Coherent unit actions on braided operads and Hopf algebras
aa r X i v : . [ m a t h . QA ] M a y COHERENT UNIT ACTIONS ON BRAIDED OPERADS AND HOPF ALGEBRAS
LI GUO AND YUNNAN LIA bstract . The notion of a coherent unit action on algebraic operads was first introduced by Lodayfor binary quadratic nonsymmetric operads and generalized by Holtkamp, to ensure that the freeobjects of the operads carry a Hopf algebra structure. There was also a classification of suchoperads in the binary quadratic nonsymmetric case. We generalize the notion of coherent unitaction to braided operads and show that free objects of braided operads with such an action carriesa braided Hopf algebra structure. Under the conditions of binary, quadratic and nonsymmetric, wegive a characterization and classification of the braided operads that allow a coherent unit actionand thus carry a braided Hopf algebra structure on their free objects. C ontents
1. Introduction 12. Braided algebraic operads 23. Coherent unit actions and braided P -Hopf algebras 84. Classification of braided binary quadratic nonsymmetric operads with coherent unitactions 11References 131. I ntroduction This paper explores braided Hopf algebra structures on free objects from braided operads bythe method of coherent unit actions.As the importance of Hopf algebras continued to show in more and more areas of mathematics,ad hoc instances of Hopf algebras sometimes turned out to be just special cases of a general con-struction. One such general construction is the free objects of various (nonassociative) algebras(operads). The first instance is the dendriform algebra introduced by Loday [16] who showedthat, even though a dendriform algebra is not an associative algebra, free dendriform algebras hasa natural Hopf algebra structure[19]. This Hopf algebra is realized on planar binary trees and wasfound to be isomorphic to the noncommutative analog (by Foissy and Holtkamp [5, 10]) of theHopf algebra of rooted trees in the Connes-Kreimer theory of renormalization of quantum fieldtheory [2]. After Hopf algebra structures were discovered for several related algebraic structures,such as tridendriform algebra and quadri-algebras [1, 20], Loday showed in [16] that a nonsym-metric (also called regular) operad with a so-called coherent unit action endows a natural Hopfalgebra structure on its free objects. Similar results hold for the corresponding algebraic struc-tures with certain commutativity [17, 18]. This uniform approach not only included as special
Date : May 5, 2020.2010
Mathematics Subject Classification.
Key words and phrases. operad, unit action, braided Hopf algebra, braided dendriform algebra, braided tridendri-form algebra. cases many Hopf algebras obtained case by case during that period of time, it also recovered theclassical Hopf algebra structures on the shu ffl e product algebra and quasi-shu ffl e algebra, as thealgebras are shown to be the free objects of commutative dendriform and tridendriform algebras.This work was further generalized to nonsymmetric operads by Holtkamp [11]. Given the greatinterest in such Hopf algebras, a characterization and classification were achieved for binary qua-dratic nonsymmetric operads which have a coherent unit action, and hence allow for Hopf algebrastructures on their free objects [3].Operads have been extended to various broader contexts. We are particularly interested inbraided operads [4, 7] for their connection with Yang-Baxter equations and quantum theory[15].In particular, in this context quantum shu ffl e algebras, quantum quasi-shu ffl e algebras and gen-erally quantum multi-brace algebras have been obtained by Rosso, Jian-Rosso-Zhang and Jian-Rosso [22, 14, 12, 13] respectively. Recently, with motivation from braided construction of rootedtrees from the work of Connes-Kreimer on renormalization of quantum field theory, braided struc-tures for dendriform algebras and tridendriform algebras have attracted attention [6, 8, 9]. Thereagain the free objects can be equipped with braided Hopf algebra structures.Thus it appears to be the time to provide a uniform approach in the context of braided op-erads, in the direction of coherent unit actions on algebraic operads developed by Loday andHoltkamp [17, 11] as noted above. In particular, it is interesting to study the coherent unit actionssuch that the free objects of the braided operads are braided Hopf algebras. This is the purposeof this article. We also extend the classification of coherent unit actions on binary quadratic non-symmetric operads in [3] to the braided context, thereby determining such braided operads thatgive rise to braided Hopf algebras from their free objects.Here is the layout of the paper. In Section 2, we provide background on braided operads withsome details on the dendriform and tridendriform cases. In Section 3, we extend the notion ofcoherent unit actions on nonsymmetric operads to braided operads and show that such braidedoperads have a braided Hopf algebra structure on their free objects. We consider the two casesof braided nonsymmetric and completely commutative operads, so that we can cover for instanceboth the braided dendriform operad and braided commutative dendriform (Zinbiel) operad. InSection 4, we give a classification of braided binary quadratic nonsymmetric operads with coher-ent unit actions. Notations.
In this paper, we fix a ground field k of characteristic 0. All the objects underdiscussion, including vector spaces, algebras and tensor products, are taken over k by default.Also denote S n (resp. B n ) the n -th symmetric group (resp. braid group) for any n ≥ S = ∅ (resp. B = ∅ ). For the convenience of discussion, we also fix a braided tensor category C over k to involve all the forthcoming braided objects.2. B raided algebraic operads We start with a basic notion.
Definition 2.1. A braided vector space is a vector space V together with a linear operator σ on V ⊗ , called the Yang-Baxter operator characterized by the equation(1) ( σ ⊗ id V )(id V ⊗ σ )( σ ⊗ id V ) = (id V ⊗ σ )( σ ⊗ id V )(id V ⊗ σ ) . For any braided vector space ( V , σ ) and n ≥
1, the tensor space V ⊗ n becomes a representation of B n with its usual generators b i acting as σ i : = id ⊗ ( i − ⊗ σ ⊗ id ⊗ ( n − − i ) for 1 ≤ i ≤ n −
1. For braidedvector spaces ( V , σ ) , ( V ′ , σ ′ ), a linear map f : V → V ′ is called a homomorphism of braided NIT ACTION ON BRAIDED OPERADS AND HOPF ALGEBRAS 3 vector spaces if ( f ⊗ f ) σ = σ ′ ( f ⊗ f ). Let VS (resp. BV ) denote the category of (braided) vectorspaces.Let π n : B n → S n be the natural projection mapping b i to transpositions s i , i = , . . . , n − ι n : S n → B n , satisfying π n ι n = id S n andcalled the Matsumoto-Tits section . If w = s i · · · s i r is any reduced expression of w ∈ S n , then ι n ( w ) = b i · · · b i r , which is uniquely determined by w . Given any braided vector space ( V , σ ), wedenote the action of ι n ( w ) on V ⊗ n by T σ w : = σ i · · · σ i r . In particular, for the usual flip map τ of V ,we have T τ w ( v ⊗ · · · ⊗ v n ) = v w − (1) ⊗ · · · ⊗ v w − ( n ) . Definition 2.2.
Let A be an algebra with product µ , and σ be a braiding on A . We call the triple( A , µ, σ ) a braided algebra if it satisfies the conditions(2) ( µ ⊗ Id A ) σ σ = σ (Id A ⊗ µ ) , (Id A ⊗ µ ) σ σ = σ ( µ ⊗ Id A ) . Moreover, if A is unital with unit 1 A and satisfies(3) σ ( a ⊗ A ) = A ⊗ a , σ (1 A ⊗ a ) = a ⊗ A for all a ∈ A , then A is called a unital braided algebra .Dually, we call ( C , ∆ , ε, σ ) a braided coalgebra if ( C , ∆ , ε ) is a coalgebra with braiding σ andsatisfies σ σ ( ∆ ⊗ Id C ) = (Id C ⊗ ∆ ) σ, σ σ (Id C ⊗ ∆ ) = ( ∆ ⊗ Id C ) σ, (4) ( ε ⊗ Id C ) σ = Id C = (Id C ⊗ ε ) σ. (5)Also, a quintuple ( H , µ, ∆ , ε, σ ) is called a braided bialgebra , if ( H , µ, σ ) (resp. ( H , ∆ , ε, σ ))is a braided algebra (resp. coalgebra) such that(6) ∆ µ = ( µ ⊗ µ ) σ ( ∆ ⊗ ∆ ) . When H has unit 1 H and antipode S : H → H such that µ ( S ⊗ Id H ) ∆ = ε H = µ (Id H ⊗ S ) ∆ , thenthe septuple ( H , µ, ∆ , ε, S , σ ) is a braided Hopf algebra . Similar to the case of braided vectorspaces, the homomorphisms for these braided objects are the usual homomorphisms commutatingwith braidings.The notion of braided operads was originally introduced by Fiedorowicz in [4]. Especiallywhen the Yang-Baxter operators are chosen to be the flip maps, the framework recovers the usualdefinition of operads. For the study of algebraic operads and braided operads, we refer the readerto [7, 21]. Definition 2.3. A braided (algebraic) operad over k is an analytic functor P : BV → BV suchthat P (0) = γ : P◦P → P and a unit η : id BV → P such that the following diagrams commute:( P ◦ P ) ◦ P = / / γ ◦ id P (cid:15) (cid:15) P ◦ ( P ◦ P ) id P ◦ γ / / P ◦ P γ (cid:15) (cid:15) P ◦ P γ / / P , id BV ◦ P η ◦ id P / / = % % ❏❏❏❏❏❏❏❏❏❏❏❏❏ P ◦ P γ (cid:15) (cid:15) P ◦ id BV id P ◦ η o o = y y ttttttttttttt P . Namely, γ V γ P ( V ) = γ V P ( γ V ) , γ V η P ( V ) = id P ( V ) = γ V P ( η V ) for any ( V , σ ) ∈ BV . LI GUO AND YUNNAN LI
For two braided operads ( P , γ P , η P ) , ( Q , γ Q , η Q ), a natural transformation α : P → Q is called a morphism of braided operads , if γ Q ◦ ( α, α ) = α ◦ γ P and η Q = α ◦ η P . Definition 2.4. A B -module over k is a family M = { M ( n ) } n ≥ of right B n -modules M ( n ). Amorphism of B -modules f : M → N is a family of homomorphisms of B n -modules f n : M ( n ) → N ( n ).Given a B -module M and a braided vector space ( V , σ ), we define a functor e M : BV → BV by e M ( V ) : = M n ≥ M ( n ) ⊗ B n V ⊗ n , whose Yang-Baxter operator, denoted by σ M , is determined by the following equalities,(7) σ M (( µ, u , . . . , u i ) ⊗ ( ν, v , . . . , v j )) : = X ( ν, v ′ , . . . , v ′ j ) ⊗ ( µ, u ′ , . . . , u ′ i ) , for any ( µ, u , . . . , u i ) ∈ M ( i ) ⊗ B i V ⊗ i and ( ν, v , . . . , v j ) ∈ M ( j ) ⊗ B j V ⊗ j , where we denote T σβ ij ( u ⊗ · · · ⊗ u i ⊗ v ⊗ · · · ⊗ v j ) : = X v ′ ⊗ · · · ⊗ v ′ j ⊗ u ′ ⊗ · · · ⊗ u ′ i , with β i j ∈ S i + j such that β i j ( k ) = j + k if 1 ≤ k ≤ i , and k − i if i + ≤ k ≤ i + j .Define the tensor product of B -modules M and N to be the B -module M ⊗ N with( M ⊗ N )( n ) : = M i + j = n Ind B n B i × B j M ( i ) ⊗ N ( j ) = M i + j = n ( M ( i ) ⊗ N ( j )) ⊗ B i × B j k [ B n ] , n ≥ . Define the composite of B -modules M and N to be the B -module M ◦ N : = M k ≥ M ( k ) ⊗ B k N ⊗ k . More precisely,( M ◦ N )( n ) : = M k ≥ M ( k ) ⊗ B k M i + ··· + i k = n Ind B n B i ×···× B ik N ( i ) ⊗ · · · ⊗ N ( i k ) , n ≥ , where the left module action of B k on N ⊗ k is defined as follows. Any braid b ∈ B k sends( µ , . . . , µ k , c ) ∈ ( N ( i ) ⊗ · · · ⊗ N ( i k )) ⊗ B i ×···× B ik k [ B n ]to ( µ b − (1) , . . . , µ b − ( k ) , b ( i , . . . , i k ) c ) ∈ ( N ( i b − (1) ) ⊗ · · · ⊗ N ( i b − ( k ) )) ⊗ B ib − ×···× B ib − k ) k [ B n ] , where b ∈ B k acts on { , . . . , k } by permutations via the natural projection π k , and b ( i , . . . , i k ) isthe braid obtained from b by replacing its j -th strand with i j parallel strands for any j = , . . . , k . Proposition 2.5.
For any B -modules M , N and braided vector space V, we have ( ] M ⊗ N )( V ) = e M ( V ) ⊗ e N ( V ) , ( ] M ◦ N )( V ) = e M ( e N ( V )) . Proof.
By the definition of e M and M ⊗ N , we find( ] M ⊗ N )( V ) = M n ≥ ( M ⊗ N )( n ) ⊗ B n V ⊗ n = M n ≥ M i + j = n ( M ( i ) ⊗ N ( j )) ⊗ B i × B j k [ B n ] ⊗ B n V ⊗ n NIT ACTION ON BRAIDED OPERADS AND HOPF ALGEBRAS 5 = M i , j ≥ ( M ( i ) ⊗ N ( j )) ⊗ B i × B j V ⊗ ( i + j ) = M i ≥ M ( i ) ⊗ B i V ⊗ i ⊗ M j ≥ N ( j ) ⊗ B j V ⊗ j = e M ( V ) ⊗ e N ( V ) . On the other hand, by the definition of e M and M ◦ N , we have( ] M ◦ N )( V ) = M n ≥ ( M ◦ N )( n ) ⊗ B n V ⊗ n = M n ≥ M k ≥ M ( k ) ⊗ B k M i + ··· + i k = n ( N ( i ) ⊗ · · · ⊗ N ( i k )) ⊗ B i ×···× B ik k [ B n ] ⊗ B n V ⊗ n = M k ≥ M ( k ) ⊗ B k M i ,..., i k ≥ ( N ( i ) ⊗ · · · ⊗ N ( i k )) ⊗ B i ×···× B ik V ⊗ ( i + ··· + i k ) = M k ≥ M ( k ) ⊗ B k M i ≥ N ( i ) ⊗ B i V ⊗ i ⊗ · · · ⊗ M i k ≥ N ( i k ) ⊗ B ik V ⊗ i k = M k ≥ M ( k ) ⊗ B k e N ( V ) ⊗ k = e M ( e N ( V )) , where the third equality is due to the fact that the left action of B k on Ind B n B i ×···× B ik N ( i ) ⊗· · · ⊗ N ( i k )commutes with the right one of B n , and the fifth equality is based on the representation e N ( V ) ⊗ k of B k via the Yang-Baxter operator σ N of e N ( V ). (cid:3) According to [4, Definition 3.2] or the construction in [7, § P consists ofa B -module {P ( n ) } n ≥ , with the composition maps γ ( i , . . . , i k ) : P ( k ) ⊗ P ( i ) ⊗ · · · ⊗ P ( i k ) → P ( i + · · · + i k )and the unit map η satisfying the associativity and the unity conditions, and also the followingequivalence conditions: γ ( µ b ; µ , . . . , µ k ) = γ ( µ ; µ b − (1) , . . . , µ b − ( k ) ) b ( i , . . . , i k ) ,γ ( µ ; µ b , . . . , µ k b k ) = γ ( µ ; µ , . . . , µ k )( b × · · · × b k ) , for any µ ∈ P ( k ) , µ ∈ P ( i ) , . . . , µ k ∈ P ( i k ), where b ∈ B k acts on { , . . . , k } by permutationsvia the natural projection π k , b ( i , . . . , i k ) is the braid obtained from b by replacing its j -th strandwith i j parallel strands for any j = , . . . , k , and b × · · · × b k denotes the direct product of braids b , . . . , b k . It is easy to see that γ ( i , . . . , i k ) factors through P ( k ) ⊗ B k Ind B i + ··· + ik B i ×···× B ik P ( i ) ⊗ · · · ⊗ P ( i k ) = P ( k ) ⊗ B k ( P ( i ) ⊗ · · · ⊗ P ( i k )) ⊗ B i ×···× B ik k [ B i + ··· + i k ]by the homomorphism of B i + ··· + i k -modules sending any µ ⊗ B k ( µ ⊗ · · · ⊗ µ k ) ⊗ B i ×···× B ik b to γ ( µ ; µ , . . . , µ k ) b , as we have b ( i , . . . , i k )( b × · · · × b k ) = ( b b − (1) × · · · × b b − ( k ) ) b ( i , . . . , i k ). LI GUO AND YUNNAN LI
Given any ( V , σ ) ∈ BV , we have P ( V ) : = M n ≥ P ( n ) ⊗ B n V ⊗ n , with its Yang-Baxter operator σ P defined as in Eq. (7). If P is nonsymmetric (also called regular ), then P ( n ) = P n ⊗ k [ B n ] , n ≥
0, where L n ≥ P n is a graded vector space. Hence, P ( V ) = L n ≥ P n ⊗ V ⊗ n . Definition 2.6.
Given any (braided) operad P , a (braided) vector space A is called an algebraover P or P -algebra , if it is equipped with a homomorphism θ A : P ( A ) → A of (braided) vectorspaces such that the following diagrams commute:( P ◦ P )( A ) = / / γ A (cid:15) (cid:15) P ( P ( A )) θ P ( A ) / / P ( A ) θ A (cid:15) (cid:15) P ( A ) θ A / / A , A η A / / = ! ! ❈❈❈❈❈❈❈❈❈❈❈ P ( A ) θ A (cid:15) (cid:15) A . For two P -algebras A and B , a homomorphism ϕ : A → B of (braided) vector spaces is called a morphism of P -algebras if ϕθ A = θ B P ( ϕ ).In particular, the free P -algebra generated by a (braided) vector space V is given by ( P ( V ) , γ V )with η V : V → P ( V ), such that for any P -algebra A and homomorphism f : V → A of (braided)vector spaces, there exists a unique morphism ˜ f : P ( V ) → A of P -algebras satisfying f = ˜ f η V . Example 2.7.
A dendriform algebra D is a vector space with two binary operators ≺ , ≻ such that( a ≺ b ) ≺ c = a ≺ ( b ≺ c + b ≻ c ) , ( a ≻ b ) ≺ c = a ≻ ( b ≺ c ) , a ≻ ( b ≻ c ) = ( a ≺ b + a ≻ b ) ≻ c , for any a , b , c ∈ D . Intrinsically, any dendriform algebra is an algebra over the nonsymmetricoperad ( P D , γ, I ) generated by P D , = k I , P D , = k ≺ ⊕ k ≻ , with relations γ ( ≺ ; ≺ , I ) = γ ( ≺ ; I , ≺ + ≻ ) ,γ ( ≺ ; ≻ , I ) = γ ( ≻ ; I , ≺ ) ,γ ( ≻ ; I , ≻ ) = γ ( ≻ ; ≺ + ≻ , I ) ,γ ( ≺ ; I , I ) = ≺ , γ ( ≻ ; I , I ) = ≻ ,γ ( I ; ≺ ) = ≺ , γ ( I ; ≻ ) = ≻ . As a braided analogue, we interpret the braided dendriform algebras in [8] as the braided P D -algebras. A braided dendriform algebra ( D , ≺ , ≻ , σ ) is a braided vector space ( D , σ ) endowedwith the dendriform algebra structure ( ≺ , ≻ ) such that σ (Id D ⊗ ≺ ) = ( ≺ ⊗ Id D ) σ σ , σ ( ≺ ⊗ Id D ) = (Id D ⊗ ≺ ) σ σ , (8) σ (Id D ⊗ ≻ ) = ( ≻ ⊗ Id D ) σ σ , σ ( ≻ ⊗ Id D ) = (Id D ⊗ ≻ ) σ σ . (9)Since the map θ D : P D ( D ) → D is a homomorphism of braided vector spaces, we clearly have( θ D ⊗ θ D ) σ P D = σ ( θ D ⊗ θ D ) . NIT ACTION ON BRAIDED OPERADS AND HOPF ALGEBRAS 7
Then for any ( I , a ) ∈ P D , ⊗ D and ( ≺ , b , c ) ∈ P D , ⊗ D ⊗ , we have( θ D ⊗ θ D ) σ P D (( I , a ) ⊗ ( ≺ , b , c )) = ( θ D ⊗ θ D ) (cid:16)X ( ≺ , b ′ , c ′ ) ⊗ ( I , a ′ ) (cid:17) = X ( b ′ ≺ c ′ ) ⊗ a ′ , where we set T σβ ( a ⊗ b ⊗ c ) = σ σ ( a ⊗ b ⊗ c ) = P b ′ ⊗ c ′ ⊗ a ′ . On the other hand, σ ( θ D ⊗ θ D )(( I , a ) ⊗ ( ≺ , b , c )) = σ ( a ⊗ ( b ≺ c )) . Hence, σ ( a ⊗ ( b ≺ c )) = X ( b ′ ≺ c ′ ) ⊗ a ′ , that is, σ (Id D ⊗ ≺ ) = ( ≺ ⊗ Id D ) σ σ . The other conditions in Eq. (8), (9) can be verified similarly.In particular, the free P D -algebra over a braided vector space ( V , σ ) is the free braided den-driform algebra, realized as the braided analogue of the Loday-Ronco algebra of planar binaryrooted trees; see [9, Theorem 2.8]. Example 2.8.
A tridendriform algebra T is a vector space with three binary operators ≺ , ≻ , ∗ suchthat ( a ≺ b ) ≺ c = a ≺ ( b ≺ c + b ≻ c + b ∗ c ) , ( a ≻ b ) ≺ c = a ≻ ( b ≺ c ) , a ≻ ( b ≻ c ) = ( a ≺ b + a ≻ b + a ∗ b ) ≻ c , ( a ∗ b ) ≺ c = a ∗ ( b ≺ c ) , ( a ≺ b ) ∗ c = a ∗ ( b ≻ c ) , ( a ≻ b ) ∗ c = a ≻ ( b ∗ c ) , ( a ∗ b ) ∗ c = a ∗ ( b ∗ c ) , for a , b , c ∈ T . Intrinsically, any tridendriform algebra is an algebra over the nonsymmetric operad( P T , γ, I ) generated by P T , = k I , P T , = k ≺ ⊕ k ≻ ⊕ k ∗ , with relations γ ( ≺ ; ≺ , I ) = γ ( ≺ ; I , ≺ + ≻ + ∗ ) ,γ ( ≺ ; ≻ , I ) = γ ( ≻ ; I , ≺ ) ,γ ( ≻ ; I , ≻ ) = γ ( ≻ ; ≺ + ≻ + ∗ , I ) ,γ ( ≺ ; ∗ , I ) = γ ( ∗ ; I , ≺ ) ,γ ( ∗ ; ≺ , I ) = γ ( ∗ ; I , ≻ ) ,γ ( ∗ ; ≻ , I ) = γ ( ≻ ; I , ∗ ) ,γ ( ∗ ; ∗ , I ) = γ ( ∗ ; I , ∗ ) ,γ ( ≺ ; I , I ) = ≺ , γ ( ≻ ; I , I ) = ≻ , γ ( ∗ ; I , I ) = ∗ ,γ ( I ; ≺ ) = ≺ , γ ( I ; ≻ ) = ≻ , γ ( I ; ∗ ) = ∗ . A braided tridendriform algebra [9], denoted ( T , ≺ , ≻ , ∗ , σ ) is a braided vector space ( T , σ )endowed with a tridendriform algebra structure ( ≺ , ≻ , ∗ ) such that σ (Id T ⊗ ≺ ) = ( ≺ ⊗ Id T ) σ σ , σ ( ≺ ⊗ Id T ) = (Id T ⊗ ≺ ) σ σ , (10) σ (Id T ⊗ ≻ ) = ( ≻ ⊗ Id T ) σ σ , σ ( ≻ ⊗ Id T ) = (Id T ⊗ ≻ ) σ σ , (11) σ (Id T ⊗ ∗ ) = ( ∗ ⊗ Id T ) σ σ , σ ( ∗ ⊗ Id T ) = (Id T ⊗ ∗ ) σ σ . (12) LI GUO AND YUNNAN LI
Then the braided tridendriform algebras are exactly the braided P T -algebras, as the conditionsin (10)–(12) can be recovered by the fact that the map θ T : P T ( T ) → T is a homomorphismof braided vector spaces. In particular, the free P T -algebra over a braided vector space ( V , σ ) isthe free braided tridendriform algebra, constructed as the braided analogue of the Loday-Roncoalgebra of planar rooted trees; see [9, Theorem 4.5].3. C oherent unit actions and braided P -H opf algebras Let P be any (braided) operad such that P (0) = P (1) = k = k I . Let I be a 0-ary elementadjoined to P by P ′ ( i ) : = P ( i ) , i ≥ , k I , i = . A unit action on the operad ( P , γ, I ) is a partial extension of the composition map γ on P ′ with γ ( i , . . . , i k ) : P ′ ( k ) ⊗ B k Ind B i + ··· + ik B i ×···× B ik P ′ ( i ) ⊗ · · · ⊗ P ′ ( i k ) → P ′ ( i + · · · + i k )defined for all i j ≥ j = , . . . , k and i + · · · + i k > k ≥
2. Note that γ ( i , · · · , i k ) is notdefined when i = · · · = i k = k ≥ γ (1 ,
0) : P ⊗ k I ⊗ k I → k I and γ (0 ,
1) : P ⊗ k I ⊗ k I → k I in thenonsymmetric case, there exist linear maps α, β : P → k such that(13) γ ( µ ; I , I ) = α ( µ ) I , γ ( µ ; I , I ) = β ( µ ) I for any µ ∈ P . Given a P -algebra A , we define A + : = k ⊕ A , called a unitary P -algebra, with structure map θ A + : P ′ ( A + ) → A + extending θ A by sending I ∈ P ′ (0) to 1 ∈ A + . Definition 3.1.
For any (braided) nonsymmetric operad ( P , γ, I ) generated by operation sets M k ⊆P k , k ≥
2, with a unit action, suppose that there are operations ⋆ n ∈ P n for all n ≥ γ ( ⋆ n ; I , . . . , i th I , . . . , I ) = ⋆ n − , ≤ i ≤ n , (14) γ ( ⋆ ; I , I ) = γ ( ⋆ ; I , I ) = I , (15)particularly ⋆ = I , ⋆ = I and α ( ⋆ ) = β ( ⋆ ) =
1. Then one can further extend γ by requiring γ ( ⋆ n ; n times z }| { I , . . . , I ) = I . In this case, such a unit action is called coherent , if for any two P -algebras A and B , A ⊠ B : = ( A ⊗ k ) ⊕ ( k ⊗ B ) ⊕ ( A ⊗ B )is again a P -algebra defined as follows. Let c be the braiding in the braided tensor category C ;see [15, Ch. XIII]. We let σ : = σ A , B : = c A + ⊕ B + , A + ⊕ B + denote the corresponding braiding on A + ⊕ B + when A , B ∈ C . For any p ∈ M n , a i ∈ A + and b i ∈ B + , ≤ i ≤ n with n ≥
2, let(16) p ( a ⊗ b , . . . , a n ⊗ b n ) : = P ⋆ n ( a ′ , . . . , a ′ n ) ⊗ p ( b ′ , . . . , b ′ n ) , if at least one b i ∈ B , p ( a , . . . , a n ) ⊗ , if all b i = , p ( a , . . . , a n ) is defined , undefined , otherwise , NIT ACTION ON BRAIDED OPERADS AND HOPF ALGEBRAS 9 where we denote T σ w n ( a ⊗ b ⊗ · · · ⊗ a n ⊗ b n ) = P a ′ ⊗ · · · ⊗ a ′ n ⊗ b ′ ⊗ · · · ⊗ b ′ n with w n ∈ S n suchthat w n (2 i − = i , w n (2 i ) = n + i for i = , . . . , n . Then Eq. (16) actually defines the structuremap θ A ⊠ B of A ⊠ B as a P -algebra. Lemma 3.2.
For a unit action on a braided operad ( P , γ, I ) , let ⋆ : = I , ⋆ : = I, and ⋆ ∈ P satisfying Eq. (15) . If ⋆ is associative, i.e. γ ( ⋆ ; ⋆ , I ) = γ ( ⋆ ; I , ⋆ ) , then one can recursivelydefine a sequence of operations ⋆ n , n ≥ , by ⋆ n : = γ ( ⋆ ; ⋆ n − , I ) ∈ P n , satisfying Eq. (14) . Moreover, (17) γ ( ⋆ ; ⋆ i , ⋆ n − i ) = ⋆ n , for all i = , , . . . , n with n ≥ . Then we call an operad ( P , γ, I ) associative , if it has such anassociative operation ⋆ : = ⋆ ∈ P .Proof. First we inductively prove that ⋆ n , n ≥
0, satisfy Eq. (14). It is clear when n = ,
2. If n >
2, then by the definition of ⋆ n and the induction hypothesis we have γ ( ⋆ n ; I , . . . , i th I , . . . , I ) = γ ( γ ( ⋆ ; ⋆ n − , I ); I , . . . , i th I , . . . , I ) = γ ( ⋆ ; γ ( ⋆ n − ; I , . . . , I , . . . , I ) , I ) , ≤ i < n ,γ ( ⋆ ; γ ( ⋆ n − ; I , . . . , I ) , I ) , i = n , = γ ( ⋆ ; ⋆ n − , I ) , ≤ i < n ,γ ( ⋆ ; ⋆ n − , I ) , i = n , = γ ( ⋆ ; ⋆ n − , I ) , ≤ i < n ,γ ( γ ( ⋆ ; I , I ); ⋆ n − ) , i = n , = ⋆ n − . Further, Eq. (17) can also be proved by induction on n . Indeed, it is obviously true when n = , ,
2. If n >
2, then γ ( ⋆ ; ⋆ i , ⋆ n − i ) = γ ( ⋆ ; γ ( ⋆ ; ⋆ i − , I ) , ⋆ n − i ) = γ ( γ ( ⋆ ; ⋆ , I ); ⋆ i − , I , ⋆ n − i ) = γ ( γ ( ⋆ ; I , ⋆ ); ⋆ i − , I , ⋆ n − i ) = γ ( ⋆ ; ⋆ i − , γ ( ⋆ ; I , ⋆ n − i )) = γ ( ⋆ ; ⋆ i − , ⋆ n + − i ) , for any i = , . . . , n −
1. It then follows that ⋆ n = γ ( ⋆ ; ⋆ n , I ) = γ ( ⋆ ; ⋆ n − , I ) = · · · = γ ( ⋆ ; I , ⋆ n ) . Hence, Eq. (17) holds for any n ≥ (cid:3) Example 3.3.
Let P D be the braided dendriform operad given in Example 2.7. There is a unitaction on P D defined as follows. First extend P D to be P ′D such that P ′D (0) = k I , P ′D , = k I , P ′D , = k ≺ ⊕ k ≻ , and then extend γ to P ′D such that γ ( ≺ ; I , I ) = I , γ ( ≺ ; I , I ) = , γ ( ≻ ; I , I ) = , γ ( ≻ ; I , I ) = I . Let ⋆ : = I , ⋆ : = I , ⋆ : = ≺ + ≻ and ⋆ n : = γ ( ⋆ ; ⋆ n − , I ) inductively for any n ≥
3. Then ⋆ clearly satisfies Eq. (15). By the relations of P ′D , we know that ⋆ is also associative. Accordingto Lemma 3.2, the sequence of operations { ⋆ n } n ≥ satisfies Eqs. (14) and (17).Now for the braided dendriform operad case, we further have the following property. Proposition 3.4.
The stated unit action on P D generated by M = {≺ , ≻} is coherent.Proof. It is enough to check that the operations ≺ and ≻ on A ⊠ B defined by Eq. (16) satisfythe dendriform relations and the compatibility conditions (8), (9) for any P D -algebras A , B . Theproof is the same as the case of the algebraic operads proved in [9, Proposition 4.9] to which werefer the reader for details. (cid:3) Let P be a (braided) nonsymmetric operad equipped with a coherent unit action and A + : = k ⊕ A be a unitary P -algebra. Do the same for A + ⊗ A + (cid:27) ( A ⊠ A ) + via Eq. (16). Let ∆ : A + → ( A ⊠ A ) + (cid:27) A + ⊗ A + be a coassociative linear map, such that ∆ (1) = ⊗ ∆ ′ ( a ) : = ∆ A ( a ) − a ⊗ − ⊗ a ∈ A ⊗ A for any a ∈ A .If ∆ is a morphism of unitary P -algebras, we call A + together with ∆ a P -bialgebra . Moreover,if there is a direct sum A + = L n ≥ A ( n ) of vector spaces, such that A (0) = k and ∆ ( A ( n ) ) ⊆ n X i = A ( i ) ⊗ A ( n − i ) for all n ≥
0, we call A + a connected graded P -Hopf algebra .As a braided analogue of [11, Lemma 6], we provide the following theorem. Theorem 3.5.
Let P be a braided nonsymmetric operad equipped with a coherent unit action andlet A + : = k ⊕ A with A : = P ( V ) be the free unitary P -algebra generated by a braided vector spaceV. Then there is a coassociative morphism ∆ A : A + → A + ⊗ A + of P -algebras defined by ∆ A ( x ) = x ⊗ + ⊗ x , x ∈ V , which provides A + with a connected graded P -bialgebra algebra structure. It further providedA + with a connected braided bialgebra structure and hence a Hopf algebra structure.Furthermore, if the Yang-Baxter operator σ A on A + becomes a P -algebra isomorphism, then ∆ A is twisted cocommutative, i.e. σ A ∆ A = ∆ A .Proof. Since P ( V ) is the free P -algebra on V and P ( V ) ⊠ n , n = , , are P -algebras by the coherentunit action, there is a unique P -algebra morphism ∆ A : P ( V ) → P ( V ) ⊠ , v ⊗ v + v ⊗ , v ∈ V , identifying V with P ⊗ V . It then induces an braided (associative) algebra homomorphism ∆ :( P ( V ) , ⋆ ) → ( P ( V ) ⊠ , ⋆ ) . By the same argument, we obtain the compatibility requirement ∆ A θ A = θ A ⊠ A P ( ∆ A ) , where θ A = γ V and θ A ⊠ A are defined in Eq. (16). Then it also guarantees the commutativitybetween ∆ A and the braidings, ( ∆ A ⊗ ∆ A ) σ A = β ( ∆ A ⊗ ∆ A ) . On the other hand, since( ∆ A ⊗ id A + ) ∆ A ( x ) = x ⊗ ⊗ + ⊗ x ⊗ + ⊗ ⊗ x = (id A + ⊗ ∆ A ) ∆ A ( x ) NIT ACTION ON BRAIDED OPERADS AND HOPF ALGEBRAS 11 for any x ∈ V , by the universal property of P ( V ) again, ∆ A as a morphism of P -algebras iscoassociative. Also, ∆ A (1) = ⊗ ∆ ′ A ( a ) : = ∆ A ( a ) − a ⊗ − ⊗ a ∈ A ⊗ A for any a ∈ A .Furthermore, the free unitary P -algebra A + allows a grading A ( n ) : = P n ⊗ V ⊗ n which is respectedby all P -operations. The same holds for A + ⊗ A + . Thus ∆ A makes A + into a connected graded P -bialgebra algebra. Thus ( A + , ⋆, ∆ ) is a connected graded braided bialgebra and hence a braidedHopf algebra.Finally, the stated cocommutativity of ∆ a follows from its definition. (cid:3) Example 3.6.
For the braided dendriform operad, by Proposition 3.4 and Theorem 3.5, we knowthat the free unitary P D -algebra over a braided vector space is a connected graded P D -Hopfalgebra. In particular, we recover the braided Hopf algebra structures of free braided dendriformalgebras, namely, the braided analogue of the Loday-Ronco Hopf algebra of planar binary rootedtrees, directly verified in [9]. Proposition 3.7.
The conclusion of Theorem 3.5 holds when the associativity condition statedin Lemma 3.2 is replaced by a completely commutativity condition for the braided operad ( P , γ, I ) , in the sense that there are operations ⋆ n ∈ P ( n ) , n ≥ , invariant under the right B n -module action and satisfying Eqs. (14) and (15) .Proof. Under this condition, P -operations on A ⊠ B are still well-defined via Eq. (16), thus againendow A ⊠ B with a P -algebra structure given by the same θ A ⊠ B from P ( A ⊠ B ) = L n ≥ P ( n ) ⊗ B n ( A ⊠ B ) ⊗ n to A ⊠ B in Eq. (16). In short, there is again a coherent unit action on P . ThenProposition 3.7 follows from the same argument. (cid:3) Example 3.8.
Typical examples of completely commutative (braided) operads with coherent unitactions are the (braided) commutative dendriform (identified with Zinbiel) operad and the braidedcommutative tridendriform (CTD) operad. Thus Proposition 3.7 in particular implies the braidedHopf algebra structures on free braided commutative (tri)dendriform algebras. Free braided com-mutative dendriform algebras are shown [8] to be isomorphism to the quantum shu ffl e algebrasdefined in [22] for involutive braidings. Thus the coherent unit action approach of braided Hopfalgebras in Proposition 3.7 puts the braided Hopf algebra structure on quantum shu ffl e algebras ina broader context. A similar approach can be given to the quantum quasi-shu ffl e algebra in [14].4. C lassification of braided binary quadratic nonsymmetric operads with coherent unitactions In [3] the authors classified all the associative, binary, quadratic and nonsymmetric (ABQR)operads with coherent unit actions. In this section we extend this classification to the braidedcontext.
Theorem 4.1.
Let ( P , γ, I , ⋆ ) be a braided ABQR operad. A unit action on P is coherent if andonly if, for every ( P i ⊙ (1) i ⊗ ⊙ (2) i , P j ⊙ (3) j ⊗ ⊙ (4) j ) ∈ P ⊗ ⊕ P ⊗ from the quadratic relations of P , thefollowing coherence equations hold in terms of the linear maps α, β defined in Eq. (13) . (C1) P i β ( ⊙ (1) i ) ⊙ (2) i = P j β ( ⊙ (3) j ) ⊙ (4) j , (C2) P i α ( ⊙ (1) i ) ⊙ (2) i = P j β ( ⊙ (4) j ) ⊙ (3) j , (C3) P i α ( ⊙ (2) i ) ⊙ (1) i = P j α ( ⊙ (4) j ) ⊙ (3) j , (C4) P i β ( ⊙ (2) i ) ⊙ (1) i = P j β ( ⊙ (3) j ) β ( ⊙ (4) j ) ⋆, (C5) P i α ( ⊙ (1) i ) α ( ⊙ (2) i ) ⋆ = P j α ( ⊙ (3) j ) ⊙ (4) j . Proof.
A braided ABQR operad P is generated by P = k I , P = k ω with relations X i γ ( ⊙ (1) i ; ⊙ (2) i , I ) = X j γ ( ⊙ (3) j ; I , ⊙ (4) j ) , where ( P i ⊙ (1) i ⊗ ⊙ (2) i , P j ⊙ (3) j ⊗ ⊙ (4) j ) ∈ P ⊗ ⊕ P ⊗ .A unit action on P is coherent if and only if, for any P -algebra A and B and for any a , a ′ , a ′′ ∈ A + and b , b ′ , b ′′ ∈ B + , we have the equation(18) X i (cid:16) ( a ⊗ b ) ⊙ (1) i ( a ′ ⊗ b ′ ) (cid:17) ⊙ (2) i ( a ′′ ⊗ b ′′ ) = X j ( a ⊗ b ) ⊙ (3) j (cid:16) ( a ′ ⊗ b ′ ) ⊙ (4) j ( a ′′ ⊗ b ′′ ) (cid:17) . Then there are eight mutually disjoint cases for eight subsets of b , b ′ , b ′′ which are in k (and hencenot in B ). Note that when all of b , b ′ , b ′′ are in k or all of b , b ′ , b ′′ are in B , Eq. (18) just means that( P i ⊙ (1) i ⊗ ⊙ (2) i , P j ⊙ (3) j ⊗ ⊙ (4) j ) is a relation for P , so is automatic true. Thus to prove the theoremwe only need to proveCase 1. Eq. (18) holds for b ∈ k , b ′ , b ′′ ∈ B if and only if (C1) is true;Case 2. Eq. (18) holds for b ′ ∈ k , b , b ′′ ∈ B if and only if (C2) is true;Case 3. Eq. (18) holds for b ′′ ∈ k , b , b ′ ∈ B if and only if (C3) is true;Case 4. Eq. (18) holds for b , b ′ ∈ k , b ′′ ∈ B if and only if (C4) is true;Case 5. Eq. (18) holds for b ′ , b ′′ ∈ k , b ∈ B if and only if (C5) is true;Case 6. Eq. (18) holds for b , b ′′ ∈ k , b ′ ∈ B if (C1) is true.All these cases are routinely checked as in the original paper [3] using binary operations on A ⊠ B defined in Eq. (16). (cid:3) We now apply Theorem 4.1 to classify all braided ABQR operads with coherent unit actions.
Theorem 4.2.
Let ( P , γ, I , ⋆ ) be a braided ABQR operad of dimension n (that is, dim P = n). (i) There is a coherent unit action ( α, β ) on P with α , β if and only if there is a basis {⊙ i } of P with ⋆ = P i ⊙ i such that the space Λ of quadratic relations for P are contained inthe subspace Λ ′ n , coh of P ⊗ ⊕ P ⊗ with the basis (19) λ ′ n , coh : = ( ⋆ ⊗ ⊙ , ⊙ ⊗ ⊙ ) , ( ⊙ ⊗ ⊙ , ⊙ ⊗ ⋆ ) , ( ⊙ i ⊗ ⊙ , ⊙ i ⊗ ⊙ ) , ≤ i ≤ n , ( ⊙ ⊗ ⊙ j , ⊙ ⊗ ⊙ j ) , ≤ j ≤ n , ( ⊙ ⊗ ⊙ i , ⊙ i ⊗ ⊙ ) , ≤ i ≤ n , ( ⊙ i ⊗ ⊙ j , , (0 , ⊙ i ⊗ ⊙ j ) , ≤ i , j ≤ n . (ii) There is a coherent unit action ( α, β ) on P with α = β if and only if there is a basis {⊙ i } of P with ⋆ = P i ⊙ i such that the space Λ of quadratic relations for P are contained inthe subspace Λ ′′ n , coh of P ⊗ ⊕ P ⊗ with the basis (20) λ ′′ n , coh : = ( ( ⊙ ⊗ ⋆, ⊙ ⊗ ⋆ ) + ( ⋆ ⊗ ⊙ , ⋆ ⊗ ⊙ ) − ( ⊙ ⊗ ⊙ , ⊙ ⊗ ⊙ ) , ( ⊙ i ⊗ ⊙ j , , (0 , ⊙ i ⊗ ⊙ j ) , ≤ i , j ≤ n . ) . Proof.
The proof is the same as the one for the algebraic operads in [3, Theorem 4.9], so we referthe reader there for details. (cid:3)
NIT ACTION ON BRAIDED OPERADS AND HOPF ALGEBRAS 13
By inspection, the braided dendriform algebra (see Proposition 3.4) and braided tridendriformalgebras have coherent unit actions.
Acknowledgments.
We thank the organizers for the hospitality during the International Work-shop on Hopf Algebras and Tensor Categories held in Nanjing University, September 2019. Thiswork is supported by Natural Science Foundation of China (Grant Nos. 11501214, 11771142,11771190). R eferences
1. M. Aguiar and J.-L. Loday, Quadri-algebras,
J. Pure Appl. Algebra (2004), 205-221. 12. A. Connes and D. Kreimer, Hopf algebras, renormalization, and noncommutative geometry,
Comm. Math. Phys. (1998), 203–242. 13. K. Ebrahimi-Fard and L. Guo, Coherent unit actions on operads and Hopf algebras,
Theory Appl. Categ. (2007), 348–371. 2, 11, 124. Z. Fiedorowicz, The symmetric bar construction, Preprint, , 1992. 2, 3, 55. L. Foissy, Les alg´ebres de Hopf des arbres enracin´es d´ecor´es. I, II, Bull. Sci. Math. (2002), 193–239, 249–288. 16. L. Foissy, Quantization of the Hopf algebras of decorated planar rooted trees, preprint, 2008. 27. B. Fresse, Homotopy of operads and Grothendieck-Teichm¨uller groups. Part 1: The algebraic theory and its topo-logical background, Mathematical Surveys and Monographs , American Mathematical Society, Providence,RI (2017). 2, 3, 58. L. Guo and Y. Li, Braided Rota-Baxter algebras, quantum quasi-shu ffl e algebras and braided dendriform alge-bras, arXiv:1901.02843. 2, 6, 119. L. Guo and Y. Li, Braided dendriform and tridendriform algebras and braided Hopf algebras of planar trees,arXiv:1906.06454, published online at J. Algebraic Combin., https: // doi.org / / s10801-020-00957-0 2, 7,8, 10, 1110. R. Holtkamp, Comparison of Hopf algebras on trees, Archiv der Mathematik , Vol. , (2003), 368-383. 111. R. Holtkamp, On Hopf algebra structures over operads, Adv. Math. (2006), 544–565. 2, 1012. R.-Q. Jian, Quantum quasi-shu ffl e algebras II, J. Algebra (2017), 480–506. 213. R.-Q. Jian and M. Rosso, Braided cofree Hopf algebras and quantum multi-brace algebras,
J. Reine Angew.Math. (2012) 193–220. 214. R.-Q. Jian, M. Rosso and J. Zhang, Quantum quasi-shu ffl e algebras, Lett. Math. Phys. (2010), 1–16. 2, 1115. C. Kassel, Quantum Groups, Graduate Texts in Mathematics , Springer-Verlag, New York, 1995. 2, 816. J.-L. Loday, Dialgebras, in “Dialgebras and related operads”,
Lecture Notes in Math. , (2002), 7-66. 117. J.-L. Loday, Scindement d’associativit´e et alg`ebres de Hopf, Actes des Journ´ees Math´ematiques `a la M´emoirede Jean Leray, S´emin. Congr., (2004), 155–172. 1, 218. J.-L. Loday, On the algebra of quasi-shu ffl es, Manuscripta Mathematica , (2007), 79–93. 119. J.-L. Loday and M. O. Ronco, Hopf algebra of the planar binary trees, Adv. Math. , , (1998), 293-309. 120. J.-L. Loday and M. Ronco, Trialgebras and families of polytopes, in “Homotopy Theory: Relations with Alge-braic Geometry, Group Cohomology, and Algebraic K-theory” Contemporary Mathematics (2004), 369–398. 121. J.-L. Loday and B. Vallette, Algebraic Operads, Gr¨undlehren der mathematischen Wissenschaften, , Springer,Heidelberg, 2012. 322. M. Rosso, Quantum groups and quantum shu ffl es, Invent. Math. (1998), 399–416. 2, 11D epartment of M athematics and C omputer S cience , R utgers U niversity , N ewark , NJ 07102, USA E-mail address : [email protected] S chool of M athematics and I nformation S cience , G uangzhou U niversity , W aihuan R oad W est uangzhou hina E-mail address ::