L-algebras, triplicial-algebras, within an equivalence of categories motivated by graphs
aa r X i v : . [ m a t h - ph ] A p r L-algebras, triplicial-algebras,within an equivalence of categories motivatedby graphs Philippe
Leroux
Abstract:
In a previous work, we gave a coalgebraic framework of directed graphsequipped with weights (or probability vectors) in terms of (Markov) L-coalgebras. Theyare K -vector spaces equipped with two co-operations, ∆ M , ˜∆ M verifying,( ˜∆ M ⊗ id )∆ M = ( id ⊗ ∆ M ) ˜∆ M . In this paper, we study the category of L-algebras (dual of L-coalgebras), prove that thefree L-algebra on one generator is constructed over rooted planar symmetric ternary treeswith odd numbers of nodes and the L-operad is Koszul. We then introduce triplicial-algebras: vector spaces equipped with three associative operations verifying three entan-glement relations. The free triplicial-algebra is computed and turns out to be relatedto even trees. Via a general structure theorem (`a la Cartier-Milnor-Moore) proved inSection 4, the category of L-algebras turns out to be equivalent to a much more struc-tured category called connected coassociative triplicial-bialgebras (coproduct linked tooperations via infinitesimal relations), that is the triple of operads (
As, T rip,
L) is good.Bidirected graphs, related to
N AP -algebras (L-commutative algebras), are briefly evokedand postponed to another paper.
In the sequel, K will be a characteristic zero field and its unit will always be denoted by1 K . The symmetric group over n elements is denoted by S n and if P denotes a regular As c -L-bialgebras, triplicial-algebras, As c − T rip -bialgebras, Structuretheorems (Cartier-Milnor-Moore), Good triples, L-commutative algebras,
N AP -algebras.
Email : ph ler [email protected];
Mail : 27, Rue Roux Soignat 69003 Lyon, France. P ( n ) = P n ⊗ KS n , the K -vector space of the n -ary operations of P , see for instance [20] for notation and basic definitions in operad theory. The symbol ◦ stands for the composition of maps and the notation v . . . v n for v ⊗ . . . ⊗ v n with v i ∈ V ,where V is a K -vector space.In [14], we introduced a coalgebraic framework to code any weighted directed graphswhich are row and locally finite. This coding leads to the so-called L-coalgebras setting.Recall that a L -coalgebra ( L, ∆ , ˜∆) is a K -vector space equipped with a right co-operation∆ : L −→ L ⊗ and a left co-operation ˜∆ : L −→ L ⊗ , verifying what we call now the entanglement relation : ( ˜∆ ⊗ id )∆ = ( id ⊗ ∆) ˜∆ . A L -coalgebra may have two partial counits. The right counit ǫ : L −→ K verifying( id ⊗ ǫ )∆ = id and the left counit ˜ ǫ : L −→ K verifying, (˜ ǫ ⊗ id ) ˜∆ = id. It has been provedin [14] that directed graphs having no source and sink but weighted by probability vectorsyield (Markov) L-coalgebras with same counits. The co-operation ∆ M codes the future ofa given vertex: ∆ M (Present) := Present ⊗ Future , and the co-operation ˜∆ codes its past:˜∆ M (Present) := Past ⊗ Present . The entanglement relation means that Past, Present andFuture are related together as expected,(
P ast ⊗ P resent ) ⊗ F uture = P ast ⊗ ( P resent ⊗ F uture ) . Let ( A, · ) be a unital associative algebra. Two convolutions products can be definedover Hom K ( KG , A ), where G is the set of vertices of a given graph: f ≺ g := · ( f ⊗ g )∆ and f ≻ g := · ( f ⊗ g ) ˜∆ . We get for any maps f, g, h ∈ Hom K ( KG , A ), what willbe also called an entanglement relation:( f ≻ g ) ≺ h = f ≻ ( g ≺ h ) . The K -vector space Hom K ( KG , A ) equipped with these two operations turns out tobe a so-called L-algebra. Set η : K ֒ → A , 1 K A . It has a “unit” 1 := η ◦ ǫ = η ◦ ˜ ǫ , verifying: f ≺ f = 1 ≻ f, if left and right counits are supposed to beequal. From [14], it has also been proved that bidirected graphs yield to the so-calledL-cocommutative coalgebras, that is, K -vector spaces equipped with two co-operationsverifying the entanglement relations and the following extra condition: ∆ = τ ◦ ˜∆ , where τ is the usual flip map. The entanglement relation becomes,( id ⊗ τ )(∆ M ⊗ id )∆ M = (∆ M ⊗ id )∆ M . Such coalgebras have also been found by M. Livernet [18] under the name
N AP -coalgebras.The case of bidirected graphs is postponned to another paper.In this paper, we propose a study of the category of L-algebras. In Section 2, weexplicit the dual of the L-operad and find the free L-algebra over a given vector space V thanks to rooted planar symmetric ternary trees with odd numbers of nodes coded bywords. We prove the existence of an involution on the free L-algebra over V and computethe free L-monoid over a given set. We also propose a new coding for rooted planar2inary trees. The L-operad happens to be Koszul, hence generating functions of the L-operad and its dual are inverse one another for the composition of functions. This givesan algebraic interpretation of the sequence A n operations [ n ] − M ag equipped with a coassociative coproduct linked tooperations via nonunital infinitesimal relations. We show that entanglement equationsyield primitive relations. Dividing out by such relations yields many good triples, one ofthem being the second main results of this paper. Indeed, using this theorem, we provein Section 5 that the triple of operad (
As, T rip,
L) is good, where the operad
T rip isassociated with triplicial-algebras,
T rip -algebras for short. The free
T rip -algebra over V is computed and turns out to be related to even trees. We then obtain that the categoryof L-algebras is equivalent to the category of connected coassociative triplicial-bialgebras.In case of walks over a graph, operations of L-algebras are nonassociative and just codethe past or the future of the walk. Requiring that Past, Present and Future are orderedleads to a much more structured objects which are the coassociative triplicial-bialgebras!We explicit relations between these two objects via idempotents (Subsection 4.2.) and theuniversal envelopping functor U (Subsection 5.3.). Reversing time leads to reverse walkover a graph. This is coded through our objects by an involution. The free commutative T rip -algebra over a K -vector space is also given. It turns out to be related to commutativealgebras and permutative algebras. Section 6 shows that the L-operad cannot be ananticyclic operad although [ n ] − M ag is as soon as n >
1. We present here a picture ofour main results applied in the particular case of weighted directed graphs.
Coassociative triplicial bialgebras:3 associative operations + 3 relations;1 coassociative coproduct; (Markov) L−algebras2 (nonassociative) operations1 relation coding Past, Present, Future
IdempotentsUniversal envelopingfunctor: Utime reversal
Involution Involution Free object: Even trees Free object: Symmetric rooted planar of nodes.
Weighted directed graphs
Walks
Infinitesimal relations
Functor: Prim ternary trees with odd number The free L-algebra and its dual
Definition 2.1
A L-algebra is a K -vector space L equipped with two binary operations ≺ , ≻ : L ⊗ → L verifying the so-called entanglement relation:( x ≻ y ) ≺ z = x ≻ ( y ≺ z ) , for all x, y, z ∈ L . A L-algebra is said to be involutive if it exists an involution † : L → L such that ( x ≺ y ) † = y † ≻ x † and ( x ≻ y ) † = y † ≺ x † . The opposite of a L-algebra L isthe K -vector space L equipped with the operations: x ≺ op y := y ≻ x ; x ≻ op := y ≺ x, for all x, y ∈ L . A L-algebra is said to be commutative if it coincides with its oppo-site. Therefore, a commutative L-algebra is a K -vector space equipped with one binaryoperation ≺ verifying: ( x ≺ y ) ≺ z = ( x ≺ z ) ≺ y. As mentioned in the introduction, bidirected graphs lead to such structures. They havebeen also introduced independently by M. Livernet [18] under the name
N AP -algebras.
Example 2.2
As seen in [14], L-algebras arise from coding weighted directed graphs.But numerous types of algebras are in fact L-algebras. Associative algebras (the twooperations coincide with the associative product), magmatic algebras [12] (take the secondoperation to be zero), dendriform algebras and dialgebras [21], quadri-algebras [1], ennea-algebras [15] and all the types of algebras coming from [17]. In [14], associative L-algebrashave been considered. These are L-algebras whose two operations are associative. Suchstuctures appear in the previous works of A. Brouder and A. Frabetti [3] and J.-L. Lodayand M. Ronco [24]. They have been renamed in [19] as duplicial-algebras. We note alsothat L-algebras appear in [6] without citations to our previous works.L-algebras give birth to the category
L-alg and the so called L-operad which is binary,quadratic and regular. Consequently, it admits a dual, in the sense of V. Ginzburg andM. Kapranov [11], the so called L ! -operad. One can check that L ! -algebras are defined asfollows. Definition 2.3
A L ! -algebra is a K -vector space L ′ equipped with two binary operations ⊣ , ⊢ : L ′⊗ → L ′ such that: ( x ⊢ y ) ⊣ z = x ⊢ ( y ⊣ z ) , ( x ⊢ y ) ⊢ z = 0 , x ⊣ ( y ⊣ z ) = 0 , ( x ⊣ y ) ⊣ z = 0 , x ⊢ ( y ⊢ z ) = 0 , ( x ⊣ y ) ⊢ z = 0 , x ⊣ ( y ⊢ z ) = 0 , hold for all x, y, z ∈ L ′ . 4his category is denoted by L ! -alg . Observe that both ⊣ and ⊢ are associative and anylinear combinations of these two operations as well. If As , Dend , Dias , Dup denote respec-tively the category of associative algebras, dendriform algebras, dialgebras [21], duplicial-algebras [19], then we get the following canonical functors: L ! -alg → As , L ! -alg → Dend , L ! -alg → Dias and L ! -alg → Dup . ! -algebra Theorem 2.4
Let V be a K -vector space. Let Ψ : ( V ⊕ K ) ⊗ → K be the canonicalprojection. The free L ! -algebra over V is the K -vector space, L ! ( V ) := ( V ⊕ K ) ⊗ V ⊗ ( K ⊕ V ) , equipped with the following operations: v ⊗ v ⊗ v ⊢ v ′ ⊗ v ′ ⊗ v ′ := Ψ( v ⊗ v ⊗ v ′ ) v ⊗ v ′ ⊗ v ′ ,v ⊗ v ⊗ v ⊣ v ′ ⊗ v ′ ⊗ v ′ := Ψ( v ⊗ v ′ ⊗ v ′ ) v ⊗ v ⊗ v ′ . Moreover the generating function of the L ! -operad is: f L ! ( x ) = x + 2 x + x . Proof:
Showing that L ! ( V ) is a L ! -algebra is left to the reader. We use the embedding i : V → K ⊗ V ⊗ K , v ⊗ v ⊗
1. Let A be a L ! -algebra and f : V → A be a linearmap. We construct its extension Φ : L ! ( V ) → A as follows:Φ( v ⊗ v ⊗ v ) := f ( v ) ⊢ f ( v ) ⊣ f ( v ) , with the convention that f ( v ) ⊢ (resp. ⊣ f ( v )) disappears if v = 1 or v = 1. That isΦ( v ⊗ v ⊗
1) := f ( v ) ⊢ f ( v ), Φ(1 ⊗ v ⊗ v ) := f ( v ) ⊣ f ( v ) and Φ(1 ⊗ v ⊗
1) := f ( v ).The map Φ is a L ! -algebra morphism. Indeed, on the one hand, Φ( v ⊗ v ⊗ v ⊢ v ′ ⊗ v ′ ⊗ v ′ ) = Ψ( v ⊗ v ⊗ v ′ ) f ( v ) ⊗ f ( v ′ ) ⊗ f ( v ′ ). On the other hand, Φ( v ⊗ v ⊗ v ) ⊢ Φ( v ′ ⊗ v ′ ⊗ v ′ ) = [ f ( v ) ⊢ f ( v ) ⊣ f ( v )] ⊢ [ f ( v ′ ) ⊢ f ( v ′ ) ⊣ f ( v ′ )] . If v ′ = 1, then we get( . . . ) ⊢ [ f ( v ′ ) ⊢ ( . . . )] which vanishes. The case v = 1 gives the same result. Suppose now v = v ′ = 1 and v = 1, we get [ f ( v ) ⊣ f ( v )] ⊢ [ f ( v ′ ) ⊣ f ( v ′ )] which again vanishes. Ifnow v = v ′ = 1 = v , then we get f ( v ) ⊢ ( f ( v ′ ) ⊣ f ( v ′ )) showing that Φ is a morphismfor the ⊢ operation. The same computation for the other operation shows that Φ is aL ! -algebra morphism which obey Φ ◦ i = f . Hence the unicity of Φ since such a morphismhas to coincide on V with f . (cid:3) Before entering the description of the free L-algebra over a K -vector space V , we needto introduce a combinatorial object, the so-called planar rooted ternary symmetric trees5ith odd degrees. These rooted trees have internal vertices (or children) with one input,three outputs and have a reflexive symmetry around the axis passing through the rootand its middle child. Let W n , n >
0, be the set of the so-called rooted ternary symmetrictrees with 2n-1 vertices. Here are W and W : symmetric axis WW
12: :
It has been proved by E. Deutsch, S. Feretic and M. Noy [8] that the cardinality of W n is n (cid:0) n − n − (cid:1) and is registered under the name A006013 in the On-Line Encyclopedia of IntegerSequences. For the purpose of the next subsection, we will use a coding of these trees.In [7], E. Deutsch asks for the cardinalities of the following sets: Let W n − , n >
0, bethe set of sequences of integers (called words) of length 2 n − ω := ω ω . . . ω n − ∈ W n − , we have,1. ω = 1;2. ω i >
0, for all i ∈ { , , , . . . , n − } ;3. ω i − ω i − ∈ { , − , − , − , − , . . . } , for all i ∈ { , , , . . . , n − } .For instance, W = { } , W = { , } and W = { , , , , , , } .It has been proved by D. Callan [7] that W n − has the same cardinality than W n (infact more can be proved if we include planar rooted symmetric ternary trees with evendegrees). To enumerate W n − , he introduces the following bijection which will be crucialin the next subsection. Bijection: from words to lattice paths.
In the sequel, the symbol ¯2 will stand for the integer −
2. We map a word ω of W n − intoa word l of length 3 n − l i ∈ { , ¯2 } for all i . The word l is built as follows:1. l = ω ;2. for 2 ≤ i ≤ n − l i = ω i − ω i − ;3. if l i <
0, then replace it by (1 − l i ) / l such that the sum of the entriesof l , P i l i , equals 1.We denote by L n the set of words coding trees of W n − . For instance, we get 1 N × N and map 1 into thevector (1; 1) and ¯2 into the vector (2; − n −
3; 1). Consequently, we get:
Proposition 2.5
A word l = l l . . . l n − ∈ L n if and only if for all ≤ k < n , P ki =1 l i ≥ and P ni =1 l i = 1 . There exists a very simple bijection between words l and the planar symmetric ternaryrooted trees with odd degrees found by M. Bousquet and C. Lamathe [2]. Set K L := L n> K L n . Let l ∈ L n and l ′ ∈ L m . Define two operations ≻ , ≺ on K L firstby l ≻ l ′ := 1 l ¯2 l ′ ,l ≺ l ′ := l l ′ ¯2 , then by bilinearity. Observe that our operations respect the canonical graduation of K L since, ≻ , ≺ : K L n ⊗ K L n K L n + m . Proposition 2.6
The K -vector space K L equipped with the two previous operations isa L-algebra generated by the word 1.Proof: This computation ( l ≻ l ′ ) ≺ l ′′ = 1 l ¯2 l ′ l ′′ ¯2 = 1 l ¯2( l ′ l ′′ ¯2) = l ≻ ( l ′ ≺ l ′′ ) showsthat K L is a L-algebra. By hand, one can check that L , L , L are generated by 1.Fix n >
0. Suppose this holds up to L n . Let l = 1 l l . . . l n +1 ∈ L n +1 . First of all,there exists an integer k ∈ { , . . . , n + 1 } such that l k = ¯2 and l ′ := l . . . l k − is aword. Indeed, if l n +1 = ¯2, then set l ′ := l . . . l n . If l n +1 = 1, then l n = ¯2 because ofProposition 2.5. Set l ′ := l . . . l n − to conclude. Let k be the smallest integer realizingthe previous assertion. Using Proposition 2.5, if k < n + 1, then there exists a word l ′′ such that l = l ′ ≻ l ′′ , otherwise l = 1 ≺ l ′′ . Therefore, by induction K L is generated bythe word 1 as a L-algebra. (cid:3) Theorem 2.7
The unique L-algebra map L ( K ) → ( K L , ≺ , ≻ ) sending the generator x of L ( K ) to the word 1 of K L is an isomorphism, i.e., ( K L , ≺ , ≻ ) is the free L-algebra onone generator. roof: Consider the map i : K L( K ), 1 K x and the map f : K → K L , 1 K
1. AsL( K ) is the free L-algebra on one generator, there exists a unique L-algebra morphism χ : L( K ) → K L such that χ ◦ i = f . For n = 1 , ,
3, one can check by hand that therestriction of χ to L n into K L n is an isomorphism. We suppose this result holds up to aninteger n −
1. As the L-operad is binary, any monomials of L n can be written as X ≺ Y or X ≻ Y , with X ∈ L k and Y ∈ L n − k for a k ∈ { , . . . , n − } . We get, χ ( X ≺ Y ) := χ ( X )1 χ ( Y )¯2 , χ ( X ≻ Y ) := 1 χ ( X )¯2 χ ( Y ) . Let X ′ ≺ Y ′ and X ≺ Y be two monomials of L n , with at least X = X ′ or Y = Y ′ .Suppose χ ( X ′ ≺ Y ′ ) = χ ( X ≺ Y ). Then χ ( X ′ )1 χ ( Y ′ )¯2 = χ ( X )1 χ ( Y )¯2. If χ ( X ′ ) and χ ( X ′ ) have the same length then by induction X = X ′ and thus Y = Y ′ which is notpossible by assumption. Otherwise one of them has a greater length. Suppose this is χ ( X ′ ). Then there exists u made of 1 and ¯2 such that, χ ( X ′ ) = χ ( X )1 u, and whose the sum of its entries is -1. Therefore, χ ( Y ) = uχ ( Y ′ ) which is impossiblebecause of Proposition 2.5. Hence, χ ( X ′ ≺ Y ′ ) = χ ( X ≺ Y ) does not hold with ourassumption. Suppose now χ ( X ′ ≻ Y ′ ) = χ ( X ≺ Y ) with at least X = X ′ or Y = Y ′ .Then, 1 χ ( X ′ )¯2 χ ( Y ′ ) = χ ( X )1 χ ( Y )¯2. If χ ( X ′ ) and χ ( X ′ ) have same length then the word χ ( Y ) starts with a ¯2, which is not possible. Suppose the length of χ ( X ′ ) is greater than χ ( X ). Then, set 1 χ ( X ′ ) = χ ( X )1 u, with the sum of the entries equals to 0. Hence, χ ( Y )¯2 = u ¯2 χ ( Y ′ ) which is impossiblesince the sum of the entries of u and ¯2 equals -2 and χ ( Y ) is a word. Suppose now thatthe length of χ ( X ) is greater than χ ( X ′ ). Set, χ ( X ) = 1 χ ( X ′ ) u, with the sum of the entries of u equals to -1. Therefore, u χ ( Y )¯2 = ¯2 χ ( Y ′ ). Hence, u starts with a ¯2. Hence, there exists an integer k , such that χ ( Y ′ ) := u u . . . u k χ ( Y )¯2.But P ki =2 u i = +1. Since χ ( Y ′ ) is a word, Proposition 2.5 claims that u u . . . u k is aword too. By induction, there exists a unique monomial Z of smaller degree such that χ ( Z ) := u u . . . u k . But χ ( X ≺ Y ) = χ ( X ′ ≻ Y ′ ) = χ ( X ′ ≻ Z ≺ Y ). Therefore, X ≺ Y = X ′ ≻ Y ′ = X ′ ≻ Z ≺ Y . Hence, for each n >
0, the restriction of χ to L n into K L n maps differents monomials into different words, so is injective. HoweverProposition 2.6 show that χ is surjective, so χ is an isomorphism. (cid:3) Remark:
In the sequel, we denote by ̟ the inverse of χ . Remark: [Involution and time reversal]
There exists a natural involution † over K L built by induction. As any word of K L n is generated uniquely by 1 one defines † asfollows. First of all 1 † = 1, ( l ≺ l ′ ) † = l ′† ≻ l † and ( l ≻ l ′ ) † = l ′† ≺ l † , by linearity then.This involution is important since reversing time when dealing with walks over a weighteddirected graph, viewed as a (Markov) L-algebra, can be coded through this involution.Because the operad L is regular, we get the following result.8 heorem 2.8 Let V be a K -vector space. Then, the K -vector space, M n> K L n ⊗ V ⊗ n , equipped with the following binary operations: ( l ⊗ v . . . v n ) ≺ ( l ′ ⊗ v . . . v n ′ ) = ( l ≺ l ′ ) ⊗ v . . . v n v . . . v n ′ , ( l ⊗ v . . . v n ) ≻ ( l ′ ⊗ v . . . v n ′ ) = ( l ≻ l ′ ) ⊗ v . . . v n v . . . v n ′ , is the free L-algebra over V . Otherwise stated, the unique L-algebra map L ( V ) → L n> K L n ⊗ V ⊗ n sending v ∈ V to ⊗ v is an isomorphism. Moreover, the generating function of theL-operad is, f L ( x ) = 43 sin ( 13 asin ( r x , = X n> n (cid:18) n − n − (cid:19) x n = x + 2 x + 7 x + 30 x + . . . . The next result is a consequence of the theory developed by V. Ginzburg and M. Kapranov[11].
Proposition 2.9
Let ( L, ≺ , ≻ ) be a L-algebra and ( M, ⊣ , ⊢ ) be a L ! -algebra. Then thebinary operation ∗ on M ⊗ L defined by ( x ⊗ a ) ∗ ( y ⊗ b ) := ( x ⊣ y ) ⊗ ( a ≺ b ) + ( x ⊢ y ) ⊗ ( a ≻ b ) , where x, y ∈ M and a, b ∈ L turns M ⊗ L into an associative algebra. Magmatic algebras consist of K -vector spaces equipped with one binary operation. Theyhave been investigated by R. Holkamp [12], see also [13]. The operad is called M ag . Ithas been proved that the free magmatic algebra on one generator,
M ag ( K ) is constructedover rooted planar binary trees and the grafting, denoted by ∨ , as operation. By Y n , wemean the set of rooted planar binary trees with n nodes. Recall card( Y n ) = c n , theCatalan numbers. In low dimensions, these sets are: Y := { } , Y := { Y := } , Y := { , } , Y := { , , , , } . For instance the grafting of by itself yields ∨ = . Therefore, as a K -vectorspace M ag ( K ) = L n ≥ KY n . Forgetting the operation ≻ , the space ( K L , ≺ ) can beseen as a magmatic algebra. Conversely, the space ( M ag ( K ) , ∨ ,
0) can be viewed as aL-algebra. Consequently, if we denote the canonical injections i : K → M ag ( K ); 1 K i ′ : K → K L ; 1 K
1, there exist a unique L-algebra morphism Φ and a uniquemagmatic morphism Ψ such that the following diagrams commute: iK → M ag ( K ) i ′ ց ↓ Ψ K L , i ′ K → K L i ց ↓ Φ M ag ( K ) . As Φ is also a magmatic morphism, we get Φ ◦ Ψ = id Mag ( K ) . Otherwise stated, rootedplanar binary trees can be coded in a unique way by rooted symmetric planar ternarytrees or simpler, by words made of 1 and ¯2. In low dimensions, we get:Ψ( ) = Ψ( ∨ ) = Ψ( ) ≺ Ψ( ) = 1 ≺ . Ψ( ) = Ψ( ∨ ) = Ψ( ) ≺ Ψ( ) = 111¯2 ≺ . Ψ( ) = Ψ( ∨ ) = Ψ( ) ≺ Ψ( ) = 111¯2 ≺ . Here is a way to code straigthforwardly a binary trees into a word of K L . Observe thatin a word of L n +1 , there are 2 n + 1 times 1 and n times ¯2 and in a binary tree of Y n , thereare n + 1 leaves and n nodes. Therefore, assign to each leaf or node of a binary tree t a1 and to each node a ¯2. Start with the node giving the most left leaf. This will give you(1) 1 (word) ¯2 (we put into parentheses the code of the left leaf and the right leaf). Nowgo a step below to meet another node. This will give you ((1) 1 (word) ¯2) 1 (word’) ¯2 andso on, because Ψ( t = t left ∨ t right ) = Ψ( t left )1Ψ( t right )¯2.For instance, Pursuing an idea of J.-L. Loday [23, 22, 16], we can propose an arithmetics from the L -operad . Indeed since the L -operad comes from a set operad, one can define over S n> L n (disjoint unions) two nonassociative gradded additions and a multiplication as follows: l + ≻ l ′ := 1 l ¯2 l ′ , l + ≺ l ′ := l l ′ ¯2 , l ⋉ l ′ := ̟ ( l ) ← l ′ , In [16], such an arithmetics over the free associative L -algebra (now called duplicial-algebra) has beenstudied in relations with dendriform algebras. ̟ ( l ) ← l ′ means that the word 1 has to be replaced by l ′ in ̟ ( l ). For instanceif l := 1 ≺
1, then l ⋉ l ′ = l ′ ≺ l ′ . As expected + ≻ , + ≺ : L n × L m → L n + m and ⋉ : L n × L m → L nm , thus words from L p , with p a prime number, are prime for thisarithmetics. Observe that ⋉ is associative and left distributive with regards to additions.L-monoids being straightforward to define, we get the following. Theorem 2.10
The free L-monoid over a set X is given by, L = ( [ n> L n × ( X × . . . × X ) | {z } n copies , + ≻ , + ≺ ) . Moreover, for any words l, l ′ , l ′′ , l + • l ′ = l + • l ′′ ⇔ l ′ = l ′′ ; l ′ + • l = l ′′ + • l ⇔ l ′ = l ′′ , where • = ≺ , ≻ .l ′ ⋉ l = l ′′ ⋉ l ⇔ l ′ = l ′′ . Proof:
The first two assertions are a straightforward consequence of the structure of freeL-algebra. As the L-algebra K L is free, there exists a unique L-algebra automorphismsending the generator 1 to l , therefore since the multiplication is left distributive, the map − ⋉ l : K L → K L is this automorphism, hence the third assertion. (cid:3) Observe that associative algebras are L-algebras hence a functor inc : As → L-alg . Let( L, ≺ , ≻ ) be a L-algebra and I be the L-ideal generated by the relations x ≻ y − x ≺ y for x, y ∈ L . Then L/I is an associative algebra, hence a functor F As : L-alg → As .Denote by π : L ։ L/I the canonical surjection. Let A be an associative algebra. Forany morphism of L-algebras f : L → A there exists a unique morphism of associativealgebras ˜ f : L/I → A such that f = ˜ f ◦ π . Similarly, if ˜ f : L/I → A is a morphism ofassociative algebras then f := ˜ f ◦ π : L → inc ( A ) is a morphism of L-algebras. Hence, Hom As ( F As ( L ); A ) ∼ = Hom L − alg ( L ; inc ( A )) , the functor F As is left adjoint to the functor inc . By L ! ( n ), we mean the K -vector space spanned by n -ary operations of the L ! -operad madeout of our two generating binary operations ⊢ and ⊣ . We proved that L ! ( n ) = 0 as soonas n >
3. We have L ! (1) = K.id , L ! (2) = K. ⊢ ⊕ K. ⊣ and L ! (3) = K. ( ⊢ ) ⊣ . Following V.Ginzburg and M. Kapranov, the chain-complex over a L-algebra A is restricted to L ! (3) ⊗ A ⊗ d −→ L ! (2) ⊗ A ⊗ d −→ L ! (1) ⊗ A, d is the differential operator which agrees in low dimensions with the L-algebrastructure of A .Therefore, so as to give explicitly an homology theory for L-algebras let us define C to be the set { } ; C to be the set { , } ; C to be another copy of the set { } . Let A be a L-algebra. The module of n -chains, for n = 1 , ,
3, is C Ln ( A ) := KC n ⊗ A ⊗ n . Thedifferential operator d is defined as follows: d (1; x ⊗ y ⊗ z ) := (1; ( x ≻ y ) ⊗ z ) − (2; x ⊗ ( y ≺ z )); d (1; x ⊗ y ) := (1; x ≺ y ); d (2; x ⊗ y ) := (1; x ≻ y ) , for all x, y, z ∈ A . We do have d = 0 since, d (1; x ⊗ y ⊗ z ) := d (1; ( x ≻ y ) ⊗ z ) − d (2; x ⊗ ( y ≺ z )) = (1; ( x ≻ y ) ≺ z ) − (1; x ≻ ( y ≺ z )) = 0 . Hence the complex, CL ∗ ( A ) : 0 d −→ KC ⊗ A ⊗ d −→ KC ⊗ A ⊗ d −→ KC ⊗ A d −→ . By definition, the homology of the L-algebra A is the homology of our short chain-complex CL ∗ ( A ) and, HL n ( A ) := H n ( CL ∗ ( A ) , d ); n = 1 , , . We get HL ( A ) := A/J , where J is the ideal generated by the x ≺ y and x ≻ y , for x, y ∈ A . Theorem 2.11
Let V be a K -vector space and L ( V ) be the free L-algebra over V . Then, HL ( L ( V )) ≃ V,HL n ( L ( V )) = 0 , for n > . Therefore, the L-operad is Koszul.Proof:
As the L-operad is regular, we restrict the proof to the free L-algebra on onegenerator, i.e. , K L . To ease notation, we rename d as follows: CL ∗ ( K L ) : 0 d −→ KC ⊗ K L ⊗ d −→ KC ⊗ K L ⊗ d −→ KC ⊗ K L d −→ . The first assertion is trivial since Im d = L n ≥ K L n and ker d = K L , thus ker d / Im d ≃ K . Let us show that HL ( K L ) = 0 , that is d is injective. Let l, l ′ , l ′′ be three words suchthat d (1; l ⊗ l ′ ⊗ l ′′ ) = 0. Hence, we get both (1; ( l ≻ l ′ ) ⊗ l ′′ ) = 0 and (2; l ⊗ ( l ′ ≺ l ′′ )) = 0.As operations respect the graduation, one of them has to be equal to zero. Let us showthat HL ( K L ) = 0 , that is Im d = ker d . We know that ker d ⊃ Im d . Let l , l , l , l
12e four words and λ, µ ∈ K such that d ( λ (1; l ⊗ l ) − µ (2; l ⊗ l )) = 0. We get λ (1; l ≺ l ) = µ (1; l ≻ l ). Hence, µ = λ and l ≺ l = l ≻ l , that is: w := l l ¯2 = 1 l ¯2 l . The proof of Theorem 2.7 shows that such an equality holds if and only if there exists aword u such that l = l ≻ u. Consequently, the word l can be written as: l = u ≺ l . Therefore, the expression we started with, λ (1; l ⊗ l ) − µ (2; l ⊗ l ) ∈ ker d can be written λd (1; l ⊗ u ⊗ l ). Our complex is exact and the L-operad is Koszul. (cid:3) As the L-operad is Koszul, applying results of [11] gives the following.1. The L ! -operad is Koszul.2. The generating functions of the L ! -operad and the L-operad are inverse one anotherfor the composition, that is: f L ( − f L ! ( − x )) = x. This gives an algebraic interpretation of the fact that these two series have beendiscovered to be inverse one another in A Recall 1 K denotes the unit of the field K . Let ( L, ≺ , ≻ ) be a L-algebra. Recall [14],directed graphs equipped with probability vectors, viewed as (Markov) L-coalgebras nat-urally have a left counit equals to a right one. Dually, left and right units can be intro-duced. We focus on the case when they coincide. Over K ⊕ L a structure of L -algebracan be constructed as follows. 1 K ≺ K = 1 K = 1 K ≻ K ; ∀ x ∈ L ; 1 K ≺ x = t ( x ); x ≻ K = s ( x ); 1 K ≻ x = x = x ≺ K , where s, t : L → L are linear maps such that, ∀ x, y ∈ L ; t ( x ) ≺ y = x ≻ s ( y ) . A L-algebra is said to be unital if it has an element denoted by 1 and a pair ( t, s ) verifyingthe above equations. If L and L ′ are (unital) L-algebras then the following operations,( x ⊗ x ′ ) ≺ ( y ⊗ y ′ ) = ( x ≺ y ) ⊗ ( x ′ ≺ y ′ );( x ⊗ x ′ ) ≻ ( y ⊗ y ′ ) = ( x ≻ y ) ⊗ ( x ′ ≻ y ′ ) , for x, y ∈ L and x ′ , y ′ ∈ L ′ , turns L ⊗ L ′ into a (unital) L-algebra.13 roposition 3.1 Let V be a K -vector space. There exits a cocommutative coassociativecoproduct and a counit over the augmented free L-algebra K ⊕ L ( V ) which are unitalL-algebra morphisms.Proof: Fix l ∈ L( V ). Define the map s, t : L( V ) → L( V ) by, s ( l ) := l ≺ l, and t ( l ) := l ≻ l . Following J.-L. Loday [23], since L( V ) is free, the map v K ⊗ v + v ⊗ K , for any v ∈ V ,has a natural extention, morphism of L-algebras, ∆ : K ⊕ L( V ) → ( K ⊕ L( V )) ⊗ whichis coassociative. As the flip operator is a L-morphism and leaves v K ⊗ v + v ⊗ K invariant, the coproduct ∆ will be cocommutative. The counit ǫ is such that ǫ (L( V )) = 0. (cid:3) Remark:
For instance, δ ( l ≺ l ′ ) = 1 K ⊗ ( l ≺ l ′ ) + ( l ≺ l ′ ) ⊗ K + l ⊗ t ( l ′ ) + t ( l ) ⊗ l ′ .Equipped with this coproduct, K ⊕ K L ( V ) is not connected in the sense of Quillen exceptif s = t = 0. In this case, the K -vector space of the primitive elements P rim ( K L ( V )) isa L-algebra isomorphic to K L ( V ). The aim of this section is to obtain a general structure theorem which will be useful inSection 5. [ n ] − M ag -bialgebras
We first start with recalling what the nonunital infinitesimal compatibility relation overa binary operad P is, allowing the definition of coassociative P -bialgebras. We generalizethe magmatic operad M ag to [ n ] − M ag the magmatic operad with n operations and someresults obtained in [13]. Inspired by proofs from [13], we show the existence of a structuretheorem for the triple ( As, [ n ] − M ag, P rim [ n ] − M ag ) and obtain, as a consequence,a structure theorem for the triple (
As, L , P rim L). Let us start with the definition ofcoassociative P -bialgebras. Definition 4.1
Let P be a binary, quadratic operad. A coassociative P -bialgebras, As c − P -bialgebra for short, is a P -algebra P equipped with a coassociative coproduct δ : P → P ⊗ verifying the so-called nonunital infinitesimal compatibility relation,( ∗ ) δ ( x • y ) = δ ( x ) • y + x • δ ( y ) + x ⊗ y, i.e., ∗ ) δ ( x • y ) = x (1) ⊗ ( x (2) • y ) + ( x • y ) ⊗ y + x ⊗ y, where δ ( x ) = x (1) ⊗ x (2) (Sweedler’s notation), for any x, y ∈ P and any generatingoperations • ∈ P (2). An As c − P -bialgebra H is said to be connected if H = S r ≥ F r H , where F r H is the coradical filtration of H defined recursively by, F H := K. K , F r H := { x ∈ H | δ ( x ) ∈ F r − H ⊗ F r − H} . By definition the space of primitive elements is defined as,
P rim H := ker δ. On the cofree coalgebra.
Let V be a K -vector space. Recall that As c ( V ) := L m> V ⊗ m as a K -vector space and equipped with the deconcatenation coproduct ˆ δ , defined by,ˆ δ ( v ) := 0 , ∀ m > , ˆ δ ( v ⊗ . . . ⊗ v m ) := m − X i =1 ( v ⊗ . . . ⊗ v i ) ⊗ ( v i +1 ⊗ . . . ⊗ v m ) , is the cofree connected coassociative coalgebra in the corresponding category. Recall alsothat ˆ δ verifies Formula ( ∗ ). The magmatic operad with n operations. Fix an integer n >
0. Let [ n ] − M ag be the (free) operad generated by n binary operations • i , i = 1 , . . . n , that is [ n ] − M ag (2) = K {• i , i = 1 , . . . n } . We set [1] − M ag := M ag . Its dual is the operad [ n ] − N il also generated by n binary operations • ∨ i , i = 1 , . . . , n and such that any nontrivialcompositions vanish, i.e., [ n ] − N il ( m ) = 0 for m >
2. Therefore, [ n ] − M ag is a Koszuloperad. For p ≥
0, let
Col [ n ] Y p be the set of rooted planar binary trees whose nodes arecolored by a color i = 1 , . . . , n . For instance, Col [ n ] Y = { } , Col [ n ] Y = { i , i = 1 . . . n } .The n -grafting operations we are looking for are denoted by ∨ i , i = 1 , . . . , n . Hence t ∨ i t ′ means that the tree t is grafted to t ′ via a root colored by i . The free [ n ] − M ag -algebraover a K -vector space V is the K -vector space,[ n ] − M ag ( V ) = M p ≥ KCol [ n ] Y p ⊗ V ⊗ ( p +1) , equipped with the n -operations,( t ; v ⊗ . . . ⊗ v p +1 ) ∨ i ( s ; w ⊗ . . . ⊗ v q +1 ) = ( t ∨ i s ; v ⊗ . . . ⊗ v p +1 ⊗ w ⊗ . . . ⊗ v q +1 ) . Proposition 4.2
Let [ n ] − M ag ( V ) be the free [ n ] − M ag -algebra over the K -vectorspace V . Then, there exists a unique coproduct δ vanishing on K ⊗ V ≃ V and turning [ n ] − M ag ( V ) into an As c − [ n ] − M ag -bialgebra. roof: Equip [ n ] − M ag ( V ) ⊗ [ n ] − M ag ( V ) with the following n operations,( x ⊗ y ) ∨ i ( x ′ ⊗ y ′ ) = ( x ∨ i x ′ ) ⊗ ( y ∨ i y ′ ) , hence [ n ] − M ag ( V ) ⊗ [ n ] − M ag ( V ) is also a [ n ] − M ag -algebra. Define now the coproduct δ : [ n ] − M ag ( V ) → [ n ] − M ag ( V ) ⊗ [ n ] − M ag ( V ) as follows. δ ( ; v ) = 0 , for any v ∈ V and recursively by the Formula ( ∗ ). For instance, δ (( i ; v ⊗ w )) = δ (( ; v ) ∨ i ( ; w )) = v ⊗ w and so on. It is straightforward to prove recursively that Formula( ∗ ) implies the coassociativity of δ . Hence, the uniqueness of δ and [ n ] − M ag ( V ) has As c − [ n ] − M ag -bialgebra structure. (cid:3)
To show that the As c − [ n ] − M ag -bialgebra is connected, we use the following lemmaadapted from [13]. First of all, let t be a colored planar binary rooted tree with p + 1leaves, numbered from left to right by 1 , , . . . , p + 1. Let j = 1 , . . . , p . We split the tree t into two colored trees t j (1) and t j (2) as follows. The tree t j (1) is the part of t at the left handside of the path from the leaf j to the root, the path being included. The tree t j (2) is thepart of t at the right hand side of the path going from the leaf j + 1 to the root. The colorassigned to the node “father” of the leaves j and j + 1 is then ignored. For instance, t = i=1i=2i=3i=4 t =t = t =t = t = t =t = t = Proposition 4.3
Let [ n ] − M ag ( V ) be the free [ n ] − M ag -algebra over the K -vector space V . Then, the coproduct δ can be written as a sum of co-operations P ≤ j ≤ p δ j , where, δ j ( t ; v ⊗ . . . ⊗ v p +1 ) = ( t j (1) ; v ⊗ . . . ⊗ v j ) ⊗ ( t j (2) ; v j +1 ⊗ . . . ⊗ v p +1 ) . Hence, ([ n ] − M ag ( V ) , δ ) is a connected As c − [ n ] − M ag -bialgebra.Proof:
We adapt the proof from [13]. Define the co-operation ∆ by ∆ := P ≤ j ≤ p δ j ,where, δ j ( t ; v ⊗ . . . ⊗ v p +1 ) = ( t j (1) ; v ⊗ . . . ⊗ v j ) ⊗ ( t j (2) ; v j +1 ⊗ . . . ⊗ v p +1 ) . ⊗ v , for all v ∈ V . Let us show that ∆ verifies Formula( ∗ ). For p >
2, there exist a unique k = 1 , . . . p −
1, a unique i = 1 , . . . , n , unique coloredrooted planar binary trees t l ∈ Col [ n ] Y k and t r ∈ Col [ n ] Y p − k − such that t = t l ∨ i t r .Hence, ∆(( t l ; v ⊗ . . . ⊗ v k +1 ) ∨ i ( t r ; v k +2 ⊗ . . . ⊗ v p +1 )) = ∆( t ; v ⊗ . . . ⊗ v p +1 ) = k − X j =1 (( t l ) j (1) ; v ⊗ . . . ⊗ v j ) ⊗ (( t l ) j (2) ∨ i t r ; v j +1 ⊗ . . . ⊗ v p +1 )+( t l ; v ⊗ . . . ⊗ v k +1 ) ⊗ ( t r ; v k +2 ⊗ . . . ⊗ v p +1 )+ p X j = k +2 ( t l ∨ i ( t r ) j − k − ; v ⊗ . . . ⊗ v j ) ⊗ (( t r ) j − k − ; v j +1 ⊗ . . . ⊗ v p +1 ) =∆(( t l ; v ⊗ . . . ⊗ v k +1 )) ∨ i ( t r ; v k +2 ⊗ . . . ⊗ v p +1 ) + ( t l ; v ⊗ . . . ⊗ v k +1 ) ∨ i ∆( t r ; v k +2 ⊗ . . . ⊗ v p +1 )+( t l ; v k +2 ⊗ . . . ⊗ v p +1 ) ⊗ ( t r ; v k +2 ⊗ . . . ⊗ v p +1 ) . Therefore, ∆ verifies the formula ( ∗ ). Consequently, the previous proposition shows that∆ is coassociative and thus has to coincide on the whole [ n ] − M ag ( V ) with δ . Moreover,this formula shows that [ n ] − M ag ( V ) is connected. (cid:3) We now refer to notation, definitions and results of J.-L. Loday [19]. Since the nonuni-tal infinitesimal relation ( ∗ ) is distributive (Hypothesis H n ] − M ag ( V ) isequipped with a coassociative [ n ] − M ag ( V )-bialgebra (Hypothesis H P rim [ n ] − M ag is also an operad, suboperad of [ n ] − M ag . Therefore, it makes sense to deal with
P rim [ n ] − M ag -algebras and the forgetful functor, F : [ n ] − M ag P rim [ n ] − M ag, has a left adjoint, the so-called universal enveloping algebra functor, U : P rim [ n ] − M ag [ n ] − M ag.
Theorem 4.4
For any As c − [ n ] − M ag -bialgebra, H , the following are equivalent:1. The As c − [ n ] − M ag -bialgebra H is connected.2. There is an isomorphism of bialgebras H ≃ U ( P rim H ) .3. There is an isomorphism of connected coalgebras H ≃ As c ( P rim H ) ,That is, the triple of operads ( As, [ n ] − M ag, P rim ([ n ] − M ag )) is good. roof: We will apply Theorem 2.5.1. from [19] by checking that the so-called Hypothesis( H epi ) [19] holds. Let V be a K -vector space. By the previous proposition, [ n ] − M ag ( V )has a natural As c − [ n ] − M ag -bialgebra structure. Consequently, the projection map proj V : [ n ] − M ag ( V ) ։ V and the cofreness of As c ( V ) give a unique coalgebra map: φ ( V ) : [ n ] − M ag ( V ) → As c ( V ) , such that π ◦ φ ( V ) = proj V , where π : As c ( V ) ։ V is the canonical projection. Theformula giving the coproduct δ implies that φ ( V )( t ) = 1 K for any colored tree t (recall As cn = K ). Thus, φ ( V ) is surjective. Fix now a color i and denote by comb m , m ≥
0, theleft comb obtained recursively by comb := and comb m := comb m − ∨ i . Define the map s ( V ) : As c ( V ) → [ n ] − M ag ( V ) by 1 K ∈ As cm comb m − ∈ [ n ] − M ag m . Then, s ( V ) isa coalgebra map since, δ ( comb ; v ) = 0 , ∀ m > , δ ( comb m ; v ⊗ . . . ⊗ v m +1 ) = m − X j =1 ( comb j − ; v ⊗ . . . ⊗ v j ) ⊗ ( comb m − j ; v j +1 ⊗ . . . ⊗ v m +1 ) , and φ ( V ) ◦ s ( V ) = id As c ( V ) . Hence, the natural coalgebra map φ ( V ) is surjective andadmits a natural coalgebra splitting s ( V ), hence Hypothesis ( H epi ) holds. Therefore,applying [19] Theorem 2.5.1., the triple ( As, [ n ] − M ag, P rim ([ n ] − M ag )) is a good tripleof operads. (cid:3)
Theorem 4.5
Let P be a binary, quadratic operad such that for any generating opera-tions • i ∈ P (2) , there exist relations only in P (3) of the form, (++) X i,j ; σ i,j ∈ S λ i,j • j ( • i ⊗ id ) σ i,j = X i,j ; σ i,j ∈ S λ ij • i ( id ⊗ • j ) σ i,j , for any i, j ∈ { , . . . , dim P (2) } and λ ij ∈ K . Then, the triple ( As, P , P rim P ) is good.Considering only binary quadratic operad coming from a set operad, quadratic relationsof the form: (+ + +) • j ( • i ⊗ id ) σ i,j = ( id ⊗ • j ) σ i,j , are the only ones giving such good triples.Proof: Suppose first P to be regular. Set n = dim P . In the operad [ n ] − M ag , theoperations • j ( • i ⊗ id ) − • i ( id ⊗ • j ) ∈ [ n ] − M ag are primitive operations. Indeed, let x, y, z be primitive elements of a As c − [ n ] − M ag -bialgebra ( H , δ ). Then, δ (( x • i y ) • j z ) = x ⊗ ( y • j z ) + ( x • i y ) ⊗ z, which is compensated with, δ ( x • i ( y • j z )) = ( x • i y ) ⊗ z + x ⊗ ( y • j z ) . Therefore, ( x • i y ) • j z − x • i ( y • j z ) ∈ ker δ and any linear combinations of such relationsremains primitive operations. This result still stands if one permutates the entries in the18ame way on each side of Formula (++). The relations (+ + +) are the only ones to giveprimitive operations when only operads coming from set operads are considered since tocharacterize P , linear combinations of ternary operations are not allowed. Denote by J the operadic ideal generated by the primitive operations, X i,j ; σ i,j ∈ S λ i,j • j ( • i ⊗ id ) σ i,j − X i,j ; σ i,j ∈ S λ ij • i ( id ⊗ • j ) σ i,j . Applying Proposition 3.1.1. of J.-L. Loday [19], one gets that ( As c , [ n ] − M ag/J, P rim [ n ] − M ag/J ) is still a good triple of operads. As, [ n ] − M ag/J ≃ P , the triple of operads,(
As, P , P rim P ) is good. (cid:3) Example 4.6
The structure theorem holds for the As c -duplicial-bialgebras as provedin a different way in [19]. As another example, in an associative algebra, consider twoassociative products ⋆ and ⋆ . Requiring the product ⋆ = ⋆ + ⋆ to be associative leadsto the so-called Hochschild 2-cocycle [9]:( ∗∗ ) ( x ⋆ y ) ⋆ z + ( x ⋆ y ) ⋆ z = x ⋆ ( y ⋆ z ) + x ⋆ ( y ⋆ z ) . Consider the regular, binary, quadratic operad G made out with two associative productsverifying the relation ( ∗∗ ), then Theorem 4.5 claims the existence of a notion of As c − G -bialgebras and a good triple ( As, G, P rimG ). The triple (
As, P re − Lie, P rimP re − Lie )is another (well-known) example. The relations (++) are not the only ones to give suchgood triples (if the operad P does not come from a set operad). For instance the quadraticbinary operad defined to be [4] − M ag (4 binary operations • , • , • , • ) divided out bythe operadic ideal generated by,( • − • )(( • − • ) ⊗ id ) , will also provide a good triple. We now improve a result of [25]. Let P be a binary operad for which the notion ofconnected As c − P -bialgebras stands. Let ( H , δ ) be such a bialgebra. For each generatingoperation • ∈ P (2), define the linear map, e r • : H → H , recursively by, x e r • ( x ) := x − x (1) • e r • ( x (2) ) , where using Sweedler’s notation δ ( x ) := x (1) ⊗ x (2) . As H is connected, the map e r • iswell defined. Similarly, define the linear map, e l • : H → H , recursively by, x e l • ( x ) := x − e l • ( x (1) ) • x (2) . roposition 4.7 For each generating operation • ∈ P (2) , the maps e r • and e l • areidempotents from H to P rim H . Suppose the existence of an associative generating oper-ation ⋆ ∈ P (2) . Then, ker e ⋆ = H ⋆ H . Proof:
We focus on e r • renamed in e • . We will prove these claims by induction on thefiltration F n H of H . Let x ∈ P rim H , then δ ( x ) = 0, thus e ( x ) = x and e ( x ) ∈ P rim H .Let x ∈ F n H and suppose e • ( y ) ∈ P rim H for any y ∈ F r H , r < n . As H is connected, δ ( e • ( x )) = δ ( x ) − δ ( x (1) • e • ( x (2) )) = δ ( x ) − δ ( x (1) ) • e • ( x (2) ) − x (1) ⊗ e • ( x (2) ) , = δ ( x ) − x (11) ⊗ ( x (12) • e • ( x (2) )) − x (1) ⊗ e • ( x (2) ) . As δ is coassociative, this is equal to: δ ( x ) − x (1) ⊗ ( x (21) • e • ( x (22) )) − x (1) ⊗ e • ( x (2) ) = δ ( x ) − x (1) ⊗ ( id − e • )( x (2) ) − x (1) ⊗ e • ( x (2) ) = 0 . Let x ∈ H , then e • ( x ) is primitive, therefore e • ( e • ( x )) = e • ( x ) and e • is an idempotent.Suppose the existence of an associative generating operation ⋆ ∈ P (2). Let x, y ∈ H .To prove that e ⋆ ( x ⋆ y ) = 0, we proceed by induction on the sum of the filtration-degrees of x and y . If x, y ∈ P rim H , then δ ( x ⋆ y ) = x ⊗ y . Therefore, e ⋆ ( x ⋆ y ) = x ⋆ y − x ⋆ e ⋆ ( y ) = x ⋆ y − x ⋆ y = 0 since e ⋆ is an idempotent. Suppose this resultholds when the sum of the filtration-degrees is strictly less than the one of x and y . As δ ( x ⋆ y ) := x ⊗ y + x (1) ⊗ ( x (2) ⋆ y ) + ( x ⋆ y (1) ) ⊗ y (2) , and ⋆ is associative, e ⋆ ( x⋆y ) = x⋆y − [ x⋆e ⋆ ( y )+( x⋆y (1) ) ⋆e ⋆ ( y (2) )] = x⋆y − x⋆ ( e ⋆ ( y ) − y (1) e ⋆ ⋆ ( y (2) )) = x⋆y − x⋆y = 0 , holds. Suppose x ∈ ker e ⋆ , then e ⋆ ( x ) = 0 and x = x (1) ⋆ e ⋆ ( x (2) ). Therefore, x ∈ H ⋆ H . (cid:3) For each n >
0, denote by e • ; n the restriction of e • to P ( n ) ⊗ S n V ⊗ n . Proposition 4.8
Let V be a K -vector space and a generating operation • ∈ P (2) . Then,for all n , P ( n ) ⊗ S n V ⊗ n / ker e • ; n ≃ ( P rim P )( n ) ⊗ S n V ⊗ n , as K -vector spaces.Proof: We show that, e • ; n : P ( n ) ⊗ S n V ⊗ n → ( P rim P )( n ) ⊗ S n V ⊗ n , is well-defined and surjective. We proceed by induction on the degree of the involved K -vector spaces. We have e • ;1 ( V ) = V . Let ⊥∈ P (2) be another generating operation.For n = 2, if x, y ∈ V , then e • ;2 ( x ⊥ y ) = x ⊥ y − x • y ∈ ( P rim P )(2). Suppose theresult holds up to a n −
1. Then, for a monomial of P ( n ) ⊗ S n V ⊗ n , as P is binary, there20xits two monomials of smaller degrees X and Y and a generating operation ⋄ ∈ P (2)such that it can be written X ⋄ Y . Therefore, e • ( X ⋄ Y ) := X ⋄ Y − X • e • ( Y ) , hence maps e • ; n are well-defined by induction. To prove the restriction of the idempotent e • on the homogeneous coponents is surjective, let x ∈ ( P rim P )( n ) ⊗ S n V ⊗ n . As e • issurjective, there exists a y ∈ P ( V ) such that e • ( y ) = x . Therefore y and x have the samedegree and e • ; n : P ( n ) ⊗ S n V ⊗ n ։ ( P rim P )( n ) ⊗ S n V ⊗ n , factors through ker e • . (cid:3) Remark:
This result can be of assistance when searching a presentation of the operad
P rim P .Let V be a K -vector space and P ( V ) be the free P -algebra over V . Suppose theexistence of an involution † over V . The P -algebra P ( V ) is said to be involutive if thereexists a map still denoted by † , † : P (2) → P (2) , • i
7→ • i † , given by induction by, ( t • i s ) † = s † • i † t † , for all t, s ∈ P ( V ) and leaving the relations in P (3) defining the operad P globally invari-ant (recall we suppose the notion As c − P -bialgebras holds so there is no relation betweenthe generating operations of P (2)). This involution is the only anti-homomorphism of P -algebras which agrees with the involution over V . Extend this involution on P ( V ) ⊗ by the formula, ( t ⊗ s ) † := s † ⊗ t † , and operations as follows:( t ⊗ s ) • i ( t ′ ⊗ s ′ ) := ( t • i t ′ ) ⊗ ( s • i s ′ ) , for all s, t ∈ P ( V ) and • i ∈ P (2). Lemma 4.9
Denote by δ the coassociative coproduct of the As c − P -bialgebra P ( V ) .Then, δ commutes with † , that is: δ ( t † ) = δ ( t ) † , for any t ∈ P ( V ) .Proof: Observe that the involution † preserves the gradding. Let t ∈ V . Then t † ∈ V and δ ( t ) † = 0 = δ ( t † ). Suppose δ ( r ) † = δ ( r † ) for any element of degree up to n . Let r be anelement of degree n + 1. As the operad P is binary, there exist t and s of smaller degreesand an operation • ∈ P (2) such that r = t • s . δ ( t • s ) † = s † ⊗ t † + [ s (2) ] † ⊗ ([ s (1) ] † • † t † ) + ( s † • † [ t (2) ] † ) ⊗ [ t (1) ] † , = s † ⊗ t † + δ ( s ) † • † t † + s † • † δ ( t ) † , = s † ⊗ t † + δ ( s † ) • † t † + s † • † δ ( t † ) , (by induction) , = δ ( s † • † t † ) = δ (( t • s ) † ) . Proposition 4.10
Let V be a K -vector space. Let ( As, P , P rim P ) be a good triple ofoperads from Theorem 4.5. Suppose the existence of an involution † on V which extendsto an involution † on P ( V ) . Then, P rim P ( V ) is invariant under † .Proof: Let t be a primitive element. Then, δ ( t † ) = δ ( t ) † = 0. Hence the space of primitiveelements of P rim P ( V ) is invariant under the involution † . (cid:3) We present a result linking right and left idempotents associated with an operation.
Proposition 4.11
Let V be a K -vector space. The idempotents e r • , e l • : P ( V ) → P rim P ( V ) , verify: ∀ t ∈ P ( V ) , e r • ( t † ) = e l †• ( t ) † , e l • ( t † ) = e r †• ( t ) † . Proof:
We have e r • ( t † ) = t † = e l • ( t ) † , for any t ∈ V . we proceed by induction on thedegree and suppose the result holds for any element of degree equals at most n . As P is binary, any element of degree n + 1 is of the form: t ⋆ s with t, s elements of smallerdegrees and ⋆ ∈ P (2). On the one hand, e r • (( t ⋆ s ) † ) = e r • ( s † ⋆ † t † ) , = s † ⋆ † t † − [ s † • e r • ( t † ) + [ s † ] (1) • e r • ([ s † ] (2) ⋆ † t † ) + ( s † ⋆ † [ t † ] (1) ) • e r • ([ t † ] (2) )] . On the other hand, e l †• ( t ⋆ s ) = t ⋆ s − [ e l †• ( t ) • † s + e l †• ( t (1) ) • † ( t (2) ⋆ s ) + e l †• ( t ⋆ s (1) ) • † s (2) ] . Therefore, taking the involution on both sides leads to: e l †• ( t ⋆ s ) † = s † ⋆ † t † − [ s † • e l †• ( t ) † + ( s † ⋆ † [ t (2) ] † ) • e l †• ( t (1) ) † + [ s (2) ] † • e l †• ( t ⋆ s (1) ) † ] . Applying induction, e l †• ( t ⋆ s ) † = s † ⋆ † t † − [ s † • e r • ( t † ) + ( s † ⋆ † [ t (2) ] † ) • e r • ([ t (1) ] † ) + [ s (2) ] † • e r • (( t ⋆ [ s (1) ]) † )] . However, Lemma 4.9 shows that the decomposition of an element r by δ leads to thefollowing equalities: [ r (1) ] † = [ r † ] (2) and [ r (2) ] † = [ r † ] (1) . Therefore, e l †• ( t ⋆ s ) † = s † ⋆ † t † − [ s † • e r • ( t † ) + ( s † ⋆ † [ t † ] (1) ) • e r • ([ t † ] (2) ) + [ s † ] (1) • e r • ([ s † ] (2) ⋆ † t † )] . Hence the first equality. For the second one, let t be an element of P ( V ). Then, e r †• ( t ) † = e r †• ( t †† ) † ) = e l • ( t † ) . (cid:3) .3 The structure theorem for L-algebras We now come back to the L-operad and obtain:
Theorem 4.12
For any As c -L-bialgebra, H , the following are equivalent:1. The As c -L-bialgebra H is connected.2. There is an isomorphism of bialgebras H ≃ U ( P rim H ) .3. There is an isomorphism of connected coalgebras H ≃ As c ( P rim H ) . Because of Theorem 4.5, we get an isomorphism of Schur functors L ≃ As c ◦ P rim
L. Asexplained in [19], one can deduce the generating function of the operad
P rim
L. Indeedsince the generating function of the operad As c is f As c ( x ) = x − x and the generatingfunction of the L-operad is f L ( x ) = sin ( asin( q x )) , we claim that the generatingfunction of the operad P rim
L is, f Prim L ( x ) = sin ( asin( q x )) + sin ( asin( q x )) . Its Taylor series starts with, f Prim L ( x ) = x + x + 4 x + 17 x + 81 x + 412 x + 2192 x . . . . For instance, dim
P rim L = 1 so P rim L is spanned by x ⊲⊳ y := x ≻ y − x ≺ y anddim P rim L = 4 so P rim L is spanned by,[ x, y, z ] := ( x ⊲⊳ y ) ⊲⊳ z ; [ x, y, z ] := x ⊲⊳ ( y ⊲⊳ z );[ x, y, z ] := ( x ≻ y ) ≻ z − x ≻ ( y ≻ z ); [ x, y, z ] := ( x ≺ y ) ≺ z − x ≺ ( y ≺ z ) . We call a triplicial-algebra a K -vector space equipped with 3 operations verifying thefollowing constraints: ∀ ≤ i ≤ j ≤
3; ( x • i y ) • j z = x • i ( y • j z ) .
23n particular, all our operations are associative. We denote by
Trip the correspondingcategory and by
T rip the associated operad. Note that
T rip is a binary quadratic regularand set-theoretic operad. Kill one of the three products to recover the definition ofduplicial-algebras. The main interest of triplicial-algebras (
T rip -algebras for short) liesin the following theorem.
Theorem 5.1
The triple ( As, T rip, L ) is a good triple of operads. Therefore, The cate-gory of connected As c − T rip -bialgebras and the category of L-algebras are equivalent. { conn. As c − T rip − bialg. } U ⇆ P rim { L − alg. } Proof: As T rip is a regular quadratic binary operad with only entanglement relations,Theorem 4.5 claims that (
As, T rip, P rim T rip ) is a good triple of operads. Let T be a T rip -algebra. The following operations, x ≻ y := x • y − x • y, x ≺ y := x • y − x • y, for all x, y ∈ T , verify: ( x ≻ y ) ≺ z = x ≻ ( y ≺ z ) , and turns the K -vector space T into a L-algebra. Moreover, if T is a As c − T rip -bialgebrathen these operations are primitive operations. Consequently, its primitive part is a L-algebra. Theorem 5.6 in the next section shows that
T rip ( V ) is isomorphic to As ( L ( V ))as triplicial-algebras. Since the coproduct in As ( L ( V )) is the usual deconcatenation andthe triple ( As, As, V ect ) endowed with the infinitesimal relation is good [25], its primitivepart is L ( V ). Hence P rim T rip = L . Apply now [19], Theorem 2.6.3. to conclude. (cid:3) Remark:
This theorem is important since it allows to consider the (Markov) L-algebraassociated with a given weighted directed graph as a connected As c − T rip -bialgebra viathe universal enveloping functor U , see Subsection 5.3. H om K ( T , T ) We give few words on the K -vector space, Hom K ( T, T ), of endomorphisms of a given As c − T rip -bialgebra ( T, • , • , • ). We introduce three convolution product defined asfollows: f ¯ • i g := • i ( f ⊗ g ) δ, for all f, g ∈ Hom K ( T, T ) and i = 1 , ,
3. Then, observe that (
Hom K ( T, T ) , ¯ • , ¯ • , ¯ • )is also a T rip -algebra and for each i = 1 , ,
3, (
Hom K ( T, T ) , ¯ • i ) can be endowed witha Ennea-algebra structure [15]. There are at least 6 Rota-Baxter maps of weight 1 on Hom K ( T, T ) given by the shift operators: β i ( f ) := id • i f, γ i ( f ) := f • i id, for all f ∈ Hom K ( T, T ) and i = 1 , ,
3. They obey the following commutation rules: ∀ ≤ i ≤ j ≤ , β i ◦ γ j = γ j ◦ β i . .3 Even trees and the triplicial-algebras An even tree of size n is an ordered tree with 2 n edges in which each node has an evenoutput. Here are even trees of size 1 and 2. n= 2 :n= 1 : The set of even trees of size n is denoted by E n . It has been shown in [8] that thecardinality of E n is n +1 (cid:0) nn (cid:1) which is also the number of planar rooted ternary trees on n nodes, and also the number of symmetric planar rooted ternary trees on 2 n nodes!Define the gluing operation for any positive integers n, m as follows, ⌣ : E n × E m → E n + m , ( t, s ) t ⌣ s, where t ⌣ s is the even tree of size n + m obtained by gluing the root of t to the root of s . For instance ⌣ = . Define also the following operations: ր , տ : E n × E m → E n + m , as follows: t ր s is the even tree of size n + m obtained by setting t on the most left leafof s and t տ s is the even tree of size n + m obtained by setting s on the most right leafof s . Extend these three operations by K -bilinearity to get three binary operations, ⌣, ր , տ : K E n ⊗ K E m → K E n + m . Theorem 5.2
Let V be a K -vector space. Then, the K -vector space, E ven ( V ) := M n> K E n ⊗ V ⊗ n , equipped with the three binary operations still denoted by ⌣, ր , տ and defined by: ( t ⊗ v ⊗ . . . ⊗ v n ) • ( s ⊗ v ′ ⊗ . . . ⊗ v ′ m ) := ( t • s ) ⊗ v ⊗ . . . ⊗ v n ⊗ v ′ ⊗ . . . ⊗ v ′ m , for • ∈ { ⌣, ր , տ} is the free T rip -algebra over V . Its generating function is: f T rip ( x ) = 2 √ x sin( 13 asin ( r x − . roof: As the operad
T rip is regular, we need only to prove the theorem for a one di-mensional vector space. Observe first that E ven ( K ) is a T rip -algebra. Using ⌣, ր , տ ,one easily check that K E is generated by . Suppose this is the case up to K E n . Let t ∈ E n +1 . Then, suppose there exist k > t i , . . . , t i k , with i + . . . + i k = n and t i j ∈ E i j such that, t = t i ⌣ . . . ⌣ t i k . By induction, these even trees will be generated by . If this is not possible, then t is ofthe form t ր տ t , or t ր or տ t . Therefore by induction E n +1 is generatedby . Let ( A, • , • , • ) be a T rip -algebra and f : K → A , a map such that f (1 K ) = a .Define i : K → E ven ( K ) , K and ˜ f : E ven ( K ) → A inductively as follows:˜ f ( ) := a, ˜ f ( t ր s ) := ˜ f ( t ) • ˜ f ( s ) , ˜ f ( t ⌣ s ) := ˜ f ( t ) • ˜ f ( s ) , ˜ f ( t տ s ) := ˜ f ( t ) • ˜ f ( s ) . Then, ˜ f is by construction the unique T rip -algebra morphism verifying ˜ f ◦ i = f . Hence, E ven ( K ) is the free T rip algebra on one generator. As dim K E n = n +1 (cid:0) nn (cid:1) , the generat-ing function of the operad T rip is the generating function of even trees or planar rootedternary trees, hence the last claim. (cid:3)
Remark: [The universal enveloping functor U ] Recall notation of Subsection 5.1.The functor U acts as follows. Let ( L, ≺ , ≻ ) be a L-algebra. Then U ( L ) is given by T rip ( L ) / ∼ , where the equivalence relation ∼ consists in identifying, x ≻ y := x ր y − x ⌣ y, x ≺ y := x տ y − x ⌣ y, for all x, y ∈ L . An involutive
T rip -algebra ( T, • , • , • ) is a triplicial-algebra equipped with an involution ι : T → T verifying ι ( x • y ) = ι ( y ) • ι ( x ), ι ( x • y ) = ι ( y ) • ι ( x ), ι ( x • y ) = ι ( y ) • ι ( x ).Observe that once • is given so is • and conversely. The free T rip -algebra on onegenerator E ven ( K ) is an involutive T rip -algebra. Indeed, consider the involution over E ven ( K ) still denoted by † and defined inductively for any even trees t, s by, † = , ( s ր t ) † := t † տ s † , ( s տ t ) † := t † ր s † , ( s ⌣ t ) † := t † ⌣ s † . An even tree is said to be symmetric if it is invariant under the involution † . Extend nowthis involution on E ven ( K ) ⊗ by the formula,( t ⊗ s ) † := s † ⊗ t † . heorem 5.3 The L-algebra of primitive elements of the As c − T rip -bialgebra E ven ( K ) is invariant under the involution defined on even trees. Moreover, this involution coincideswith the involution introduced on rooted planar ternary trees with odd degrees.Proof: Use Proposition 4.10 for the first part. Recall that
P rim E ven ( K ) is a L-algebragenerated by and the two operations over even trees: t ≻ s := t ր s − t ⌣ s, t ≺ s := t տ s − t ⌣ s. Therefore,( t ≻ s ) † = ( t ր s ) † − ( t ⌣ s ) † = s † ≺ t † and ( t ≺ s ) † = ( t տ s ) † − ( t ⌣ s ) † = s † ≻ t † . Identify to the word 1 coding the symmetric planar rooted tree on one node . Thenthe involution on even trees does coincide with the involution on rooted planar symmetricternary trees we introduced in Subsection 2.3 since both of the involution are definedrecursively and agree with the generator 1. (cid:3)
Here is a way to construct other involutive triplicial-algebras. Let V be a K -vector space.Recall As ( V ) is the free associative algebra over V . The concatenation will be used todenote the usual associative product in As ( V ). Proposition 5.4
Let ( A, ∗ ) be an involutive associative algebra with involution denotedby ι : A → A , that is ι ( a ∗ a ′ ) = ι ( a ′ ) ∗ ι ( a ) , for all a, a ′ ∈ A . Extend this involution on A ⊗ As ( A ) as follows, ι ( a ⊗ a . . . a n ) := ι ( a n ) ⊗ ι ( a n − ) . . . ι ( a ) ι ( a ) , for any tensor ω := a ⊗ a . . . a n ∈ A ⊗ As ( A ) with the a i ∈ A . Then, A ⊗ As ( A ) equippedwith the following operations, ( a ⊗ a . . . a n ) • ( a ′ ⊗ a ′ . . . a ′ m ) := [ a ∗ a ′ ] ⊗ a . . . a n a ′ . . . a ′ m , ( a ⊗ a . . . a n ) • ( a ′ ⊗ a ′ . . . a ′ m ) := a ⊗ a . . . a n a ′ a ′ . . . a ′ m , ( a ⊗ a . . . a n ) • ( a ′ ⊗ a ′ . . . a ′ m ) := a ⊗ a . . . a n − a ′ a ′ . . . a ′ m − [ a n ∗ a ′ m ] , where a, a ′ ∈ A and a i , a ′ i ∈ As ( A ) , is an involutive T rip -algebra.Proof:
Straightforward. (cid:3)
Remark:
The
T rip -monoid can be easily defined and its free
T rip -monoid over a set X is of course given by the even trees.Diving out by the T rip -ideal generated by t ⌣ s for all t, s ∈ E ven ( K ) gives the freeduplicial-algebra on one generator Dup ( K ) since only planar rooted binary trees willsurvive.For another operad whose associated free object is constructed over ternary trees thereader should read [17]. We get also a coassociative coproduct over the free object butthe infinitesimal relation linking the coproduct and binary operations has to be replacedby the so-called semi-Hopf relation. 27 .5 Another presentation of the free triplicial-algebra We give here another presentation of the free triplicial-algebra over V inspired by [19]Prop. 5.2.7. Proposition 5.5
Let ( L, ≻ , ≺ ) be a L-algebra. Define on As ( L ) the following opera-tions: (1) x ◦ y := x ⊗ y + x ≻ y, where the extension of the operation ≻ , still denoted by ≻ , is defined by induction asfollows: ( r ) : x ≻ ( y ⊗ l ′ ) := ( x ≻ y ) ⊗ l ′ , ( x ⊗ l ) ≻ l ′ = − ( x ≻ l ) ≻ l ′ + x ≻ ( l ≻ l ′ ) + x ⊗ ( l ≻ l ′ ) , (2) x ◦ y := x ⊗ y, (3) x ◦ y := x ⊗ y + x ≺ y, where the extension of the operation ≺ , still denoted by ≺ , is defined by induction asfollows: ( r ) : ( l ⊗ x ) ≺ y = l ⊗ ( x ≺ y ) ,l ≺ ( l ′ ⊗ y ) = ( l ≺ l ′ ) ≺ y − l ≺ ( l ′ ≺ x ) + ( l ≺ l ′ ) ⊗ x, for all x, y ∈ As ( L ) and l, l ′ ∈ L . Then, As ( L ) is a triplicial-algebra.Proof: Because of the construction of the extension of ≻ and ≺ and the relations ( r )and ( r ), we have: ( x ⊗ y ) ≺ z = x ⊗ ( y ≺ z ) and x ≻ ( y ⊗ z ) = ( x ≻ y ) ⊗ z, for any x, y, z ∈ As ( L ). Therefore, we have the following equalities: ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) , and ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) , for any x, y, z ∈ As ( L ). We now establish the equality: E : ( x ◦ y ) ◦ z = x ◦ ( y ◦ z ) . Suppose degree of y greater than 2. Set y := y ′ ⊗ l , with l ∈ L . We have:( x ◦ y ) ◦ z = ( x ◦ ( y ′ ⊗ l )) ◦ z, = ( x ◦ ( y ′ ◦ l )) ◦ z, = (( x ◦ y ′ ) ◦ l ) ◦ z, = ( x ◦ y ′ ) ◦ ( l ◦ z ) , = x ◦ ( y ′ ◦ ( l ◦ z )) , = x ◦ (( y ′ ◦ l ) ◦ z ) , = x ◦ ( y ◦ z ) , E ) holds if degree of y is greater than 2. We now fix degree of y equalto 1 and set y := l ′ ∈ L . If degree x and z are equal to 1, then ( E ) holds because in aL-algebra the relation ( l ≻ l ′ ) ≺ l ′′ = l ≻ ( l ′ ≺ l ′′ ) holds. We suppose degre of z greaterthan 2 and degree of x equal to 1. We set z = l ⊗ z ′ . On the one hand:( x ◦ l ′ ) ◦ z = ( x ◦ l ′ ) ◦ ( l ⊗ z ′ ) , = ( x ⊗ l ′ + x ≻ l ′ ) ◦ ( l ⊗ z ′ ) , = x ⊗ l ′ ⊗ l ⊗ z ′ + ( x ⊗ l ′ ) ≺ ( l ⊗ z ′ ) + ( x ≻ l ′ ) ⊗ l ⊗ z ′ + ( x ≻ l ′ ) ≺ ( l ⊗ z ′ ) , = x ⊗ l ′ ⊗ l ⊗ z ′ + x ⊗ [ l ′ ≺ ( l ⊗ z ′ )] + ( x ≻ l ′ ) ⊗ l ⊗ z ′ + ( x ≻ l ′ ) ≺ ( l ⊗ z ′ ) , = x ⊗ l ′ ⊗ l ⊗ z ′ + x ⊗ [ l ′ ≺ ( l ⊗ z ′ )]+( x ≻ l ′ ) ⊗ l ⊗ z ′ +(( x ≻ l ′ ) ≺ l ) ≺ z ′ − ( x ≻ l ′ ) ≺ ( l ≺ z ′ ) + (( x ≻ l ′ ) ≺ l ) ⊗ z ′ , On the other hand: x ◦ ( l ′ ◦ z ) = x ◦ ( l ′ ◦ ( l ⊗ z ′ )) , = x ◦ ( l ′ ⊗ l ⊗ z ′ + l ′ ≺ ( l ⊗ z ′ )) , = x ⊗ l ′ ⊗ l ⊗ z ′ + x ≻ ( l ′ ⊗ l ⊗ z ′ ) + x ⊗ ( l ′ ≺ ( l ⊗ z ′ )) + x ≻ ( l ′ ≺ ( l ⊗ z ′ )) , = x ⊗ l ′ ⊗ l ⊗ z ′ + ( x ≻ l ′ ) ⊗ l ⊗ z ′ + x ⊗ ( l ′ ≺ ( l ⊗ z ′ )) + x ≻ ( l ′ ≺ ( l ⊗ z ′ )) , = x ⊗ l ′ ⊗ l ⊗ z ′ +( x ≻ l ′ ) ⊗ l ⊗ z ′ + x ⊗ ( l ′ ≺ ( l ⊗ z ′ ))+ x ≻ [( l ′ ≺ l ) ≺ z ′ − l ′ ≺ ( l ≺ z ′ ) + ( l ≺ l ′ ) ⊗ z ′ ] , + x ≻ [( l ′ ≺ l ) ≺ z ′ ] − x ≻ [ l ′ ≺ ( l ≺ z ′ )] + x ≻ [( l ≺ l ′ )] ⊗ z ′ ] , use now the relation ( l ≻ l ′ ) ≺ l ′′ = l ≻ ( l ′ ≺ l ′′ ) to conclude. We suppose now ( E ) holdsfor any x of degree lower than a fixed n and take x of degree n + 1. Set x = x ′ ⊗ l .( x ◦ l ′ ) ◦ z = (( x ′ ⊗ l ) ◦ l ′ ) ◦ z, = ( x ′ ⊗ l ⊗ l ′ + ( x ′ ⊗ l ) ≻ l ′ ) ◦ z, = ( x ′ ⊗ l ⊗ l ′ + x ≻ ( l ≻ l ′ ) − ( x ≻ l ) ≻ l ′ + x ⊗ ( l ≻ l ′ )) ◦ z, = x ′ ⊗ l ⊗ ( l ′ ◦ z )+( x ≻ ( l ≻ l ′ )) ◦ z − (( x ≻ l ) ≻ l ′ ) ◦ z +( x ⊗ ( l ≻ l ′ )) ◦ z, = x ′ ⊗ l ⊗ ( l ′ ◦ z )+( x ≻ ( l ≻ l ′ )) ◦ z − (( x ≻ l ) ≻ l ′ ) ◦ z +( x ⊗ ( l ≻ l ′ )) ◦ z.
29e rewrite the last equation in terms of the triplicial operations.( x ◦ l ′ ) ◦ z = x ′ ⊗ l ⊗ ( l ′ ◦ z ) + ( x ≻ ( l ≻ l ′ )) ◦ z − (( x ≻ l ) ≻ l ′ ) ◦ z + ( x ⊗ ( l ≻ l ′ )) ◦ z = [ x ′ ◦ ( l ◦ l ′ )] ◦ z − [( x ′ ≻ l ) ◦ l ′ ] ◦ z +[( x ′ ≻ l ) ◦ l ′ ] ◦ z +[ x ′ ◦ ( l ≻ l ′ )] ◦ z − [ x ′ ◦ ( l ≻ l ′ )] ◦ z, = x ′ ◦ [( l ◦ l ′ ) ◦ z ] − ( x ′ ≻ l ) ◦ [ l ′ ◦ z ] , use induction and degree ( x ′ ≻ l ) = degree x ′ , +( x ′ ≻ l ) ◦ [ l ′ ◦ z ]+ x ′ ◦ [( l ≻ l ′ ) ◦ z ] , use induction, − x ′ ◦ [( l ≻ l ′ ) ◦ z ] , It is easy to show by induction that ( l ≻ l ′ ) ◦ z = l ≻ ( l ′ ◦ z ) whatever the degree of z is. Therefore: ( x ◦ l ′ ) ◦ z = x ′ ◦ [ l ◦ ( l ′ ◦ z )] , use induction, − ( x ′ ≻ l ) ◦ [ l ′ ◦ z ]+( x ′ ≻ l ) ◦ [ l ′ ◦ z ]+ x ′ ◦ [ l ≻ ( l ′ ◦ z )] , − x ′ ◦ [ l ≻ ( l ′ ◦ z )] , Use now the definitions of the triplicial operations and find:( x ◦ l ′ ) ◦ z = x ′ ⊗ l ⊗ ( l ′ ◦ z )+ x ≻ ( l ≻ ( l ′ ◦ z )) − ( x ≻ l ) ≻ ( l ′ ◦ z )+ x ⊗ ( l ≻ ( l ′ ◦ z )) , = x ′ ⊗ l ⊗ ( l ′ ◦ z ) + ( x ′ ⊗ l ) ≻ ( l ′ ◦ z ) , because of the relation ( r ) , = ( x ′ ⊗ l ) ◦ ( l ′ ◦ z ) , = x ◦ ( l ′ ◦ z ) . Therefore ( E ) holds. For the associativity of ◦ and ◦ , we proceed again by induction.Let x, y, z ∈ L .( x ◦ y ) ◦ z = ( x ⊗ y + x ≺ y ) ◦ z, = x ⊗ y ⊗ z + ( x ⊗ y ) ≺ z + ( x ≺ y ) ⊗ z + ( x ≺ y ) ≺ z, = x ⊗ y ⊗ z + x ⊗ ( y ≺ z ) + ( x ≺ y ) ⊗ z + ( x ≺ y ) ≺ z, ◦ ( y ◦ z ) = x ◦ ( y ⊗ z + y ≺ z ) , = x ⊗ y ⊗ z + x ⊗ ( y ≺ z ) + x ≺ ( y ⊗ z ) + x ≺ ( y ≺ z ) , = x ⊗ y ⊗ z + x ⊗ ( y ≺ z ) + [( x ≺ y ) ≺ z ) − x ≺ ( y ≺ z ) + ( x ≺ y ) ⊗ z ]+ x ≺ ( y ≺ z ) , hence the associativity of ◦ for any elements of degree 1. Suppose x := l an element ofdegree 1. Without restriction on the degree of y and z , we get:( l ◦ y ) ◦ z − l ◦ ( y ◦ z ) = ( l ≺ y ) ⊗ z + ( l ≺ y ) ≺ z − l ≺ ( y ⊗ z ) − l ≺ ( y ≺ z ) . Let us show that:( E ) : ( l ≺ y ) ⊗ z + ( l ≺ y ) ≺ z = l ≺ ( y ⊗ z ) + l ≺ ( y ≺ z ) , holds. If the degree of y is one without any assumptions on the degree of z then ( E )holds. We suppose ( E ) holds for any y of degree lower than a fixed n and set y := l ′ ⊗ y ′ .( l ≺ y ) ⊗ z + ( l ≺ y ) ≺ z = [( l ≺ l ′ ) ≺ y ′ ] ⊗ z − [ l ≺ ( l ′ ≺ y ′ )] ⊗ z + ( l ≺ l ′ ) ⊗ y ′ ⊗ z +[( l ≺ l ′ ) ≺ y ′ ] ≺ z − [ l ≺ ( l ′ ≺ y ′ )] ≺ z + [( l ≺ l ′ ) ⊗ y ′ ] ≺ z, = [( l ≺ l ′ ) ≺ y ′ ] ⊗ z − [ l ≺ ( l ′ ≺ y ′ )] ⊗ z + ( l ≺ l ′ ) ⊗ y ′ ⊗ z +[( l ≺ l ′ ) ≺ y ′ ] ≺ z − [ l ≺ ( l ′ ≺ y ′ )] ≺ z + ( l ≺ l ′ ) ⊗ ( y ′ ≺ z ) .l ≺ ( y ≺ z ) + l ≺ ( y ⊗ z ) = l ≺ (( l ′ ⊗ y ′ ) ≺ z ) + l ≺ ( l ′ ⊗ y ′ ⊗ z ) , = l ≺ ( l ′ ⊗ ( y ′ ≺ z )) + l ≺ ( l ′ ⊗ y ′ ⊗ z ) , = ( l ≺ l ′ ) ≺ ( y ′ ≺ z ) − l ≺ ( l ′ ≺ ( y ′ ≺ z )) + ( l ≺ l ′ ) ⊗ ( y ′ ≺ z )+( l ≺ l ′ ) ≺ ( y ′ ⊗ z ) − l ≺ ( l ′ ≺ ( y ′ ⊗ z )) + ( l ≺ l ′ ) ⊗ y ′ ⊗ z. Gathering terms gives: E := ( l ≺ y ) ⊗ z + ( l ≺ y ) ≺ z − l ≺ ( y ⊗ z ) − l ≺ ( y ≺ z ) , = [( l ≺ l ′ ) ≺ y ′ ] ⊗ z + [( l ≺ l ′ ) ≺ y ′ ] ≺ z − ( l ≺ l ′ ) ≺ ( y ′ ≺ z ) − ( l ≺ l ′ ) ≺ ( y ′ ⊗ z ) −{ [ l ≺ ( l ′ ≺ y ′ )] ⊗ z + [ l ≺ ( l ′ ≺ y ′ )] ≺ z − l ≺ ( l ′ ≺ ( y ′ ≺ z )) − l ≺ ( l ′ ≺ ( y ′ ⊗ z )) . } The first row vanishes because of the induction hypothesis. The second row vanishes toobecause the degree of ( l ′ ≺ y ′ ) is the same than the degree of y ′ . Apply now the inductionhypothesis twice:[ l ≺ ( l ′ ≺ y ′ )] ⊗ z + [ l ≺ ( l ′ ≺ y ′ )] ≺ z = l ≺ [( l ′ ≺ y ′ ) ⊗ z ] + l ≺ [( l ′ ≺ y ′ ) ≺ z ] , use induction, = l ≺ [( l ′ ≺ y ′ ) ⊗ z + ( l ′ ≺ y ′ ) ≺ z ] , = l ≺ [ l ′ ≺ ( y ′ ⊗ z ) + l ′ ≺ ( y ′ ≺ z )] , use again induction. x is one, the following,( x ◦ y ) ◦ z = x ◦ ( y ◦ z )holds for any y, z ∈ As ( L ). Suppose x of degree greater than 2. Set x := x ′ ⊗ l , l ∈ L .We get: ( x ◦ y ) ◦ z = (( x ′ ⊗ l ) ◦ y ) ◦ z, = (( x ′ ◦ l ) ◦ y ) ◦ z, = ( x ′ ◦ ( l ◦ y )) ◦ z, = x ′ ◦ (( l ◦ y ) ◦ z ) , = x ′ ◦ ( l ◦ ( y ◦ z )) , since degree l = 1 , = ( x ′ ◦ l ) ◦ ( y ◦ z ) , = x ◦ ( y ◦ z ) , hence the associativite of ◦ . For ◦ we proceed similarly. As ◦ is associative, As ( L ) isa triplicial-algebra. (cid:3) Theorem 5.6
Let V be a K -vector space. Then As ( L ( V )) is the free triplicial-algebraover V .Proof: The generating function of the operad
T rip is given by: f T rip ( x ) = 2 √ x sin( 13 asin( r x − . As the generating function of the operad As is f As ( x ) := x − x , the computation f − As ◦ f T rip gives the generating function of the L-operad. Indeed, dealing with the generatingfunctions of operads, we know from Koszulity of the L-operad that f L ! ( − f L ( − x )) = x .As, f L ! ( x ) = x + 2 x + x = x ( x + 1) , we get: f L ( x )( f L ( x ) − = x, that is, f L ( x ) x = 1( f L ( x ) − . But f T rip ( x ) = q f L ( x ) x − f − As ◦ f T rip ( x ) = q f L ( x ) x − q f L ( x ) x . Therefore, f − As ◦ f T rip ( x ) = f L ( x ) and for all n >
0, dim
T rip ( n ) = dim ( As ◦ L)( n ). Let V be a K -vector space. The usual inclusion map V ֒ → As ( L ( V )) induces a unique triplicial-morphism T rip ( V ) → As ( L ( V )) which turns out to be surjective by construction. As f T rip = f As ◦ f L , this map is an isomorphism. Hence the claim. (cid:3) Remark:
This theorem allows to code even trees via forests of symmetric ternary treesin a bijective way or with the help of words made out with 1 and ¯2.32 .6 Commutative
T rip -algebras
Let T be a T rip -algebra. Define new operations • op , • op , • op by, x • op y := y • x, x • op y := y • x, x • op y := y • x, for all x, y ∈ T . The K -vector space T equipped with these three new operations is a new T rip -algebra, called the opposite
T rip -algebra denoted by T op . A T rip -algebra is said tobe commutative when it coincides with its opposite structure. Therefore, a commutative
T rip -algebra T is a K -vector space equipped with two associative binary operations • and ⋆ , verifying: x • y = y • x ;( x • y ) ⋆ z = x • ( y ⋆ z ); x ⋆ y ⋆ z = x ⋆ z ⋆ y. Observe that ( T, • ) is a commutative associative algebra and that ( T, ⋆ ) is a permutativealgebra. The operad
P erm of permutative algebras is the Koszul dual of the right Pre-Lieoperad and was introduced in [4]. The operad of commutative algebras are denoted by
ComT rip . Observe that there exit three canonical functors:
ComT rip → Com, ComT rip → P erm → As, ComT rip → L . Let V be a K -vector space. Denote by Com ( V ) the free associative commutative algebraover V and by U Com ( V ) = K ⊕ Com ( V ) , its augmented version. Denote by ∨ its usualcommutative associative product. Proposition 5.7
Let ( A, ∗ ) be a commutative associative algebra. Then, A ⊗ U Com ( A ) equipped with the following operations, ( a ⊗ a ∨ . . . ∨ a n ) • ( a ′ ⊗ a ′ ∨ . . . ∨ a ′ m ) := [ a ∗ a ′ ] ⊗ a ∨ . . . ∨ a n ∨ a ′ . . . ∨ a ′ m , ( a ⊗ a ∨ . . . ∨ a n ) • ( a ′ ⊗ a ′ ∨ . . . ∨ a ′ m ) := a ⊗ a ∨ . . . a n ∨ a ′ ∨ a ′ ∨ . . . ∨ a ′ m . where a, a ′ ∈ A and a i , a ′ i ∈ U Com ( A ) , is a commutative T rip -algebra.Proof:
Straightforward. (cid:3)
Theorem 5.8
The free commutative
T rip -algebra over V is the K -vector space: ComT rip ( V ) := Com ( V ) ⊗ U Com ( Com ( V )) , equipped with the following binary operations: ( ω ⊗ ω ∨ . . . ∨ ω n ) ∨ ( ξ ⊗ ξ ∨ . . . ∨ ξ m ) := ω ⊗ ω ∨ . . . ∨ ω n ∨ ξ ∨ ξ ∨ . . . ∨ ξ m . ( ω ⊗ ω ∨ . . . ∨ ω n ) • ( ξ ⊗ ξ ∨ . . . ∨ ξ m ) := [ ω ∨ ξ ] ⊗ ω ∨ . . . ∨ ω n ∨ ξ ∨ . . . ∨ ξ m , where ω, ξ ∈ Com ( V ) and ω i , ξ i ∈ U Com ( V ) . Its generating function is: f ComTrip ( x ) = ( exp ( x ) − exp ( exp ( x ) −
1) = x + 3 x
2! + 10 x
3! + 37 x
4! + 151 x
5! + · · · . roof: Observe that equipped with such operations,
ComT rip ( V ) is a commutative T rip -algebra. Let i : V ֒ → V ⊗ K ֒ → Com ( V ) ⊗ U Com ( Com ( V )) be the usual inclusionmap defined by i ( v ) := v ⊗ K . Let ( A, • , ⋆ ) be another commutative T rip -algebra and f : V → A a linear map. Define ˜ f : ComT rip ( V ) → A by induction as follows:˜ f ( ω ⊗ ω ∨ . . . ∨ ω n ) := ˜ f ( ω ) ⋆ ˜ f ( ω ) ⋆ . . . ⋆ ˜ f ( ω n ) , and if ω := v ∨ . . . ∨ v m ,˜ f ( ω ⊗ K ) := ˜ f ( v ∨ . . . ∨ v m ) := f ( v ) • . . . • f ( v m ) . If X and Y are monomials of ComT rip ( V ), the relation ˜ f ( X ∨ Y ) = ˜ f ( X ) ⋆ ˜ f ( Y ) holdsby construction. If X := ω ⊗ K and Y := ξ ⊗ K , then ˜ f ( X • Y ) = ˜ f ([ ω ∨ ξ ] ⊗ K ) =˜ f ( X ) • ˜ f ( Y ) by construction. This will be helpful in the next computation. Set now X := ω ⊗ X ′ and Y := ξ ⊗ Y ′ , where X ′ and Y ′ are monomials of Com ( Com ( V )). Weget:˜ f ( X ) • ˜ f ( Y ) = ( ˜ f ( ω ) ⋆ ˜ f ( X ′ )) • ( ˜ f ( ξ ) ⋆ ˜ f ( Y ′ )) = [( ˜ f ( ω ) ⋆ ˜ f ( X ′ )) • ˜ f ( ξ )] ⋆ ˜ f ( Y ′ ) = [ ˜ f ( ξ ) • ( ˜ f ( ω ) ⋆ ˜ f ( X ′ ))] ⋆ ˜ f ( Y ′ ) = [( ˜ f ( ξ ) • ˜ f ( ω )) ⋆ ˜ f ( X ′ )] ⋆ ˜ f ( Y ′ ) = [ ˜ f ( ξ • ω ) ⋆ ˜ f ( X ′ )] ⋆ ˜ f ( Y ′ ) =[ ˜ f ( ω • ξ ) ⋆ ˜ f ( X ′ )] ⋆ ˜ f ( Y ′ ) = ˜ f ( ω • ξ ) ⋆ ˜ f ( X ′ ) ⋆ ˜ f ( Y ′ ) = ˜ f ([ ω • ξ ] ⊗ X ′ ∨ Y ′ ) = ˜ f ( X • Y ) . We usetwice the entanglement relation between • and ⋆ and the fact that • is commutative. Wehave shown that ˜ f is the only ComT rip -algebra morphism which extends f . Therefore, ComT rip ( V ) is the free K -vector space over V . The last claim is just a result concerningcomposition of functors (recall that f Com ( x ) := exp ( x ) − (cid:3) Remark:
Duplicial-algebras also admit an opposite structure and we can deals withcommutative duplicial-algebras. They are K -vector spaces equipped with an associativebinary operation which verify: xyz = xzy. Hence,
ComDup = P erm = ComDias, where
ComDias is the operad of commutative dialgebras [21].
Remark:
The previous theorem suggests the existence of a triple of operads (
P erm, ComT rip, Com )or (
N AP, ComT rip, Com ). Does they exist?
T rip -operad and triangular numbers
We go on our knowledge of
T rip -algebras and present here its dual. A quasi-nilpotent
T rip -algebra,
QN T rip -algebra for short, Q is K -vector space equipped with three binaryoperations • i , i = 1 , , , verifying: ∀ ≤ i ≤ j ≤ , ( x • i y ) • j z = x • i ( y • j z ) ,i > j ⇒ ( x • i y ) • j z = 0 ,i > j ⇒ x • i ( y • j z ) = 0 , for all x, y, z ∈ Q . Therefore, there is a functor QNTrip → Trip .34 heorem 5.9
Let V be a K -vector space and U As ( V ) the augmented free associativealgebra. Consider the K -vector space QN T Rip ( V ) := U As ( V ) ⊗ [ V ⊗ U As ( V )] ⊗ U As ( V ) , equipped with the following three binary operations, ( ω ⊗ [ v ⊗ ξ ] ⊗ θ ) • ( ω ′ ⊗ [ v ′ ⊗ ξ ′ ] ⊗ θ ′ ) = Υ( ξ ⊗ θ ) ( ωvω ′ ) ⊗ [ v ′ ⊗ ξ ′ ] ⊗ θ ′ , ( ω ⊗ [ v ⊗ ξ ] ⊗ θ ) • ( ω ′ ⊗ [ v ′ ⊗ ξ ′ ] ⊗ θ ′ ) = Υ( θ ⊗ ω ′ ) ω ⊗ [ v ⊗ ( ξv ′ ξ ′ )] ⊗ θ ′ , ( ω ⊗ [ v ⊗ ξ ] ⊗ θ ) • ( ω ′ ⊗ [ v ′ ⊗ ξ ′ ] ⊗ θ ′ ) = Υ( ω ′ ⊗ ξ ′ ) ω ⊗ [ v ⊗ ξ ] ⊗ ( θvθ ′ ) , where Greek letters denote elements from U As ( V ) , v, v ′ ∈ V and Υ :
U As ( V ) ⊗ U As ( V ) → K is the canonical projection map. Then, QN T Rip ( V ) is the free QN T Rip -algebra over V . The dimension of QN T Rip n is the triangular number n ( n +1)2 and its generating func-tion is, f QNT rip ( x ) = x (1 − x ) = X n> n ( n + 1)2 x n . Proof:
The K -vector space QN T Rip ( V ) equipped with these three operations is a QN T rip -algebra. Let i : V ֒ → K ⊗ ( V ⊗ K ) ⊗ K be the expected inclusion map. Let ( A, • , • , • )be another QN T rip -algebra and f : V → A be a linear map. Define ˜ f : QN T Rip ( V ) → A inductively as follows: ˜ f (1 K ⊗ [ v ⊗ K ] ⊗ K ] := f ( v );˜ f ( ω ⊗ [ v ⊗ ξ ] ⊗ θ ) := ˜ f ( ω ) • f ( v ) • ˜ f ( ξ ) • ˜ f ( θ ) , ˜ f ( v ⊗ . . . ⊗ v n ) := f ( v ) • k . . . • k f ( v n ) , where k = 1 , , ρ := v ⊗ . . . ⊗ v n belongs to the first, secondor third copy of As ( V ). We understand this definition as follows: If ω = 1 K (resp. ξ = 1 K ,resp. θ = 1 K ) then ˜ f ( ω ) • (resp. • ˜ f ( ξ ), resp. • ˜ f ( θ )) vanishes in the right hand side ofthe middle equation. It is not hard to see that ˜ f so defined is the unique QN T rip -algebramorphism extending f . Therefore, QN T Rip ( V ) is the free QN T Rip -algebra over V . Forthe last claim, recall that the generating function of the operad U As is f UAs ( x ) = − x . (cid:3) Anti-cyclic operads allow to deal with invariant antisymmetric bilinear maps. The readershould read [10, 26] for the theory and [5] for examples. We show that such maps cannotexist on L-algebras but do exist on the operad [ n ] − M ag , ( n > P be a regularbinary and quadratic operad and A be a P -algebra. An invariant antisymmetric bilinearmap on A with values in some vector space V , h · ; · i : A ⊗ → V is by definition,35. A collection of a map τ n : P n → P n , n ≥ n + 1 verifying, h γ ( x , . . . , x n ); x n +1 i = h τ n ( γ )( x , . . . , x n +1 ); x i , for any generating operation γ ∈ P n ,2. And a map τ : ( P ≃ K ) → K defined by 1 K
7→ − K . Theorem 6.1
Let A be a L -algebra. There exists no invariant antisymmetric bilinearmap on A .Proof: Suppose there exists an invariant antisymmetric bilinear map on A . Then, thereexists a map τ : L → L of order 3. As L = K {≺} ⊕ K {≻} , τ is a two by two matrixwritten in this basis as, τ = (cid:18) a bc d (cid:19) . As the L-operad is binary, the maps τ n for n >
2, if they exist, are built from τ (andthus are unique). We now show that such a τ does not exist. We compute,0 = h x ≻ ( y ≺ z ) − ( x ≻ y ) ≺ z ; t i = h τ ( ≻ )[( y ≺ z ) , t ]; x i − h τ ( ≺ )[ z, t ]; ( x ≻ y ) i , = h c ( y ≺ z ) ≺ t + d ( y ≺ z ) ≻ t ; x i + h ( x ≻ y ); τ ( ≺ )[ z, t ] i , = h c ( y ≺ z ) ≺ t + d ( y ≺ z ) ≻ t ; x i + h τ ( ≻ )[ y ; τ ( ≺ )[ z, t ]]; x i , = h c ( y ≺ z ) ≺ t + d ( y ≺ z ) ≻ t + ac y ≺ ( z ≺ t )+ bc y ≺ ( z ≻ t ) + ad y ≻ ( z ≺ t ) + bd y ≻ ( z ≻ t ); x i . Hence, c and d have to vanish. Now, τ cannot be of order 3. (cid:3) Remark:
The same proof holds for
T rip -algebras. Let us focus on the operad [ n ] − M ag , n >
1. Set, C = (cid:18) − − (cid:19) , be the matrix used in [5]. This matrix is of order 3. If n := 2 p , then construct τ as adiagonal block of p matrices C or its transpose. If n := 2( p −
1) + 3, then construct τ asa diagonal block of p − C or its transpose and the following matrix used onetime: P erm = . The map τ : [ n ] − M ag → [ n ] − M ag is thus of order three. As the operad [ n ] − M ag is binary, the other maps τ n , n >
2, will be uniquely determined by τ and τ . One couldhave also used the following map, τ = (cid:18) C Id n − (cid:19) . Hence, the following holds. 36 heorem 6.2
For a fixed τ , there exits a unique collection of maps τ n , n > , extending τ and τ and turning the regular operad [ n ] − M ag , n > , into an anticyclic operad. Remark:
For the case n = 1, so is the operad [1] − M ag := M ag , if we assume theexistence of an element j ∈ K of order three (take for instance K = C and a solution of1 + x + x = 0.). Acknowledgments:
The author is indebted to E. Deutsch for sending him papers [7, 8]and to J.-L. Loday for his comments and for improving Proposition 2.9.
References [1] M.
Aguiar and J.-L.
Loday . Quadri-algebras.
J. Pure Applied Algebra , 191:205–221, 2004.[2] M.
Bousquet and C.
Lamathe . On symmetric structures of order two.
DiscreteMath. and Theoretical Computer Science , 2006.[3] A.
Brouder and A.
Frabetti . QED Hopf algebras on planar binary trees.
J.Algebra , 267(1):298–322, 2003.[4] F.
Chapoton . Un endofoncteur de la th´eorie des op´erades.[5] F.
Chapoton . On some anticyclic operads.
Algebraic, Geometric and Topology ,5:53–69, 2005.[6] F.
Chapoton . The anticyclic operad of moulds.
International Mathematics ResearchNotices , 2007:36p, 2007.[7] E.
Deutsch , D.
Callan , S.
Cautis , and
Southwest Missouri ProblemsGroup . Another path to generalized Catalan numbers: 10751.
The American Math.Monthly , 108(9):872–873, 2001.[8] E.
Deutsch , S.
Feretic , and M.
Noy . Diagonally convex directed polyominoesand even trees: a bijection and related issues.
Discrete Math. , 256:645–654, 2002.[9] M.
Gerstenhaber and J.D.
Stasheff (Eds.).
Deformation theory and quantumgroups with applications to mathematical physics , volume 134. Proceedings of anAMS-IMS-SIAM 1990 joint summer research conference, Contemporary Mathemat-ics, 1992.[10] E.
Getzler and M.
Kapranov . Cyclic operads and cyclic homology.
Geome-try, Topology and Physics; Conf. Proc. Lecture Notes Geom. Topology, IV, Internat.Press, Cambridge, MA , pages 167–201, 1995.[11] V.
Ginzburg and M.
Kapranov . Koszul duality for operads.
Duke Math. J. 76(1994) 203–272 . 3712] R.
Holtkamp . On Hopf algebra structures over free operads.
Adv. in Maths ,207:544–565, 2006.[13] R.
Holtkamp , J.-L.
Loday , and M.
Ronco . Coassociative magmatic bialgebrasand the fine numbers. arXiv:math.RA/0609125 .[14] Ph.
Leroux . An algebraic framework of weighted directed graphs.
Int. J. Math.Math. Sci. , 58, 2003.[15] Ph.
Leroux . Ennea-algebras.
J. Algebra , 281:287–302, 2004.[16] Ph.
Leroux . Free dendriform algebras: A parenthesis setting.
Int. J. Math. Math.Sci. , Part I18:1–16, 2006.[17] Ph.
Leroux . A simple symmetry generating operads related to rooted planar m -arytrees and polygonal numbers. J. Integer Sequences , 10, 2007. article 07.4.7.[18] M.
Livernet . A rigidity theorem for Pre-Lie algebras.
J.P.A.A. , 207:1–18, 2006.[19] J.-L.
Loday . Generalized bialgebras and triples of operads. arXiv:math.QA/0611885 .[20] J.-L.
Loday . La renaissance des op´erades.
Ast´erisque , 237:Exp. No. 792, 3, 47–74,1996.[21] J.-L.
Loday . Dialgebras.
Dialgebras and related operads, Lecture Notes in Math.Springer. , 1763:7–66, 2001.[22] J.-L.
Loday . Arithmetree.
J. Algebra , 258:275–309, 2002.[23] J.-L.
Loday . Scindement d’associativit´e et alg`ebres de Hopf.
Actes des journ´eesmath´ematiques `a la m´emoire de Jean Leray, Nantes (2002), S´eminaire et Congr`es(SMF) , 9:155–172, 2004.[24] J.-L.
Loday and M.
Ronco . Order structure on the algebra of permutations andof planar binary trees.
Journal of Algebraic Combinatorics , 15(3):253–270, 2002.[25] J.-L.
Loday and M.
Ronco . On the structure of cofree Hopf algebras.
J. reineangew. Math. , 592:123–155, 2006.[26] M.
Markl . Cyclic operads and homology of graph complexes.