x տ y ) = l տ ( x ) l ⋆ ( y ) , (4.2) r տ ( y ) r ր ( x ) = r ր ( x ≺ y ) , r տ ( y ) l ր ( x ) = l ր ( x ) r ≺ ( y ) , l տ ( x ր y ) = l ր ( x ) l ≺ ( y ) , (4.3) r ր ( y ) r ∧ ( x ) = r ր ( x ≻ y ) , r ր ( y ) l ∧ ( x ) = l ր ( x ) r ≻ ( y ) , l ր ( x ∧ y ) = l ր ( x ) l ≻ ( y ) , (4.4) r տ ( y ) r ւ ( x ) = r ւ ( x ∧ y ) , r տ ( y ) l ւ ( x ) = l ւ ( x ) r ∧ ( y ) , l տ ( x ւ y ) = l ւ ( x ) l ∧ ( y ) , (4.5) r տ ( y ) r ց ( x ) = r ց ( x տ y ) , r տ ( y ) l ց ( x ) = l ց ( x ) r տ ( y ) , l տ ( x ց y ) = l ց ( x ) l տ ( y ) , (4.6) r ր ( y ) r ∨ ( x ) = r ց ( x ր y ) , r ր ( y ) l ∨ ( x ) = l ց ( x ) r ր ( y ) , l ր ( x ∨ y ) = l ց ( x ) l ր ( y ) , (4.7) r ւ ( y ) r ≺ ( x ) = r ւ ( x ∨ y ) , r ւ ( y ) l ≺ ( x ) = l ւ ( x ) r ∨ ( y ) , l ւ ( x ≺ y ) = l ւ ( x ) l ∨ ( y ) , (4.8) r ւ ( y ) r ≻ ( x ) = r ց ( x ւ y ) , r ւ ( y ) l ≻ ( x ) = l ց ( x ) r ւ ( y ) , l ւ ( x ≻ y ) = l ց ( x ) l ւ ( y ) , (4.9) r ց ( y ) r ⋆ ( x ) = r ց ( x ց y ) , r ց ( y ) l ⋆ ( x ) = l ց ( x ) r ց ( y ) , l ց ( x ⋆ y ) = l ց ( x ) l ց ( y ) . In fact, (
V, l տ , r տ , l ր , r ր , l ւ , r ւ , l ց , r ց ) is a bimodule of a quadri-algebra A if and only if thedirect sum A ⊕ V of the underlying vector spaces of A and V is turned into a quadri-algebra (the semidirect sum ) by defining multiplications in A ⊕ V by (we still denote them by տ , ր , ւ , ց ):(4.10)( x + u ) ◦ ( x + u ) := x ◦ x + ( l ◦ ( x ) u + r ◦ ( x ) u ) , ∀ x , x ∈ A, u , u ∈ V, ◦ ∈ {տ , ր , ւ , ց} . We denote it by A ⋉ l տ ,r տ ,l ր ,r ր ,l ւ ,r ւ ,l ց ,r ց V or simply A ⋉ V . Lemma 4.2. ([Bai3])
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra. If ( V, l տ , r տ , l ր , r ր , l ւ , r ւ , l ց , r ց ) is a bimodule of A , then ( V ∗ , r ∗ց , l ∗ ⋆ , − r ∗∨ , − l ∗≺ , − r ∗≻ , − l ∗∧ , r ∗ ⋆ , l ∗տ ) is a bimodule of A . UADRI-BIALGEBRAS 9
Definition 4.3.
Let ( A, տ A , ր A , ւ A , ց A ) and ( B, տ B , ր B , ւ B , ց B ) be two quadri-algebras.Suppose that there exist linear maps l տ A , r տ A , l ր A , r ր A , l ւ A , r ւ A , l ց A , r ց A : A → gl ( B ) and l տ B , r տ B , l ր B , r ր B , l ւ B , r ւ B , l ց B , r ց B : B → gl ( A ) such that for all x, y ∈ A, a, b ∈ B ,(4.11) ( x + a ) ⋄ ( y + b ) := x ⋄ A y + l ⋄ B ( a ) y + r ⋄ B ( b ) x + a ⋄ B b + l ⋄ A ( x ) b + r ⋄ A ( y ) a, ⋄ ∈ {տ , ր , ւ , ց} , define a quadri-algebra structure on A ⊕ B . Then ( A, B, l տ A , r տ A , l ր A , r ր A , l ւ A , r ւ A , l ց A , r ց A , l տ B , r տ B , l ր B , r ր B , l ւ B , r ւ B , l ց B , r ց B ) is called a matched pair of quadri-algebras andthe quadri-algebra structure on A ⊕ B is denoted by A ⊲⊳ l տ B ,r տ B ,l ր B ,r ր B ,l ւ B ,r ւ B ,l ց B ,r ց B l տ A ,r տ A ,l ր A ,r ր A ,l ւ A ,r ւ A ,l ց A ,r ց A B orsimply A ⊲⊳ B . Remark 4.4.
Similar to Proposition 3.4 one can also write down the necessary and sufficientconditions that the above linear maps make A ⊕ B into a quadri-algebra. We omit the detailssince we will not use such a conclusion in this paper directly. In particular, in this case, A and B are bimodules of B and A respectively. Proposition 4.5.
Let ( A, B, l տ A , r տ A , l ր A , r ր A , l ւ A , r ւ A , l ց A , r ց A , l տ B , r տ B , l ր B , r ր B , l ւ B , r ւ B , l ց B , r ց B ) be a matched pair of quadri-algebras. Then (( A ) v , ( B ) v , l տ A + l ր A , r տ A + r ր A , l ւ A + l ց A , r ւ A + r ց A , l տ B + l ր B , r տ B + r ր B , l ւ B + l ց B , r ւ B + r ց B ) is a matched pair of dendriformdialgebras.Proof. It is straightforward. (cid:3)
Proposition 4.6.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra. Suppose that there is a quadri-algebra structure ( տ ∗ , ր ∗ , ւ ∗ , ց ∗ ) on the dual space A ∗ . Then ( A v , ( A ∗ ) v , − R ∗ւ , L ∗≻ , R ∗≺ , − L ∗ր , − R ∗ւ ∗ , L ∗≻ ∗ , R ∗≺ ∗ , − L ∗ր ∗ ) is a matched pair of dendriform dialgebras if and only if ( A, A ∗ , R ∗ց , L ∗ ⋆ , − R ∗∨ , − L ∗≺ , − R ∗≻ , − L ∗∧ , R ∗ ⋆ , L ∗տ , R ∗ց ∗ , L ∗ ⋆ ∗ , − R ∗∨ ∗ , − L ∗≺ ∗ , − R ∗≻ ∗ , − L ∗∧ ∗ , R ∗ ⋆ ∗ , L ∗տ ∗ ) is a matched pairof quadri-algebras.Proof. By Proposition 4.5, we only need to prove the “only if” part of the proposition. Infact, if ( A v , ( A ∗ ) v , − R ∗ւ , L ∗≻ , R ∗≺ , − L ∗ր , − R ∗ւ ∗ , L ∗≻ ∗ , R ∗≺ ∗ , − L ∗ր ∗ ) is a matched pair of dendriformdialgebras, then from Proposition 3.5, we show that ( A v ⊕ ( A ∗ ) v , A v , ( A ∗ ) v , B S ) is a Manin tripleof dendriform dialgebras. Hence there exists a quadri-algebra structure ( տ • , ր • , ւ • , ց • ) on A v ⊲⊳ − R ∗ւ∗ ,L ∗≻∗ ,R ∗≺∗ , − L ∗ր∗ − R ∗ւ ,L ∗≻ ,R ∗≺ , − L ∗ր ( A ∗ ) v which is given by Proposition 2.11. Moreover, A and A ∗ are thesub-quadri-algebras. For any x, y ∈ A, a ∗ , b ∗ ∈ A ∗ , we have h x տ • a ∗ , y i = B S ( x, a ∗ ⋆ • y ) = h x, L ∗ց ( y ) a ∗ i = h R ∗ց ( x ) a ∗ , y i , h x տ • a ∗ , b ∗ i = B S ( x, a ∗ ⋆ ∗ b ∗ ) = h L ∗ ⋆ ∗ ( a ∗ ) x, b ∗ i . So x տ • a ∗ = R ∗ց ( x ) a ∗ + L ∗ ⋆ ∗ ( a ∗ ) x . Similarly, we show that x ր • a ∗ = − R ∗∨ ( x ) a ∗ − L ∗≺ ∗ ( a ∗ ) x, x ւ • a ∗ = − R ∗≻ ( x ) a ∗ − L ∗∧ ∗ ( a ∗ ) x, x ց • a ∗ = R ∗ ⋆ ( x ) a ∗ + L ∗տ ∗ ( a ∗ ) x, a ∗ տ • x = R ∗ց ∗ ( a ∗ ) x + L ∗ ⋆ ( x ) a ∗ ,a ∗ ր • x = − R ∗∨ ∗ ( a ∗ ) x − L ∗≺ ( x ) a ∗ , a ∗ ւ • x = − R ∗≻ ∗ ( a ∗ ) x − L ∗∧ ( x ) a ∗ , and a ∗ ց • x = R ∗ ⋆ ∗ ( a ∗ ) x + L ∗տ ( x ) a ∗ . Therefore, (
A, A ∗ , R ∗ց , L ∗ ⋆ , − R ∗∨ , − L ∗≺ , − R ∗≻ , − L ∗∧ , R ∗ ⋆ , L ∗տ , R ∗ց ∗ , L ∗ ⋆ ∗ , − R ∗∨ ∗ , − L ∗≺ ∗ , − R ∗≻ ∗ , − L ∗∧ ∗ , R ∗ ⋆ ∗ , L ∗տ ∗ ) is a matched pair of quadri-algebras. (cid:3) In fact, the above equivalence between two matched pairs can be interpreted in terms of theircorresponding Manin triples as follows.
Definition 4.7.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let B be a symmetric bilinearform. If B satisfies Eqs. (2.14)-(2.15), then B is called invariant on A . Proposition 4.8. ([Bai3])
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let B be a symmetricbilinear form. If B is invariant on A , then B is a 2-cocycle of the associated vertical dendriformdialgebra ( A v , ∧ , ∨ ) . Conversely, if B is a nondegenerate 2-cocycle of a dendriform dialgebra,then B is invariant on the quadri-algebra given by Eqs. (2.14)-(2.15). Definition 4.9.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra. If there is a quadri-algebra structureon the direct sum of the underlying vector space of A and A ∗ such that A and A ∗ are quadri-subalgebras and the bilinear form B S on A ⊕ A ∗ given by Eq. (3.2) is invariant, then ( A ⊲⊳A ∗ , A, A ∗ , B S ) is called a (standard) Manin triple of quadri-algebras associated to a nondegenerateinvariant bilinear form .By Proposition 4.8, the following conclusion is obvious: Corollary 4.10. ( A ⊲⊳ A ∗ , A, A ∗ , B S ) is a Manin triple of quadri-algebras associated to a non-degenerate invariant bilinear form if and only if ( A v ⊲⊳ ( A v ) ∗ , A v , A ∗ v , B S ) is Manin triple ofdendriform dialgebras associated to a nondegenerate 2-cocycle. Remark 4.11.
By Proposition 4.6, it is obvious that a Manin triple of quadri-algebras associatedto a nondegenerate invariant bilinear form can be interpreted in terms of a matched pair ofquadri-algebras (cf. Theorem 5.4) . Quadri-bialgebras
Proposition 5.1.
Let ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) be a quadri-algebra equipped with four coop-erations α, β, ˜ α, ˜ β : A → A ⊗ A . Suppose that α ∗ , β ∗ , ˜ α ∗ , ˜ β ∗ : A ∗ ⊗ A ∗ ⊂ ( A ⊗ A ) ∗ → A ∗ induce a quadri-algebra structure on A ∗ . Set տ ∗ := α ∗ , ր ∗ := β ∗ , ւ ∗ := ˜ α ∗ , ց ∗ := ˜ β ∗ . Then ( A v , ( A ∗ ) v , − R ∗ւ , L ∗≻ , R ∗≺ , − L ∗ր , − R ∗ւ ∗ , L ∗≻ ∗ , R ∗≺ ∗ , − L ∗ր ∗ ) is a matched pair of dendriform dial-gebras if and only if the following equations hold: (5.1) ˜ α ( x ⋆ y ) = ( R ∧ ( y ) ⊗
1) ˜ α ( x ) + (1 ⊗ L ≻ ( x )) ˜ α ( y ) , (5.2) β ( x ⋆ y ) = ( R ≺ ( y ) ⊗ β ( x ) + (1 ⊗ L ∨ ( x )) β ( y ) , UADRI-BIALGEBRAS 11 (5.3) ( α + ˜ α )( x ∧ y ) = ( R ∧ ( y ) ⊗ α + ˜ α )( x ) + (1 ⊗ L ր ( x )) ˜ α ( y ) , (5.4) ( β + ˜ β )( x ∧ y ) = ( R տ ( y ) ⊗ β + ˜ β )( x ) + (1 ⊗ L ∧ ( x )) ˜ β ( y ) , (5.5) ( β + ˜ β )( x ∨ y ) = (1 ⊗ L ∨ ( x ))( β + ˜ β )( y ) + ( R ւ ( y ) ⊗ β ( x ) , (5.6) ( α + ˜ α )( x ∨ y ) = (1 ⊗ L ց ( x ))( α + ˜ α )( y ) + ( R ∨ ( y ) ⊗ α ( x ) , (5.7) ( α + β )( x ≻ y ) = (1 ⊗ L ց ( x ))( α + β )( y ) + ( R ≻ ( y ) ⊗ α ( x ) , (5.8) ( ˜ α + ˜ β )( x ≻ y ) = (1 ⊗ L ≻ ( x ))( ˜ α + ˜ β )( y ) + ( R ր ( y ) ⊗
1) ˜ α ( x ) , (5.9) ( α + β )( x ≺ y ) = ( R ≺ ( y ) ⊗ α + β )( x ) + (1 ⊗ L ւ ( x )) β ( y ) , (5.10) ( ˜ α + ˜ β )( x ≺ y ) = ( R տ ( y ) ⊗ α + ˜ β )( x ) + (1 ⊗ L ≺ ( x )) ˜ β ( y ) , (5.11) (1 ⊗ L ≻ ( y ) − R ∧ ( y ) ⊗ τ β ( x ) = (1 ⊗ R ≺ ( x ) − L ∨ ( x ) ⊗
1) ˜ α ( y ) , (5.12) (1 ⊗ R տ ( x ) − L ∨ ( x ) ⊗ α + ˜ α )( y ) = (1 ⊗ L ր ( y )) τ β ( x ) − ( R ∨ ( y ) ⊗ τ ˜ β ( x ) , (5.13) ( R ∧ ( y ) ⊗ − ⊗ L ց ( y ))( τ β + τ ˜ β )( x ) = ( L ∧ ( x ) ⊗ α ( y ) − (1 ⊗ R ւ ( x )) ˜ α ( y ) , (5.14) (1 ⊗ L ≻ ( x ) − R տ ( x ) ⊗ τ α + τ β )( y ) = (1 ⊗ R ≻ ( y )) ˜ β ( x ) − ( L ւ ( y ) ⊗
1) ˜ α ( x ) , (5.15) (1 ⊗ L ց ( x ) − R ≺ ( x ) ⊗ τ ˜ α + τ ˜ β )( y ) = (1 ⊗ R ր ( y )) β ( x ) − ( L ≺ ( y ) ⊗ α ( x ) , (5.16) ( α + ˜ α + β + ˜ β )( x ւ y ) = ( R ւ ( y ) ⊗ α + β )( x ) + (1 ⊗ L ւ ( x ))( β + ˜ β )( y ) , (5.17) ( α + β + ˜ α + ˜ β )( x ր y ) = ( R ր ( y ) ⊗ α + ˜ α )( x ) + (1 ⊗ L ր ( x ))( ˜ α + ˜ β )( y ) . (5.18)( L ւ ( y ) ⊗ α + ˜ α )( x )+(1 ⊗ L ր ( x ))( τ α + τ β )( y ) = ( R ւ ( x ) ⊗ τ ˜ α + τ ˜ β )( y )+(1 ⊗ R ր ( y ))( β + ˜ β )( x ) . Proof.
From Proposition 3.5, we need to prove Eqs. (5.1)-(5.18) are equivalent to Eqs. (3.3)-(3.20) respectively in the case that A = A v , B = ( A ∗ ) v and l ∧ A = − R ∗ւ , r ∧ A = L ∗≻ , l ∨ A = R ∗≺ , r ∨ A = − L ∗ր , l ∧ B = − R ∗ւ ∗ , r ∧ B = L ∗≻ ∗ , l ∨ B = R ∗≺ ∗ , r ∨ B = − L ∗ր ∗ . As an example, we give anexplicit proof that(5.19) L ∗≻ ∗ ( a ∗ )( x ∧ y ) − L ∗≻ ∗ ( R ∗տ ( y ) a ∗ ) x − x ∧ ( L ∗ց ∗ ( a ∗ ) y ) = 0 , holds if and only if Eq. (5.4) holds. The other equivalences are similar. In fact, let the left handside of Eq. (5.19) act on an arbitrary element b ∗ ∈ A ∗ . Then we have h L ∗≻ ∗ ( a ∗ )( x ∧ y ) − L ∗≻ ∗ ( R ∗տ ( y ) a ∗ ) x − x ∧ ( L ∗ց ∗ ( a ∗ ) y ) , b ∗ i = h x ∧ y, a ∗ ≻ ∗ b ∗ i − h x, ( R ∗տ ( y ) a ∗ ) ≻ ∗ b ∗ i − h y, a ∗ ց ∗ ( L ∗∧ ( x ) b ∗ ) i = h ( β + ˜ β )( x ∧ y ) − ( R տ ( y ) ⊗ β + ˜ β )( x ) − (1 ⊗ L ∧ ( x )) ˜ β ( y ) , a ∗ ⊗ b ∗ i . So the conclusion follows. (cid:3)
Definition 5.2. (1) Let (
A, α, β, ˜ α, ˜ β ) be a vector space with four comultiplications α, β, ˜ α, ˜ β : A → A ⊗ A . If ( A ∗ , α ∗ , β ∗ , ˜ α ∗ , ˜ β ∗ ) becomes a quadri-algebra, then we call ( A, α, β, ˜ α, ˜ β ) a quadri-coalgebra , where α ∗ , β ∗ , ˜ α ∗ , ˜ β ∗ : A ∗ ⊗ A ∗ ⊂ ( A ⊗ A ) ∗ → A ∗ . A homomorphism between twoquadri-coalgebras is defined as a linear map (between the two quadri-coalgebras) which preservesthe cooperations, respectively.(2) Let ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) be a quadri-algebra equipped with four cooperations α, β, ˜ α ,˜ β : A → A ⊗ A such that ( A, α, β, ˜ α, ˜ β ) is a quadri-coalgebra and α, β, ˜ α and ˜ β satisfy Eqs. (5.1)-(5.18), then ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) is called a quadri-bialgebra . A homomorphism betweentwo quadri-bialgebras is defined as a linear map (between the two quadri-bialgebras) which is ahomomorphisms of both quadri-algebras and quadri-coalgebras.Combining Proposition 5.1 and the discussion in the previous sections, we have the followingconclusion: Theorem 5.3.
Let ( A, տ A , ր A , ւ A , ց A , α, β, ˜ α, ˜ β ) be a quadri-algebra with four comultiplica-tions α, β, ˜ α, ˜ β : A → A ⊗ A , such that ( A, α, β, ˜ α, ˜ β ) is a quadri-coalgebra. Then the followingconditions are equivalent ( տ B := α ∗ , ր B := β ∗ , ւ B := ˜ α ∗ , ց B := ˜ β ∗ , and B S is given by Eq.(3.2)): (1) ( A v ⊲⊳ ( A ∗ ) v , A v , ( A ∗ ) v , B S ) is a Manin triple of dendriform dialgebras associated to anondegenerate 2-cocycle; (2) ( A ⊲⊳ A ∗ , A, A ∗ , B S ) is a Manin triple of quadri-algebras associated to a nondegenerateinvariant bilinear form. (3) ( A v , ( A ∗ ) v , − R ∗ւ A , L ∗≻ A , R ∗≺ A , − L ∗ր A , − R ∗ւ B , L ∗≻ B , R ∗≺ B , − L ∗ր B ) is a matched pair of den-driform dialgebras; (4) ( A, A ∗ , R ∗ց A , L ∗ ⋆ A , − R ∗∨ A , − L ∗≺ A , − R ∗≻ A , − L ∗∧ A , R ∗ ⋆ A , L ∗տ A , R ∗ց B , L ∗ ⋆ B , − R ∗∨ B , − L ∗≺ B , − R ∗≻ B , − L ∗∧ B , R ∗ ⋆ B , L ∗տ B ) is a matched pair of quadri-algebras; (5) ( A, տ A , ր A , ւ A , ց A , α, β, ˜ α, ˜ β ) is a quadri-bialgebra. By a standard proof (cf. [Bai2], Proposition 2.2.10), we get the following result:
Proposition 5.4.
Two Manin triples of dendriform dialgebras are isomorphic if and only if theircorresponding quadri-bialgebras are isomorphic.
Remark 5.5.
It is obvious that for a quadri-bialgebra ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ), the dual( A ∗ , α ∗ , β ∗ , ˜ α ∗ , ˜ β ∗ , γ, δ, ˜ γ, ˜ δ ) is also a quadri-bialgebra, where γ ∗ = տ , δ ∗ = ր , ˜ γ ∗ = ւ , ˜ δ ∗ = ց . UADRI-BIALGEBRAS 13 Coboundary quadri-bialgebras
Definition 6.1.
A quadri-bialgebra ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) is called coboundary if α, β, ˜ α and ˜ β satisfy(6.1) α ( x ) = ( − ⊗ L ց ( x ) + R ⋆ ( x ) ⊗ r տ , (6.2) β ( x ) = (1 ⊗ L ∨ ( x ) − R ≺ ( x ) ⊗ r ր , (6.3) ˜ α ( x ) = (1 ⊗ L ≻ ( x ) − R ∧ ( x ) ⊗ r ւ , (6.4) ˜ β ( x ) = ( − ⊗ L ⋆ ( x ) + R տ ( x ) ⊗ r ց , where r տ , r ր , r ւ , r ց ∈ A ⊗ A and x ∈ A . Proposition 6.2.
Let ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) be a quadri-algebra with four comultiplications α, β, ˜ α, ˜ β defined by Eqs. (6.1)-(6.4) respectively. If r տ = r ր = r ւ = r ց = r ∈ A ⊗ A and r isskew-symmetric, then α, β, ˜ α and ˜ β satisfy Eqs. (5.1)-(5.18).Proof. Straightforward. (cid:3)
Lemma 6.3.
Let A be a vector space and α, β, ˜ α, ˜ β : A → A ⊗ A be four cooperations. Then ( A, α, β, ˜ α, ˜ β ) is a quadri-coalgebra if and only if the linear maps R i : A → A ⊗ A ⊗ A ( i ∈{ , ..., } ) defined by the following equations are zero: (6.5) R ( x ) := ( α ⊗ α ( x ) − (1 ⊗ ( α + β + ˜ α + ˜ β )) α ( x ) , (6.6) R ( x ) := ( β ⊗ α ( x ) − (1 ⊗ ( α + ˜ α )) β ( x ) , (6.7) R ( x ) := (( α + β ) ⊗ β ( x ) − (1 ⊗ ( β + ˜ β )) β ( x ) , (6.8) R ( x ) := ( ˜ α ⊗ α ( x ) − (1 ⊗ ( α + β )) ˜ α ( x ) , (6.9) R ( x ) := ( ˜ β ⊗ α ( x ) − (1 ⊗ α ) ˜ β ( x ) , (6.10) R ( x ) := (( ˜ α + ˜ β ) ⊗ β ( x ) − (1 ⊗ β ) ˜ β ( x ) , (6.11) R ( x ) := (( α + ˜ α ) ⊗
1) ˜ α ( x ) − (1 ⊗ ( ˜ α + ˜ β )) ˜ α ( x ) , (6.12) R ( x ) := (( β + ˜ β ) ⊗
1) ˜ α ( x ) − (1 ⊗ ˜ α ) ˜ β ( x ) , (6.13) R ( x ) := (( α + β + ˜ α + ˜ β ) ⊗
1) ˜ β ( x ) − (1 ⊗ ˜ β ) ˜ β ( x ) . Proof.
It follows from the definition of a quadri-algebra. (cid:3)
Definition 6.4.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let r ∈ A ⊗ A . The followingequations are called Q ji -equations ( i = 1 , , j = 1 ,
2) respectively:(6.14) Q := r ∧ r − r ≻ r + r ւ r = 0 , (6.15) Q := r ∨ r − r ≺ r + r ր r = 0 , (6.16) Q := r ∧ r − r ≻ r − r ւ r = 0 , (6.17) Q := r ∨ r + r ≺ r + r ր r = 0 , (6.18) Q := r ∧ r + r ≻ r + r ւ r = 0 , (6.19) Q := r ∨ r − r ≺ r − r ր r = 0 . Proposition 6.5.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let r ∈ A ⊗ A . Define α, β, ˜ α and ˜ β by Eqs. (6.1)-(6.4) respectively, where r տ = r ր = r ւ = r ց = r . Then ( A, α, β, ˜ α, ˜ β ) becomes a quadri-coalgebra if and only if the following equations holds (for any x ∈ A ): (6.20) (1 ⊗ ⊗ L ց ( x ) − R ⋆ ( x ) ⊗ ⊗ Q − Q ) = 0 , (6.21) (1 ⊗ ⊗ L ց ( x ) − R ≺ ( x ) ⊗ ⊗ Q = 0 , (6.22) (1 ⊗ ⊗ L ∨ ( x ) − R ≺ ( x ) ⊗ ⊗ Q = 0 , (6.23) (1 ⊗ ⊗ L ց ( x ) − R ∧ ( x ) ⊗ ⊗ Q = 0 , (6.24) (1 ⊗ ⊗ L ց ( x ) − R տ ( x ) ⊗ ⊗ Q + Q ) = 0 , (6.25) (1 ⊗ ⊗ L ∨ ( x ) − R տ ( x ) ⊗ ⊗ Q = 0 , (6.26) (1 ⊗ ⊗ L ≻ ( x ) − R ∧ ( x ) ⊗ ⊗ Q = 0 , (6.27) (1 ⊗ ⊗ L ≻ ( x ) − R տ ( x ) ⊗ ⊗ Q = 0 , (6.28) (1 ⊗ ⊗ L ⋆ ( x ) − R տ ( x ) ⊗ ⊗ Q + Q ) = 0 . Proof.
In fact, we need to prove R i ( x ) = 0 , ≤ i ≤
9, if and only if Eqs. (6.20)-(6.28) holdrespectively. As an example we give an explicit proof that R ( x ) = 0 if and only if Eq. (6.21) UADRI-BIALGEBRAS 15 holds. The proof of the other cases is similar. In fact, after rearranging the terms suitably, wedivide R ( x ) into three parts: R ( x ) = ( R
1) + ( R
2) + ( R , where( R
1) = X i,j {− a i ⊗ a j ∨ b i ⊗ x ց b j + a i ≺ a j ⊗ b i ⊗ x ց b j + a j ⊗ a i ⊗ ( x ∨ b j ) ց b i − a j ⊗ a i ⊗ ( x ∨ b j ) ≻ b i } = (1 ⊗ ⊗ L ց ( x ))( − r ∨ r + r ≺ r − r ր r ) , ( R
2) = X i,j { a i ⊗ ( a j ⋆ x ) ∨ b i ⊗ b j − a j ⊗ a i ⋆ ( x ∨ b j ) ⊗ b i + a j ⊗ a i ∧ ( x ∨ b j ) ⊗ b i } = 0 , ( R
3) = X i,j {− a i ≺ ( a j ⋆ x ) ⊗ b i ⊗ b j − a j ≺ x ⊗ a i ⊗ b j ց b i + a j ≺ x ⊗ a i ⋆ b j ⊗ b i + a j ≺ x ⊗ a i ⊗ b j ≻ b i − a j ≺ x ⊗ a i ∧ b j ⊗ b i = ( R ≺ ( x ) ⊗ ⊗ − r ≺ r + r ր r + r ∨ r ) . So the conclusion follows. (cid:3)
By the above study, we have the following conclusion.
Theorem 6.6.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let r ∈ A ⊗ A be skew-symmetric.Then the comultiplications α, β, ˜ α and ˜ β defined by Eqs.(6.1)-(6.4) with r տ = r ր = r ւ = r ց = r respectively make ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) into a quadri-bialgebra if and only if Eqs. (6.20)-(6.28) are satisfied. Moreover, we have the following “Drinfeld’s double” construction of a quadri-bialgebra ([CP]).
Theorem 6.7.
Let ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) be a quadri-bialgebra. Then there exists a canon-ical quadri-bialgebra structure on A ⊕ A ∗ such that the inclusion i : A → A ⊕ A ∗ is a homo-morphism of quadri-bialgebras, which is from ( A, տ , ր , ւ , ց , − α, − β, − ˜ α, − ˜ β ) to A ⊕ A ∗ , andthe inclusion i : A ∗ → A ⊕ A ∗ is also a homomorphism of quadri-bialgebras, which is from thequadri-bialgebra given in Remark 5.5 to A ⊕ A ∗ .Proof. Let r = P i e i ⊗ e ∗ i correspond to the identity map id : A → A , where { e i , ..., e s } is a basisof A and { e ∗ , ...e ∗ s } is the dual basis. Suppose the quadri-algebra structure ( տ • , ր • , ւ • , ց • ) on A ⊕ A ∗ is given by QD ( A ) := A ⊲⊳ R ∗ց∗ ,L ∗ ⋆ ∗ , − R ∗∨∗ , − L ∗≺∗ , − R ∗≻∗ , − L ∗∧∗ ,R ∗ ⋆ ∗ ,L ∗տ∗ R ∗ց ,L ∗ ⋆ , − R ∗∨ , − L ∗≺ , − R ∗≻ , − L ∗∧ ,R ∗ ⋆ ,L ∗տ A ∗ , where the subscript ∗ is used to denote the quadri-algebra structure on A ∗ . Then for any x, y ∈ A, a ∗ , b ∗ ∈ A ∗ , x տ • y = x տ y, x ր • y = x ր y, x ւ • y = x ւ y, x ց • y = x ց y,a ∗ տ • b ∗ = a ∗ տ ∗ b ∗ , a ∗ ր • b ∗ = a ∗ ր ∗ b ∗ , a ∗ ւ • b ∗ = a ∗ ւ ∗ b ∗ , a ∗ ց • b ∗ = a ∗ ց ∗ b ∗ ,x տ • a ∗ = R ∗ց ( x ) a ∗ + L ∗ ⋆ ∗ ( a ∗ ) x, x ր • a ∗ = − R ∗∨ ( x ) a ∗ − L ∗≺ ∗ ( a ∗ ) x, x ւ • a ∗ = − R ∗≻ ( x ) a ∗ − L ∗∧ ∗ ( a ∗ ) x,x ց • a ∗ = R ∗ ⋆ ( x ) a ∗ + L ∗տ ∗ ( a ∗ ) x, a ∗ տ • x = R ∗ց ∗ ( a ∗ ) x + L ∗ ⋆ ( x ) a ∗ , a ∗ ր • x = − R ∗∨ ∗ ( a ∗ ) x − L ∗≺ ( x ) a ∗ ,a ∗ ւ • x = − R ∗≻ ∗ ( a ∗ ) x − L ∗∧ ( x ) a ∗ , a ∗ ց • x = R ∗ ⋆ ∗ ( a ∗ ) x + L ∗տ ( x ) a ∗ . We prove that r satisfies Eqs. (5.1)-(5.18). We give an explicit proof that r satisfy Eq. (5.13)as an example. The proof of the other cases is similar. For any µ, ν ∈ QD ( A ), Eq. (5.13) is equivalent to X k {− ( µ ∧ • e ∗ k ) ∧ • ν ⊗ e k − e ∗ k ∧ • ν ⊗ e k ւ • µ + µ ∧ • e ∗ k ⊗ ν ց • e k + e ∗ k ⊗ ν ց • ( e k ւ • µ )+ µ ∧ e k ⊗ ν ց • e ∗ k − µ ∧ • ( e k ⋆ • ν ) ⊗ e ∗ k + e k ⊗ ( ν ≻ • e ∗ k ) ւ • µ − e k ∧ • ν ⊗ e ∗ k ւ • µ } = 0 . We can prove the equation in the following cases: (I) µ, ν ∈ A ; (II) µ, ν ∈ A ∗ ; (III) µ ∈ A, ν ∈ A ∗ ;(IV) µ ∈ A ∗ , ν ∈ A . As an example, we give the proof of the last case. The proof of the othercases is similar. Let µ = e ∗ i , ν = e j , then for any m, n , the coefficient of e ∗ m ⊗ e n is X k h− ( e ∗ i ∧ • e ∗ n ) ∧ • e j , e m i − h e ∗ k ∧ • e j , e m ih e k ւ • e ∗ i , e ∗ n i + h e ∗ i ∧ • e ∗ k , e m ih e j ց • e k , e ∗ n i + h e j ց • ( e m ւ • e ∗ i ) , e ∗ n i + h e ∗ i ∧ • e k , e m ih e j ց • e ∗ k , e ∗ n i = X k −h e ∗ i ∧ ∗ e ∗ n , e j ≻ e m i + h e ∗ k , e j ≻ e m ih e k , e ∗ i ∧ ∗ e ∗ n i + h e ∗ i ∧ ∗ e ∗ k , e m ih e j ց e k , e ∗ n i−h e ∗ i , e k ≻ e m ih e j , e ∗ k տ ∗ e ∗ n i − h e m , e ∗ i ∧ ∗ e ∗ k ih e j ց e k , e ∗ n i + h e ∗ i , e k ≻ e m ih e j , e ∗ k տ ∗ e ∗ n i = 0 . Similarly, the coefficients of e m ⊗ e n , e m ⊗ e ∗ n and e ∗ m ⊗ e ∗ n are zero, too.Furthermore, we have r ∧ • r = X i,j e j ⊗ e i ∧ • e ∗ j ⊗ e ∗ i = X i,j e j ⊗ [ − R ∗ւ ( e i ) e ∗ j + L ∗≻ ∗ ( e ∗ j ) e i ] ⊗ e ∗ i = X i,j,k − e j ⊗ e ∗ k h e ∗ j , e k ւ e i i ⊗ e ∗ i + e j ⊗ e k h e i , e ∗ j ≻ ∗ e ∗ k i ⊗ e ∗ i = X i,k − e k ւ e i ⊗ e ∗ k ⊗ e ∗ i + e i ⊗ e k ⊗ e ∗ i ≻ ∗ e ∗ k = − r ւ • r + r ≻ • r . So Q = 0. Similarly, r satisfies Eqs. (6.15)-(6.19). So the cooperations α QD ( x ) = ( − ⊗ L ց ( x ) + R ⋆ ( x ) ⊗ r, β QD ( x ) = (1 ⊗ L ∨ ( x ) − R ≺ ( x ) ⊗ r, ˜ α QD ( x ) = (1 ⊗ L ≻ ( x ) − R ∧ ( x ) ⊗ r, ˜ β QD ( x ) = ( − ⊗ L ⋆ ( x ) + R տ ( x ) ⊗ r induce a quadri-bialgebra structure on A ⊕ A ∗ .On the other hand, for e i ∈ A , we have: α QD ( e i ) = X j,k {− e j ⊗ ( h e ∗ j , e k ⋆ e i i e ∗ k + h e ∗ j տ ∗ e ∗ k , e i i e k ) + e j ⋆ e i ⊗ e ∗ j } = − X j,k h e ∗ j տ ∗ e ∗ k , e i i e j ⊗ e k = − α ( e i ) . Similarly, we have β QD ( e i ) = − β ( e i ) , ˜ α QD ( e i ) = − ˜ α ( e i ) and ˜ β QD ( e i ) = − ˜ β ( e i ). So i is ahomomorphism of quadri-bialgebras. Similarly, we prove that i is also a homomorphism ofquadri-bialgebras. (cid:3) Definition 6.8.
Let ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) be a quadri-bialgebra. With the quadri-bialgebra structure given in Theorem 6.7, A ⊕ A ∗ is called a Drinfeld Q -double of A and we UADRI-BIALGEBRAS 17 denote it by QD ( A ). On the other hand, due to the symmetry between A and A ∗ (Remark 5.5),˜ r := P i e ∗ i ⊗ e i also induces a (coboundary) quadri-bialgebra structure on A ⊕ A ∗ , and we denoteit by ˜ QD ( A ). Proposition 6.9.
Let ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) be a quadri-bialgebra. Suppose that α, β, ˜ α and ˜ β are defined by Eqs. (6.1)-(6.4) with r տ = r ր = r ւ = r ց = r respectively.(1) If r satisfies Eqs. (6.14)-(6.19), then T r is a homomorphism of quadri-bialgebras which isfrom the quadri-bialgebra given in Remark 5.5 to ( A, տ , ր , ւ , ց , − α, − β, − ˜ α, − ˜ β ) .(2) If r satisfies Eqs. (6.14)-(6.19) and r is skew-symmetric, then (6.29) ˜ T r ( x + a ∗ ) := x + T r ( a ∗ ) , ∀ x ∈ A, a ∗ ∈ A , is a homomorphism of quadri-bialgebras which is from ˜ QD ( A ) to ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) .Proof. (1) Note that (1 ⊗ α ) r = r տ r , ( α ⊗ r = − r տ r . Set տ ∗ := α ∗ . Then for any a ∗ , b ∗ ∈ A ∗ we have T r ( a ∗ տ ∗ b ∗ ) = h ⊗ ( a ∗ տ ∗ b ∗ ) , r i = h ⊗ a ∗ ⊗ b ∗ , (1 ⊗ α ) r i = h ⊗ a ∗ ⊗ b ∗ , r տ r i = T r ( a ∗ ) տ T r ( b ∗ )( T r ⊗ T r ) γ ( a ∗ ) = T r ( a ∗ (1) ) ⊗ T r ( a ∗ (2) ) = X i u i ⊗ u j h a ∗ (1) , v i ih a ∗ (2) , v j i = h ⊗ ⊗ a ∗ , r տ r i = − (1 ⊗ ⊗ a ∗ )( α ⊗ r = − α ( T r ( a ∗ )) . Here we use the Sweedler’s notation: γ ( a ∗ ) = a ∗ (1) ⊗ a ∗ (2) . Similarly T r also preserves otheroperations and cooperations. So the conclusion holds.(2) We still denote the products of ˜ QD ( A ) by տ , ր , ւ , ց . First we prove that ˜ T r is ahomomorphism of quadri-algebras, that is, ˜ T r ( µ ⋄ ν ) = ˜ T r ( µ ) ⋄ ˜ T r ( ν ) for any µ, ν ∈ A ⊕ A ∗ and ⋄ ∈ {տ , ր , ւ , ց} . It is obvious when µ, ν ∈ A . Moreover, for any x ∈ A, a ∗ ∈ A ∗ we have˜ T r ( x տ a ∗ ) = ˜ T r ( R ∗ց ( x ) a ∗ + L ∗ ⋆ ( a ∗ ) x ) = T r ( R ∗ց ( x ) a ∗ ) + L ∗ ⋆ ( a ∗ ) x = (1 ⊗ a ∗ )((1 ⊗ R ց ( x )) r ) + ( a ∗ ⊗ − ⊗ L տ ( x ) + R ց ( x ) ⊗ r )= (1 ⊗ a ∗ )(( L տ ( x ) ⊗ r ) = ˜ T r ( x ) տ ˜ T r ( a ∗ ) , where we use the fact that r is skew-symmetric. Similarly, ˜ T r ( a ∗ տ x ) = ˜ T r ( a ∗ ) տ ˜ T r ( x ) andwhen ⋄ ∈ {ր , ւ , ց} , ˜ T ( µ ⋄ ν ) = ˜ T r ( µ ) ⋄ ˜ T r ( ν ) for all µ ∈ A, ν ∈ A ∗ or µ ∈ A ∗ , ν ∈ A . On theother hand, by (1), for any a ∗ , b ∗ ∈ A ∗ , we have˜ T r ( a ∗ ⋄ b ∗ ) = T r ( a ∗ ⋄ b ∗ ) = T r ( a ∗ ) ⋄ T r ( b ∗ ) = ˜ T r ( a ∗ ) ⋄ ˜ T r ( b ∗ ) , where ⋄ ∈ {տ , ր , ւ , ց} . Therefore, ˜ T r is a homomorphism of quadri-algebras.Furthermore, let { e , ..., e n } be a basis of A and let { e ∗ , ..., e ∗ n } be the dual basis. Define˜ r := P i e ∗ i ⊗ e i . Then we have( ˜ T r ⊗ ˜ T r )(˜ r ) = X i ˜ T r ( e ∗ i ) ⊗ ˜ T r ( e i ) = X i (1 ⊗ e ∗ i )( r ) ⊗ e i = r, which implies ˜ T r is a homomorphism of quadri-coalgebras. So the conclusion follows. (cid:3) Q -equation By Theorem 6.1, we have the following conclusion.
Proposition 7.1.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and r ∈ A ⊗ A be skew-symmetric.Then the comultiplications α, β, ˜ α and ˜ β defined by Eqs. (6.1)-(6.4) with r տ = r ր = r ւ = r ց = r respectively make ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) into a quadri-bialgebra if r satisfies Eqs.(6.14)-(6.19). Definition 7.2. ([Bai3]) Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let ( V, l տ , r տ , l ր , r ր , l ւ , r ւ , l ց , r ց ) be a bimodule. An O -operator of A associated to the bimodule V is a linear map T from V to A such that for all u, v ∈ V ,(7.1) T ( u ) ◦ T ( v ) = T ( l ◦ ( T ( u )) v + r ◦ ( T ( v )) u ) , ◦ ∈ {տ , ր , ւ , ց} . Proposition 7.3. ([Bai3])
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let r ∈ A ⊗ A be skew-symmetric. Then the following conditions are equivalent: (1) r satisfies Eqs. (6.14)-(6.15); (2) r satisfies Eqs. (6.16)-(6.17); (3) r satisfies Eqs. (6.18)-(6.19); (4) T r is an O -operator of A associated to the bimodule ( A ∗ , R ∗ց , L ∗ ⋆ , − R ∗∨ , − L ∗≺ , − R ∗≻ , − L ∗∧ , R ∗ ⋆ , L ∗տ ) . (5) T r is an O -operator of A v associated to the bimodule ( A ∗ , − R ∗ւ , L ∗≻ , R ∗≺ , − L ∗ր ) ; Definition 7.4. ([Bai3]) Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let r ∈ A ⊗ A . A set ofequations (6.14) and (6.15) is called Q -equation in ( A, տ , ր , ւ , ց ). Remark 7.5.
In the sense of Proposition 7.3 (in terms of O -operators), Q -equation in a quadri-algebra can be regarded as an analogue of the classical Yang-Baxter equation in a Lie algebra([Bai1, K]), which led to the introduction of Q -equation in [Bai3].The following conclusion is obvious: Corollary 7.6.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let r ∈ A ⊗ A be skew-symmetric.Then the comultiplications α, β, ˜ α and ˜ β defined by Eqs. (6.1)-(6.4) with r տ = r ր = r ւ = r ց = r respectively make ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) into a quadri-bialgebra if r is a solution of Q -equation. Proposition 7.7.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let r be a skew-symmetric solu-tion of Q -equation. Then the quadri-algebra structure ( տ • , ր • , ւ • , ց • ) on the Drinfeld Q -double UADRI-BIALGEBRAS 19 QD ( A ) can be given as follows (for any x ∈ A, a ∗ , b ∗ ∈ A ∗ ): (7.2) a ∗ տ • b ∗ = R ∗ց ( T r ( a ∗ )) b ∗ + L ∗ ⋆ ( T r ( b ∗ )) a ∗ , a ∗ ր • b ∗ = − R ∗∨ ( T r ( a ∗ )) b ∗ − L ∗≺ ( T r ( b ∗ )) a ∗ , (7.3) a ∗ ւ • b ∗ = − R ∗≻ ( T r ( a ∗ )) b ∗ − L ∗∧ ( T r ( b ∗ )) a ∗ , a ∗ ց • b ∗ = R ∗ ⋆ ( T r ( a ∗ )) b ∗ + L ∗տ ( T r ( b ∗ )) a ∗ , (7.4) a ∗ տ • x = − T r ( L ∗ ⋆ ( x ) a ∗ ) + T r ( a ∗ ) տ x + L ∗ ⋆ ( x ) a ∗ , (7.5) a ∗ ր • x = T r ( L ∗≺ ( x ) a ∗ ) + T r ( a ∗ ) ր x − L ∗≺ ( x ) a ∗ , (7.6) a ∗ ւ • x = T r ( L ∗∧ ( x ) a ∗ ) + T r ( a ∗ ) ւ x − L ∗∧ ( x ) a ∗ , (7.7) a ∗ ց • x = − T r ( L ∗տ ( x ) a ∗ ) + T r ( a ∗ ) ց x + L ∗տ ( x ) a ∗ , (7.8) x տ • a ∗ = R ∗ց ( x ) a ∗ + x տ T r ( a ∗ ) − T r ( R ∗ց ( x ) a ∗ ) , (7.9) x ր • a ∗ = − R ∗∨ ( x ) a ∗ + x ր T r ( a ∗ ) + T r ( R ∗∨ ( x ) a ∗ ) , (7.10) x ւ • a ∗ = − R ∗≻ ( x ) a ∗ + x ւ T r ( a ∗ ) + T r ( R ∗≻ ( x ) a ∗ ) , (7.11) x ց • a ∗ = R ∗ ⋆ ( x ) a ∗ + x ց T r ( a ∗ ) − T r ( R ∗ ⋆ ( x ) a ∗ ) . Proof.
Let { e , ..., e n } be a basis of A and let { e ∗ , ..., e ∗ n } be its dual basis. Suppose that e i տ e j = X i,j c kij e k , e i ր e j = X i,j d kij e k , e i ւ e j = X i,j ˜ c kij e k , e i ց e j = X i,j ˜ d kij e k , and r = P i,j a ij e i ⊗ e j , where a ij = − a ji . Then T r ( e ∗ i ) = P k a ki e k . For any k, l we have e ∗ k տ ∗ e ∗ l = X s h e ∗ k ⊗ e ∗ l , α ( e s ) i e ∗ s = X t,s [ − a kt ˜ d lst + a tl ( c kts + d kts + ˜ c kts + ˜ d kts )] e ∗ s = R ∗ց ( T r ( e ∗ k )) e ∗ l + L ∗ ⋆ ( T r ( e ∗ l )) e ∗ k . Hence for any a ∗ , b ∗ ∈ A ∗ we have that a ∗ տ • b ∗ = R ∗ց ( T r ( a ∗ )) b ∗ + L ∗ ⋆ ( T r ( b ∗ )) a ∗ . Similarly a ∗ ր • b ∗ = − R ∗∨ ( T r ( a ∗ )) b ∗ − L ∗≺ ( T r ( b ∗ )) a ∗ . So Eq. (7.2) holds. Eq. (7.3) is proved in a similarway. On the other hand, we have R ∗ց ( e ∗ k ) e l = X s h e l , e ∗ s ց ∗ e ∗ k i e s = X s h e l , R ∗ ⋆ ( T r ( e ∗ s )) e ∗ k + L ∗տ ( T r ( e ∗ k )) e ∗ s i e s = X s h e ∗ s , − T r ( L ∗ ⋆ ( e l ) e ∗ k ) + T r ( e ∗ k ) տ e l i e s = − T r ( L ∗ ⋆ ( e l ) e ∗ k ) + T r ( e ∗ k ) տ e l . Thus, e ∗ k տ • e l = R ∗ց ( e ∗ k ) e l + L ∗ ⋆ ( e l ) e ∗ k = − T r ( L ∗ ⋆ ( e l ) e ∗ k ) + T r ( e ∗ k ) տ e l + L ∗ ⋆ ( e l ) e ∗ k . Therefore Eq.(7.4) holds. Eqs. (7.5)-(7.11) are verified in a same way. (cid:3) Proposition 7.8. ([Bai3])
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra and let r ∈ A ⊗ A . Supposethat r is skew-symmetric and nondegenerate. Then r is a solution of Q -equation if and only if theinverse of the isomorphism A ∗ → A induced by r , regarded as a bilinear form ω on A , satisfies (7.12) ω ( x, y ∧ z ) = − ω ( x ւ y, z ) + ω ( z ≻ x, y ) , (7.13) ω ( x, y ∨ z ) = ω ( x ≺ y, z ) − ω ( z ր x, y ) , ∀ x, y, z ∈ A. Definition 7.9.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra. A bilinear form ω : A ⊗ A → F iscalled a 2 -cocycle of A if it satisfies Eqs. (7.12) and (7.13).It is easy to show that if ω : A ⊗ A → F is a 2-cocycle of A , then for any x, y ∈ A , B ( x, y ) := ω ( x, y ) + ω ( y, x ) is a 2-cocycle on A v . Proposition 7.10.
Let ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) be a quadri-bialgebra obtained from a skew-symmetric solution of Q -equation. Let ( A, A ∗ , R ∗ց , L ∗ ⋆ , − R ∗∨ , − L ∗≺ , − R ∗≻ , − L ∗∧ , R ∗ ⋆ , L ∗տ , R ∗ց ∗ , L ∗ ⋆ ∗ , − R ∗∨ ∗ , − L ∗≺ ∗ , − R ∗≻ ∗ , − L ∗∧ ∗ , R ∗ ⋆ ∗ , L ∗տ ∗ ) be the corresponding matched pair of quadri-algebras, wherethe subscript ∗ denotes the quadri-algebra structure on A ∗ . Set (7.14) A ⋉ A ∗ =: A ⋉ R ∗ց ,L ∗ ⋆ , − R ∗∨ , − L ∗≺ , − R ∗≻ , − L ∗∧ ,R ∗ ⋆ ,L ∗տ A ∗ . (1) A ⊲⊳ R ∗ց∗ ,L ∗ ⋆ ∗ , − R ∗∨∗ , − L ∗≺∗ , − R ∗≻∗ , − L ∗∧∗ ,R ∗ ⋆ ∗ ,L ∗տ∗ R ∗ց ,L ∗ ⋆ , − R ∗∨ , − L ∗≺ , − R ∗≻ , − L ∗∧ ,R ∗ ⋆ ,L ∗տ A ∗ is isomorphic to A ⋉ A ∗ as quadri-algebras. (2) The skew-symmetric solutions of Q -equation in A are in one-to-one correspondence withlinear maps T r : A ∗ → A whose graphs are Lagrangian sub-quadri-algebras of A ⋉ A ∗ withrespect to the bilinear form (3 . . Here the graph of a linear map T : A ∗ → A is defined as graph ( T ) := { ( T ( a ∗ ) , a ∗ ) | a ∗ ∈ A ∗ } ⊂ A ⋉ A ∗ .Proof. We denote the quadri-algebra structure on A ⋉ A ∗ by տ † , ր † , ւ † , ց † . Define a linearmap θ : A ⊲⊳ R ∗ց∗ ,L ∗ ⋆ ∗ , − R ∗∨∗ , − L ∗≺∗ , − R ∗≻∗ , − L ∗∧∗ ,R ∗ ⋆ ∗ ,L ∗տ∗ R ∗ց ,L ∗ ⋆ , − R ∗∨ , − L ∗≺ , − R ∗≻ , − L ∗∧ ,R ∗ ⋆ ,L ∗տ A ∗ → A ⋉ A ∗ by θ ( x, a ∗ ) := ( T r ( a ∗ ) + x, a ∗ ) , ∀ x ∈ A ; a ∗ ∈ A ∗ . It is straightforward to check that θ is a homomorphism of quadri-algebras. Moreover, θ isbijective. So (1) holds.Suppose that r is a skew-symmetric solution of Q -equation. Then by (1) we know that θ ( A ∗ ) = graph ( T r ) and θ ( A ) = A are isotropic complementary sub-quadri-algebras of A ⋉ A ∗ and dualto each others with respect to the bilinear form (3.2). So graph ( T r ) is a Lagrangian sub-quadri-algebra of A ⋉ A ∗ . Conversely, let T r : A ∗ → A be a linear map whose graph graph ( T r ) is aLagrangian sub-quadri-algebra of A ⋉ A ∗ . Since graph ( T r ) is Lagrangian, r is skew-symmetric.Moreover, since graph ( T r ) is a sub-quadri-algebra of A ⋉ A ∗ , we have ( T r ( a ∗ ) , a ∗ ) տ † ( T r ( b ∗ ) , b ∗ ) = ( T r ( a ∗ ) տ T r ( b ∗ ) , R ∗ց ( T r ( a ∗ )) b ∗ + L ∗ ⋆ ( T r ( b ∗ )) a ∗ )= ( T r ( R ∗ց ( T r ( a ∗ )) b ∗ + L ∗ ⋆ ( T r ( b ∗ )) a ∗ ) , R ∗ց ( T r ( a ∗ )) b ∗ + L ∗ ⋆ ( T r ( b ∗ )) a ∗ ) . UADRI-BIALGEBRAS 21
Therefore, T r ( a ∗ ) տ T r ( b ∗ ) = T r ( R ∗ց ( T r ( a ∗ )) b ∗ + L ∗ ⋆ ( T r ( b ∗ )) a ∗ ). Similarly we have T r ( a ∗ ) ր T r ( b ∗ ) = T r ( − R ∗∨ ( T r ( a ∗ )) b ∗ − L ∗≺ ( T r ( b ∗ )) a ∗ ) ,T r ( a ∗ ) ւ T r ( b ∗ ) = T r ( − R ∗≻ ( T r ( a ∗ )) b ∗ − L ∗∧ ( T r ( b ∗ )) a ∗ ) ,T r ( a ∗ ) ց T r ( b ∗ ) = T r ( R ∗ ⋆ ( T r ( a ∗ )) b ∗ + L ∗տ ( T r ( b ∗ )) a ∗ ) . So by Proposition 7.3, we know that r is a skew-symmetric solution of Q -equation. (cid:3) Construction of linear operators on some “double spaces” of quadri-algebras
An interesting (and important) feature of dendriform dialgebras, quadri-algebras and othersimilar algebra structures mentioned in [EG1] is that they have close relations with various linearoperators in combinatorics ([Ag1, Ag2, Ag3, AL, Bax, E1, E2, EG1, R1, R2]).
Definition 8.1.
Let A be a vector space with a set of bilinear operations Ω := {∗ n : A ⊗ A → A, n = 1 , ..., m } . A linear operator P on A is called a Rota-Baxter operator of weight λ ( ∈ F ) if,for each ∗ ∈ Ω, we have(8.1) P ( x ) ∗ P ( y ) = P ( P ( x ) ∗ y + x ∗ P ( y ) + λx ∗ y ) , ∀ x, y ∈ A. A linear operator N on A is called a Nijenhuis operator if, for each ∗ ∈
Ω, we have(8.2) N ( x ) ∗ N ( y ) = N ( N ( x ) ∗ y + x ∗ N ( y ) − N ( x ∗ y )) , ∀ x, y ∈ A. We have the following relationship between Rota-Baxter operators of weight λ and Nijenhuisoperators ([E1]). Proposition 8.2.
With the notations in Definition 8.1, if N : A → A is a Nijenhuis operatoron A satisfying N = λ id , then P := − λ id − N is a Rota-Baxter operator of weight λ (on A ).Proof. It is straightforward. (cid:3)
Proposition 8.3.
Let ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) be a quadri-bialgebra, which is induced froma skew-symmetric solution r of Q -equation. If, in addition, r is nondegenerate, then for all x ∈ A, a ∗ ∈ A ∗ , (8.3) N λ ,λ ,λ ,λ (( x, a ∗ )) := ( λ T r ( a ∗ ) + λ x, λ T − r ( x ) + λ a ∗ ) is a Nijenhuis operator on the Drinfeld Q -double QD ( A ) , where λ i ∈ F , i = 1 , , , .Proof. It is straightforward. (cid:3)
In the following we assume the coefficient field F is the real field R and let λ ∈ R . With theconditions in Proposition 8.3, if we consider the Nijenhuis operators satisfying N λ ,λ ,λ ,λ = λ id and apply Proposition 8.2, then we can get three families of Rota-Baxter operators of weight λ on QD ( A ):(F1) P + λ,k (( x, a ∗ )) := − λ − N ,λ,k, − λ x, a ∗ )) = ( − λx, − k T − r ( x ))or P − λ,k (( x, a ∗ )) := − λ − N , − λ,k,λ x, a ∗ )) = (0 , − k T − r ( x ) − λa ∗ ) , k = 0;(F2) ˆ P + λ, ˆ k (( x, a ∗ )) := − λ − N ˆ k,λ, , − λ x, a ∗ )) = ( − ˆ k T r ( a ∗ ) − λx, P − λ, ˆ k (( x, a ∗ )) := − λ − N ˆ k, − λ, ,λ x, a ∗ )) = ( − ˆ k T r ( a ∗ ) , − λa ∗ ) , (ˆ k, λ ) = (0 , P λ,k ,k (( x, a ∗ )) := − λ − N k ,k , λ − k k , − k x, a ∗ ))= ( − k T r ( a ∗ ) − ( k + λ )2 x, ( k − λ )2 k T − r ( x ) + ( k − λ )2 a ∗ ) , k = ± λ, k = 0 , for any x ∈ A, a ∗ ∈ A ∗ , where k, ˆ k, k , k ∈ R . Here we exclude the trivial cases that P = − λ id.Furthermore, it is easy to check that these Rota-Baxter operators are idempotents (i.e., P = P )if and only if λ = −
1, i.e., they are Rota-Baxter operators of weight −
1. We write down themexplicitly as follows (see the discussion at the end of this section for their important roles inrenormalization in quantum field theory):(G1) P + − ,k (( x, a ∗ )) := 1 − N , − ,k, x, a ∗ )) = ( x, − k T − r ( x ))or P −− ,k (( x, a ∗ )) := 1 − N , ,k, − x, a ∗ )) = (0 , − k T − r ( x ) + a ∗ ) , k = 0;(G2) ˆ P + − , ˆ k (( x, a ∗ )) := 1 − N ˆ k, − , , x, a ∗ )) = ( − ˆ k T r ( a ∗ ) + x, P −− , ˆ k (( x, a ∗ )) := 1 − N ˆ k, , , − x, a ∗ )) = ( − ˆ k T r ( a ∗ ) , a ∗ );(G3) P − ,k ,k (( x, a ∗ )) := 1 − N k ,k , − k k , − k x, a ∗ ))= ( − k T r ( a ∗ ) − ( k − x, ( k − k T − r ( x ) + ( k + 1)2 a ∗ ) , k = ± , k = 0 , for all x ∈ A, a ∗ ∈ A ∗ , where k, ˆ k, k , k ∈ R .On the other hand, the requirement that r is nondegenerate can be dropped if λ appearingin the definition of N λ ,λ ,λ ,λ is equal to zero. More precisely, we have the following conclusion. Proposition 8.4.
Let ( A, տ , ր , ւ , ց , α, β, ˜ α, ˜ β ) be a quadri-bialgebra, which is induced from askew-symmetric solution r of Q -equation. Then, for all x ∈ A, a ∗ ∈ A ∗ , (8.4) N λ ,λ ,λ (( x, a ∗ )) := ( λ T r ( a ∗ ) + λ x, λ a ∗ ) is a Nijenhuis operator on QD ( A ) , where λ i ∈ F , i = 1 , , .Proof. It is straightforward. (cid:3)
UADRI-BIALGEBRAS 23
Remark 8.5.
Similarly, we consider the case that N λ ,λ ,λ = λ id for λ ∈ R and then applyProposition 8.2 to get certain families of Rota-Baxter operators (of weight λ ) on QD ( A ). In fact,one can show that the Rota-Baxter operators (of weight λ ) are given by (F2) and the Rota-Baxteroperators which are idempotents are given by (G2), where x ∈ A, a ∗ ∈ A ∗ . Proposition 8.6.
Let ( A, տ , ր , ւ , ց ) be a quadri-algebra, then (8.5) N λ ,λ ,λ ,λ (( x, y )) := ( λ y + λ x, λ x + λ y ) is a Nijenhuis operator on A v ⋉ L ր ,R տ ,L ց ,R ւ A , where x, y ∈ A and λ i ∈ F , i = 1 , , , . More-over, if ( A v , A v , L ր , R տ , L ց , R ւ , L ր , R տ , L ց , R ւ ) is a matched pair of dendriform dialgebras,then the linear operator defined by Eq. (8.5) is also a Nijenhuis operator on A v ⊲⊳ L ր ,R տ ,L ց ,R ւ L ր ,R տ ,L ց ,R ւ A v . On the other hand, (8.6) θ (( x, y )) := ( y + x, x ) is an isomorphism of dendriform dialgebras from A v ⊲⊳ L ր ,R տ ,L ց ,R ւ L ր ,R տ ,L ց ,R ւ A v to A v ⋉ L ր ,R տ ,L ց ,R ւ A .Proof. It is straightforward. (cid:3)
Remark 8.7.
Similarly, we can also consider the case that N λ ,λ ,λ ,λ = λ id and then applyProposition 8 . λ on the “doublespaces” given in Proposition 8.6. Remark 8.8.
One can use these Nijenhuis operators and Rota-Baxter operators (of weight λ ) to construct N S -algebra and dendriform trialgebra structures (see [EG1] and the referencestherein). Moreover, it is easy to show that these Nijenhuis operators also satisfy Eq. (8.2) on theircorresponding associative algebras, that is, they are the so-called associative Nijenhuis tensors of the associated associative algebras in the sense of [CGM], where this notion was introducedin the study of Wigner problem in quantum physics.
Remark 8.9.
Obviously, if P is a Rota-Baxter operator of weight λ on a quadri-algebra, then itis a Rota-Baxter operator of weight λ on its associated associative algebra. Furthermore, Rota-Baxter operators on associative algebras which are idempotents play a key role in the algebraicBirkhoff decomposition in pQFT ([EGK1, EGK2], also see [EG2] for a good survey of this topic).As we have discussed, we have constructed certain families of Rota-Baxter operators on the“double spaces” of quadri-algebras. Acknowledgements
The authors are grateful of Professor J.-L. Loday for important suggestion. This work wassupported in part by NSFC (10921061), NKBRPC (2006CB805905), SRFDP (200800550015).
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