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On the algebras obtained by tensor product. 7 0 0 Elisabeth REMM ∗- Michel GOZE † 2 n a Université deHaute Alsace, F.S.T. J 4, ruedes Frères Lumière - 68093 MULHOUSE- France 5 2 ] 2000 Mathematics Subject Classification. Primary 18D50, 17A30, 17Bxx, Secondary A 17D25. R . Keywords. Tensor product of algebras on quadratic algebras. h t Abstract a m Let P be a quadratic operad. Wedefine an associated operad P˜ such that for any P-algebraAandP˜-algebraB,thealgebraA⊗BisalwaysaP-algebrafortheclassical [ tensor product. 3 v 1 Introduction 5 0 1 LetP be aquadraticoperadwithonegeneratingoperation(i.e. the algebrasonthisoperad 6 have only one operation) and P! its dual operad. It satisfies P! = hom(P,Lie) where Lie 0 is the quadratic operadcorrespondingto Lie algebras. For anyP-algebraA and P!-algebra 6 0 B, the vector space A⊗B is naturally provided with a Lie algebra product / h µ(a1⊗b1,a2⊗b2)=µA(a1,a2)⊗µB(b1,b2)−µA(a2,a1)⊗µB(b2,b1). (1) t a where µA (resp. µB) is the multiplication of A (resp. B). We deduce that the ”natural” m tensor product µA⊗B =µA⊗µB provides A⊗B with a Lie-admissible algebra structure. : In[2]wehavedefinedspecialclassesofLie-admissiblealgebraswithrelationsofdefinition v i determinedbyanactionofthesubgroupsGi ofthe3-degreesymmetricgroupΣ3. Weobtain X quadraticoperads,denotedbyG −AssandinthisfamilywefindoperadsofLie-admissible, i r associative, Vinberg and pre-Lie algebras. For these operads we have proved that for every a P-algebraA andP!-algebraB the tensorproductA⊗B is aP-algebra. This is nottrue for general nonassociative algebras and, in this sense, the G -associative algebras are the most i regular kind of nonassociative algebras. For example if P is the operad of Leibniz algebras orofthe nonassociativealgebrasassociatedtoPoissonalgebras[?],thenthe tensorproduct of a P-algebra and P!-algebra is not a P-algebra. So we introduce a quadratic operad, denoted by P˜, such that the tensor product of a P-algebra with a P˜-algebra is a P-algebra. In case of P =Lie or G -Ass then P˜ =P! this i explain the above remarks. ∗correspondingauthor: e-mail: [email protected] †[email protected]. 1 2 Nonassociative algebras and operads WeassumeinthisworkthatP isaquadraticoperadwithonegeneratingoperation. Thenit is defined from the free operad Γ(E) generatedby a Σ -module E placed in arity 2, and an 2 ideal (R) generated by a Σ -invariant subspace R of Γ(E)(3). Our hypothesis implies that 3 the Σ -module E is generated by one element (i.e. algebras over this operad are algebras 2 withoneoperation). Recallthat,ifweconsideranoperationwithnosymmetry,theK-vector spaceΓ(E)(2)is2-dimensionalwithbasis{x ·x ,x ·x }andΓ(E)(3)isthe12-dimensional 1 2 2 1 K-vectorspacegeneratedby{x ·(x ·x ),(x ·x )·x }withi6=j 6=k 6=i, i,j,k∈{1,2,3}. i j k i j k We have a natural action of Σ on Γ(E)(3) given by: 3 Σ × Γ(E)(3) →Γ(E)(3) 3 (σ , X) 7→σ(X) where σ(xi·(xj ·xk))=xσ−1(i)·(xσ−1(j)·xσ−1(k)) σ((xi·xj)·xk)=(xσ−1(i)·xσ−1(j))·xσ−1(k)). We denote by O(X) the orbit of X associated with this action and by K(O(X)) the Σ - 3 invariantsubspaceofΓ(E)(3)generatedbyO(X). Moregenerally,ifX ,··· ,X arevectors 1 k in Γ(E)(3), we denote by K(O(X ,..,X )) the Σ -invariant subspace of Γ(E)(3) generated 1 k 3 by O(X )∪···∪O(X ). 1 k Definition 1 We say that a Σ -invariant subspace F of Γ(E)(3) is of rank k if there exists 3 X ,...,X ∈ Γ(E)(3) linearly independent such that F = K(O(X ,...,X )) and for every 1 k 1 k p<k and Y ,...Y ∈F we have K(O(Y ,...,Y ))6=F. 1 p 1 p IfP is aquadraticoperadwithone generatingoperation,its module ofrelationsR isan invariantsubspace of Γ(E)(3). We will say that P is of rank k if and only if R is of rank k. FromtheactionofΣ onΓ(E)(3)wedefinelinearmapsonthismoduleasfollows. LetK[Σ ] 3 3 bethegroupalgebraofΣ thatisthevectorspaceofallfinitelinearcombinationsofelements 3 of Σ with coefficients in K hence of all elements of the form v =a σ +a σ +...+a σ . 3 1 1 2 2 6 6 If v =Σa σ ∈K[Σ ], let Ψ be given by i i 3 v Ψ (X)=Σa σ (X). v i i Then an invariant subspace F of Γ(E)(3) is stable for every Ψ . v Let (A,µ) be a P-algebra. This means that (A,µ) is a nonassociative algebra (by nonassociative algebra we mean algebras with non necessarily associative multiplication). We consider the maps AL(µ)= µ◦(µ⊗Id) and AR(µ)=µ◦(Id⊗µ). Then the associator of µ is written A(µ) = AL(µ)−AR(µ). For each vector v ∈ K[Σ ] we define a linear map on 3 A⊗3 denoted by ΦA and given by v ΦA :A⊗3 → A⊗3 v (x1⊗x2⊗x3) 7→ ai(xσ−1(1)⊗xσ−1(2)⊗xσ−1(3)) P The multiplication µ satisfies relations of the following type AL(µ)◦ΦAv −AR(µ)◦ΦAv′ =0, (2) 2 where v =Σa σ ,v′ =Σa′σ ∈K[Σ ]. i i i i 3 Sucha relationdefines the module R of relationsofP. Infact R is the Σ -submodule of 3 Γ(E)(3) is generated by the vectors σj(Ψv(x1·(x2·x3))−Ψv′((x1·x2)·x3)) for every σ ∈Σ . j 3 Let us note that if P is of rank 1, a P-algebra is given by one multiplication satisfying only one relation of type (2). Proposition 2 Let P be a quadratic operad with one generating operation such that the Σ -submodule R of relations is generated by vectors of the following type : 3 Σ6 alσ ((x ·x )·x −x ·(x ·x )) i=1 i i 1 2 3 1 2 3 for l=1,...,k and σ ∈Σ . Then P is of rank 1. i 3 Proof. In fact, the Σ -invariant subspace of Γ(E)(3) generated by 3 ((x ·x )·x −x ·(x ·x )) 1 2 3 1 2 3 is isomorphic to K[Σ ]. This isomorphism is given by: 3 Σa σ ((x ·x )·x −x ·(x ·x ))−→Σa σ . i i 1 2 3 1 2 3 i i We have seen in [3] that for every Σ -invariant subspace F of K[Σ ], there is a vector 3 3 v ∈ K[Σ ] such that F = F = K(O(v)) where O(v) is the orbit of v corresponding to the 3 v natural action of Σ on K[Σ ]. We deduce that the rank is 1. 3 3 In the following examples we will recall the definition of the operads G -Ass and define i some quadratic operads of rank one associated to some classes of nonassociative algebras. Examples. 1. The operads G −Ass i For i,j,k ∈ {1,2,3} and i 6= j 6= k 6= i we denote by τ the transposition (i,j) and c ,c ij 1 2 the cycles (1,2,3) and (1,3,2). The subgroups of Σ are G = {Id},G =< τ >,G =< 3 1 2 12 3 τ >,G =< τ >, G =< c >, G = Σ where < g > denotes the subgroup generated 23 4 13 5 1 6 3 by g. Eachone ofthese subgroupsG defines aninvariantsubmodule R of Γ(E)(3) of rank i i 1. In fact consider the vector X = x ·(x ·x )−(x ·x )·x of Γ(E)(3) and if G is a 1 2 3 1 2 3 i subgroup of Σ , we define the vector X of Γ(E)(3) by 3 i X = (−1)ǫ(σ)σ(X) i σX∈Gi where ǫ(σ) is the sign of the permutation σ. Let R be the subspace R =K(O(V )). Then i i i 3 R1 = VectK{(xi·xj)·xk−xi·(xj ·xk)}, R2 = VectK{(xi·xj)·xk−xi·(xj ·xk)−(xj ·xi)·xk+xj ·(xi·xk)}, R3 = VectK{(xi·xj)·xk−xi·(xj ·xk)−(xi·xk)·xj +xi·(xk·xj)}, R4 = VectK{(xi·xj)·xk−xi·(xj ·xk)−(xk·xj)·xi+xk·(xj ·xi)} R5 = VectK{(xi·xj)·xk−xi·(xj ·xk)+(xj ·xk)·xi−xj ·(xk·xi) +(x ·x )·x −x ·(x ·x ) k i j k i j R6 = VectK{(x1·x2)·x3−x1·(x2·x3)+(x2·x3)·x1−x2·(x3·x1) +(x ·x )·x −x ·(x ·x )−(x ·x )·x +x ·(x ·x )−(x ·x )·x 3 1 2 3 1 2 2 1 3 2 1 3 3 2 1 +x ·(x ·x )−(x ·x )·x +x ·(x ·x )} 3 2 1 1 3 2 1 3 2 Definition 3 ThequadraticoperadG -AssisthequadraticoperadΓ(E)/(R)where(R)(3)= i R . i Some of these operads are wellknown: - G -Ass=Ass, 1 - G -Ass=Vinb, 2 - G -Ass=Pre−Lie. 3 Let us note that the G -Ass-algebras are the Lie-admissible algebras that is if µ is the 6 product of such an algebra then [x,y]=µ(x,y)−µ(y,x) is a product of Lie algebra. As G is a subgroup of G = Σ any G -Ass-algebra is Lie- i 6 3 i admissible. For a general study of the operad G -Ass = LieAdm and G -Ass = Vinb 6 2 see [2]. Let (G -Ass)! be the quadratic dual operad of G -Ass. We denote by (R )! the sub- i i i module of Γ(E)(3) defining (G -Ass)!. It is the orthogonal of R with respect to the inner i i product on Γ(E)(3) given by 1 2 3 1 2 3 hi·(j·k),i·(j·k)i=ǫ , h(i·j)·k,(i·j)·ki=−ǫ (cid:18) i j k (cid:19) (cid:18) i j k (cid:19) where ǫ(σ) is the sign of the permutation σ. Then we have (R )! = R 1 1 (R2)! = VectK{(xi·xj)·xk−xi·(xj ·xk);(xi·xj)·xk−(xj ·xi)·xk} (R3)! = VectK{(xi·xj)·xk−xi·(xj ·xk);(xi·xj)·xk)−(xi·xk)·xj} (R4)! = VectK{(xi·xj)·xk−xi·(xj ·xk);(xi·xj)·xk−(xk·xj)·xi} (R5)! = VectK{(xi·xj)·xk−xi·(xj ·xk);(xi·xj)·xk−(xj ·xk)·xi} (R6)! = VectK{(xi·xj)·xk−xi·(xj ·xk);(xi·xj)·xk−(xj ·xi)·xk (x ·x )·x −(x ·x )·x } i j k i k j Proposition 4 For i = 1, the operad (G -Ass)! = Ass is if rank 1. For 2 ≤ i ≤ 6, the 1 operads (G -Ass)! are of rank 2. i 4 Proof. The case i=1 is trivial (it is also a consequence of Proposition2). Fori=2,3,4and5,therankof(R )! is2. Wedenotebyvj,j =1,2thegeneratorsof(R )!. i i i Then if B is a (Gi-Ass)!-algebra the multiplication µB satisfies 1) µB(µB⊗Id)−µB(Id⊗µB)=0 2) µB(µB⊗Id)−µB(Id⊗µB◦Φσi)=0 with σ =τ 2 12  σ3 =τ23  σ4 =τ13 σ =c or c 5 1 2 For i=6, the space (R )! is generated by the vectors 6 (x ·x )·x −x ·(x ·x ), (x ·x )·x −(x ·x )·x , (x ·x )·x −(x ·x )·x . i j k i j k i j k j i k i j k i k j But we can write (x ·x )·x −(x ·x )·x =(Id−τ )((x ·x )·x ) i j k j i k ij i j k and (x ·x )·x −(x ·x )·x =(Id−τ )((x ·x )·x ). i j k i k j jk i j k The Σ -invariant subspace of K[Σ ] generated by the vectors Id−τ and Id−τ is of 3 3 12 23 dimension 5, and from the classification [3], this space corresponds to F =K(O)(v) with v v =2Id−τ −τ −τ +c 12 13 23 1 and we deduce that this operad is of rank 2. 2. The 3-power associative algebras We have seen that every (G -Ass)-algebra is Lie-admissible. Moreoverthe operad LieAdm i is quadratic, of rank 1. The submodule R is of dimension 1 and corresponds to the one- 6 dimensional Σ -invariant subspace K(O(V))=F of K[Σ ], where V is given by 3 V 3 V = (−1)ǫ(σ)σ. σX∈Σ3 If we consider the natural action of Σ on K[Σ ], then there exists only two irreducible 3 3 invariant one dimensional subspaces of K[Σ ] that is F and F where 3 V W W = σ. σX∈Σ3 If the set of Lie-admissible is associated to F , the set of algebras corresponding to F V W is the set of 3-power associative algebras (see [3]) that is which satisfies x2·x = x·x2 for every x. As in the Lie-admissible case we can define classes of 3-power associative algebras 5 corresponding to the action of the subgroups of Σ . We are conduced to consider the 3 following submodules of Γ(E)(3): Rp3 = R 1 1 R2p3 = VectK{(xi·xj)·xk−xi·(xj ·xk)+(xj ·xi)·xk−xj ·(xi·xk)} R3p3 = VectK{(xi·xj)·xk−xi·(xj ·xk)+(xi·xk)·xj −xi·(xk·xj)} R4p3 = VectK{(xi·xj)·xk−xi·(xj ·xk)+(xk·xj)·xi−xk·(xj ·xi)} Rp3 = R 5 5 R6p3 = VectK{(x1·x2)·x3−x1·(x2·x3)+(x2·x3)·x1−x2·(x3·x1)+(x3·x1)·x2 −x ·(x ·x )+(x ·x )·x −x ·(x ·x )+(x ·x )·x −x ·(x ·x ) 3 1 2 2 1 3 2 1 3 3 2 1 3 2 1 +(x ·x )·x −x ·(x ·x )} 1 3 2 1 3 2 This corresponds to Rp3 =K(O(Y )) with i i Yi =Σσ∈Giσ(X) with X =x ·(x ·x )−(x ·x )·x . 1 2 3 1 2 3 We denote by (G −p3Ass) the corresponding quadratic operads. The corresponding dual i operads are discribed by the following ideals of relations: (Rp3)! = R 1 1 (R2p3)! = VectK{(xi·xj)·xk−xi·(xj ·xk);(xi·xj)·xk+(xj ·xi)·xk} (R3p3)! = VectK{(xi·xj)·xk−xi·(xj ·xk);(xi·xj)·xk)+(xi·xk)·xj} (R4p3)! = VectK{(xi·xj)·xk−xi·(xj ·xk);(xi·xj)·xk+(xk·xj)·xi} (Rp3)! = R! 5 5 (R6p3)! = VectK{(xi·xj)·xk−xi·(xj ·xk);(xi·xj)·xk+(xj ·xi)·xk (x ·x )·x +(x ·x )·x }. i j k i k j The proof is analogous to the Lie-admissible case. Let us note that these operads are also of rank 2 except for i=1. 3. The K[Σ ]-associative algebras 3 This example of nonassociative algebras generalizes the previous, considering not only the onedimensionalinvariantsubspaceofK[Σ ]butalltheinvariantsubspaces. Recallthat,for 3 everyv ∈K[Σ ],wehavedenotedbyO(v)thecorrespondingorbitandbyF =K(O(v))the 3 v linear subspace of K[Σ ] generated by O(v). Since F is a Σ -invariant subspace of K[Σ ], 3 v 3 3 by Mashke’s theorem, it is a direct sum of irreducible invariant subspaces. Moreover,given aninvariantsubspaceF ofK[Σ ],there existsv ∈K[Σ ](notnecessarilyunique)suchthat 3 3 F =F =K(O(v)). v Definition 5 1. AK-algebra(A,µ)iscalledK[Σ ]-associativeifthereexistsv ∈K[Σ ],v 6= 3 3 0, such that A(µ)◦ΦA =0 v where A(µ) is the associator of µ. 2. Let A(µ)◦ΦA =0 and A(µ)◦ΦA =0 be two identities satisfied by the algebra (A,µ). v w We say that these identities are equivalent if F =K(O(v))=F =K(O(w)). v w 6 Remark that if F is not an irreducible invariant subspace, then there exists w ∈ F such v v that F ⊂F and F 6=F . In this case the identity A(µ)◦ΦA =0 implies A(µ)◦ΦA =0 w v w v v w but these identities are not equivalent. Examples. 1. If v =Id−τ , the relation 23 A(µ)◦ΦA =0 (3) v becomes A(µ)(x ⊗x ⊗x −x ⊗x ⊗x )=0. 1 2 3 1 3 2 The corresponding algebra is a pre-Lie algebra. 2. The Lie-admissible and third-power associative algebras are K[Σ ]-associative algebras. 3 In fact an algebra (A,µ) is Lie-admissible if A(µ)◦ΦA = 0 and third-power associative if V A(µ)◦ΦA =0 with W V =Id−τ −τ −τ +c +c 12 23 13 1 2 and W =Id+τ +τ +τ +c +c . 12 23 13 1 2 3. For i=1,...,6 we denote by V and W the vectors of K[Σ ] given by i i 3 V = (−1)ǫ(σ)σ, W = σ. i i σX∈Gi σX∈Gi Then (A,µ) is a G -Ass-algebra if i A(µ)◦ΦA =0 Vi and a G -p3Ass-algebra if i A(µ)◦ΦA =0. Wi They are particular cases of K[Σ ]-associative algebras. We shall return, in the last 3 section, on the determination of the corresponding operads. ˜ 3 The operad P associated to a quadratic operad P In the previous sections we saw that for some quadratic operads, the dual operad gives a way to construct on the tensor product A⊗B of a P-algebra A and a P!-algebra B a structure of P-algebra for the usual tensor product µA⊗B = µA⊗µB. But this is not true for every quadratic operad. In this section we define from a given quadratic operad P an associated quadratic operad, denoted by P˜, whose fondamental property is to satisfy the above property on the tensor product. Let (A,µ) be a P-algebra where P is a quadratic operad. Let R be the submodule of Γ(E)(3) defining the relations of A. We denote by AL(µ) = µ◦(µ⊗Id) and AR(µ) = µ◦(Id⊗µ).Ifwesuppose thatRis ofrankk, thenthe multiplicationµsatisfiesk relations of type AL(µ)◦ΦA −AR(µ)◦ΦA =0 vi wi 7 wherevi,wi ∈K[Σ3]foranyi∈I ={1,..,k}andthevectors(vi)i∈I arelinearlyindependent as well as the vectors (wi)i∈I. Examples. 1. The associative algebras correspond to k = 1 and v = w = Id, pre-Lie algebras to 1 1 k = 1 and v = w = Id−τ and more generally G -associative algebras correpond to 1 1 23 i k =1 andv =w =V . The Lie-admissible algebrascorrepondto k =1,v =w =V and 1 1 i 1 1 the 3-power associative algebras to v =w =W. 1 1 2. The Leibniz algebras correspond to k =1 and v =Id−τ , w =Id. 1 23 1 LetP be a quadraticoperadgeneratedbyE ⊂K[Σ ]. For everyv =Σ6 Sa σ ∈K[Σ ]we 2 l=1 l l 3 consider on Γ(E)(3) the linear maps ΨL((x ·x )·x )=Σa σ ((x ·x )·x ), ΨL(x ·(x ·x ))=0 v i j k l l i j k v i j k and ΨR((x ·x )·x )=0, ΨR(x ·(x ·x ))=Σa σ (x ·(x ·x )). v i j k v i j k l l i j k Let R be the module of relations of P. If it is of rank k, it is written R=VectKn(ΨLvp((xi·xj)·xk)−ΨRwp(x1·(x2·x3)), 1≤p≤ko with v = 6 apσ and w = 6 bpσ for 1≤p≤k. p i=1 i i p i=1 i i Let E˜ be tPhe sub-module of K[ΣP] defined by 2 E if E =11⊕Sgn E˜ = 2 (cid:26) Com(2)=11 if E =11 or Sgn2 If dimE˜ =2, we denote R˜ the K[Σ ]-module generated by the vectors 3 apapΦL ((x ·x )·x ), i j σi−σj 1 2 3   bpibpjΦRσi−σj(x1·(x2·x3)), apbp(ΦL((x ·x )·x )−ΦR(x ·(x ·x )),  i j σi 1 2 3 σj 1 2 3 for 1≤p≤k. If E˜ =Comm(2), R˜ is generated also by these vectors,but modulo the relations of commu- tation. Definition 6 The operad P˜ associated to the quadratic operad P is the quadratic operad generated by E˜ and with K[Σ ]-submodule of relations R˜. 3 We have the main result : Theorem 7 Let A be a P-algebra and B a P˜-algebra. Then the algebra A⊗B with product µA⊗B is a P-algebra. 8 Proof. Let A be a P-algebra. Its multiplication µA satisfies (AL(µA)◦ΦAvp −AR(µA)◦ΦAwp)(x1⊗x2⊗x3)=0 for p=1,...,k. If B is a P˜-algebra,its multiplication µB satisfies AL(µB)◦ΦBσi−σj(y1⊗y2⊗y3)=0, if ∃p, aipajp 6=0   AR(µB)◦ΦBσi−σj(y1⊗y2⊗y3)=0, if ∃p, bipbjp 6=0  AL(µA)◦ΦBσi −AR(µB)◦ΦBσj(y1⊗y2⊗y3)=0, if ∃p, aipbjp 6=0. Now we consider the product µA⊗B. We have (AL(µA⊗B)◦ΦAvp⊗B−AR(µA⊗B)◦ΦAwp⊗B)(x1⊗y1⊗x2⊗y2⊗x3⊗y3) = (ΣapiAL(µA⊗B)◦σi−ΣbpiAR(µA⊗B)◦σi)(x1⊗y1⊗x2⊗y2⊗x3⊗y3) = Σapi(AL(µA)◦σi(x1⊗x2⊗x3)⊗AL(µB)◦σi(y1⊗y2⊗y3)) −Σbpi(AR(µA)◦σi(x1⊗x2⊗x3)⊗AR(µB)◦σi(y1⊗y2⊗y3)) = (Σapi(AL(µA)◦σi(x1⊗x2⊗x3))⊗AL(µB)◦σj(y1⊗y2⊗y3) −(Σbpi(AR(µA)◦σi(x1⊗x2⊗x3))⊗AR(µB)◦σj(y1⊗y2⊗y3)) where j is choosen in {1,···6} such that ap 6=0, j = ΣapiAL(µA)◦σi(x1⊗x2⊗x3)−ΣbpiAR(µA)◦σi(x1⊗x2⊗x3)) ⊗AR(µB)◦σj(y1⊗y2⊗y3)) = (AL(µA)◦ΦAvp −AR(µA)◦ΦAwp)(x1⊗x2⊗x3))⊗AR(µB)◦σj(y1⊗y2⊗y3)) = 0. 4 Some examples 4.1 P = Gi−Ass Proposition 8 If P is a (G −Ass) operad, then the operads P! and P˜ are equal. i Proof. In this case we have k = 1 and v = w = V . Then P˜ is defined by the module of 1 1 i relations a a ΨL ((x ·x )·x ), i j σi−σj 1 2 3   aiajΨRσi−σj(x1·(x2·x3)), a a (ΨL((x ·x )·x )−ΨR(x ·(x ·x )), where V =Σa σ . This syisjtemσiis re1duce2d to3 σj 1 2 3 i j j ΨL ((x ·x )·x )−ΨR(x ·(x ·x )),a a ΨL ((x ·x )·x ), Id 1 2 3 Id 1 2 3 i j σi−σj 1 2 3 (cid:8) 9 which corresponds to the dual operad. Inparticular,ifP =LieAdm,thenP =P˜ =Comm3whereComm3isthequadraticoperad defined from the submodule of relations : R=VectK{(xi·xj)·xk−xi·(xj ·xk),(xi·xj)·xk−(xj ·xi)·xk}. Thus, a Comm3-algebraA is 3-commutative if it is associative and satisfies x ·x ·x =x ·x ·x i j k σ(i) σ(j) σ(k) for every σ ∈ Σ . If A is unitary this implies that A is a commutative algebra. If not, 3 we have that A2 is contained in the center of A. The associated Lie algebra is two step nilpotent. 4.2 P = Lie If P =Lie then P˜ =P! =Com. In fact, in this case, k =1 and v =w =Id+c +c . As 1 1 1 2 v = w , a P˜-algebra is associative and the module of relation corresponds to the operad 1 1 Com. 4.3 P = Leib Let P =Leib be the Leibniz operad. A Leibniz algebra is defined by the relation x(yz)−(xy)z+(xz)y =0. In this case the associated P˜ operad corresponds to the relations x(yz)=(xy)z and (xy)z =(xz)y. Thus a Le˜ib-algebra is an associative algebra satisfying xyz =xzy. This relation is equivalent to x[y,z]=0 with[y,z]=yz−zy. ThislastidentityimpliesthatifxetyareinthederivedLiesubalgebra of the associated Lie algebra,then xy =0. The derived Lie algebra is then abelian and the Lie algebra is 2 step nilpotent. The dual operad, also denoted by Zinb, corresponds to the identity (xy)z−x(yz)−x(zy)=0. Thus a Le˜ib-algebra is a Zinbiel algebra (i.e. a Leib!-algebra) if x(yz) = (xy)z = 0 (every product of 3 elements of the associative algebra is zero). These algebras are nilalgebras A satisfying A3 = 0. For example, any associative commutative algebra is a Le˜ib-algebra. Every Le˜ib-algebra with unit is commutative. In dimension 3 the algebra defined by e e =e , e e =e e =e 1 1 2 1 3 3 3 2 is a noncommutative Le˜ib-algebra. 10