Nijenhuis algebras, NS algebras, and N-dendriform algebras

Transkript

1 Front. Math. Chna 2012, 7(5): DOI /s Njenhus algebras, NS algebras, and N-dendrform algebras Peng LEI 1, L GUO 1,2 1 Department of Mathematcs, Lanzhou Unversty, Lanzhou , Chna 2 Department of Mathematcs and Computer Scence, Rutgers Unversty, Newark, NJ 07102, USA c Hgher Educaton Press and Sprnger-Verlag Berln Hedelberg 2012 Abstract In ths paper, we study (assocatve) Njenhus algebras, wth emphass on the relatonshp between the category of Njenhus algebras and the categores of NS algebras and related algebras. Ths s n analogy to the well-known theory of the adjont functor from the category of Le algebras to that of assocatve algebras, and the more recent results on the adjont functor from the categores of dendrform and trdendrform algebras to that of Rota- Baxter algebras. We frst gve an explct constructon of free Njenhus algebras and then apply t to obtan the unversal envelopng Njenhus algebra of an NS algebra. We further apply the constructon to determne the bnary quadratc nonsymmetrc algebra, called the N-dendrform algebra, that s compatble wth the Njenhus algebra. As t turns out, the N-dendrform algebra has more relatons than the NS algebra. Keywords Njenhus algebras, Rota-Baxter algebras, dendrform algebras, NS algebras, N-dendrform algebras MSC 16W99 1 Introducton Through the antsymmetry bracket [x, y] :=xy yx, an assocatve algebra A defnes a Le algebra structure on A. The resultng functor from the category of assocatve algebras to that of Le algebras and ts adjont functor have played a fundamental role n the study of these algebrac structures. A smlar relatonshp holds for Rota-Baxter algebras and dendrform algebras. Receved February 4, 2012; accepted June 7, 2012 Correspondng author: L GUO, E-mal: lguo@rutgers.edu

2 828 Peng LEI, L GUO Ths paper studes a smlar relatonshp between (assocatve) Njenhus algebras and NS algebras. A Njenhus algebra s a nonuntary assocatve algebra N wth a lnear endomorphsm P satsfyng the Njenhus equaton: P (x)p (y) =P (P (x)y)+p (xp (y)) P 2 (xy), x, y N. (1) The concept of a Njenhus operator on a Le algebra orgnated from the mportant concept of a Njenhus tensor that was ntroduced by Njenhus [25] n the study of pseudo-complex manfolds n the 1950s and was related to the well-known concepts of Schouten-Njenhus bracket, the Frölcher-Njenhus bracket [13], and the Njenhus-Rchardson bracket. Njenhus operators on a Le algebra appeared n [20] n a more general study of Posson-Njenhus manfolds and then more recently n [14,15] n the context of the classcal Yang- Baxter equaton. The Njenhus operator on an assocatve algebra was ntroduced by Carnena and coauthors [4] to study quantum b-hamltonan systems. In [26], Njenhus operators are constructed by analogy wth Posson-Njenhus geometry, from relatve Rota-Baxter operators. Note the close analogue of the Njenhus operator wth the more famlar Rota-Baxter operator of weght λ (where λ s a constant), defned to be a lnear endomorphsm P on an assocatve algebra R satsfyng P (x)p (y) =P (P (x)y)+p (xp (y)) + λp (xy), x, y R. The latter orgnated from the probablty study of Baxter [2], was studed by Carter and Rota and s closely related to the operator form of the classcal Yang-Baxter equaton. Its study has experenced a qute remarkable renascence n the last decade wth many applcatons n mathematcs and physcs, most notably the work of Connes and Kremer on renormalzaton of quantum feld theory [5,10,11]. See [16] for further detals and references. The recent theoretc developments of Njenhus algebras have largely followed those of Rota-Baxter algebras. Commutatve Njenhus algebras were constructed n [7,12] followng the constructon of free commutatve Rota-Baxter algebras [17]. Another development followed the relatonshp between Rota-Baxter algebras and dendrform algebras. Recall that a dendrform algebra, defned by Loday [22], s a vector space D wth two bnary operatons and such that (x y) z = x (y z), (x y) z = x (y z), x, y, z D, (x y) z = x (y z), where := +.

3 Njenhus algebras, NS algebras, and N-dendrform algebras 829 Smlarly, a trdendrform algebra, defned by Loday and Ronco [23], s a vector space T wth three bnary operatons,, and that satsfy seven relatons. Aguar [1] showed that, for a Rota-Baxter algebra (R, P ) of weght 0, the bnary operatons x P y := xp (y), x P y := P (x)y, x, y R, defne a dendrform algebra on R. Smlarly, Ebrahm-Fard [6] showed that, for a Rota-Baxter algebra (R, P ) of non-zero weght, the bnary operatons x P y := xp (y), x P y := P (x)y, x P y := λxy, x, y R, defne a trdendrform algebra on R. As an analogue of the trdendrform algebra, the concept of an NS algebra was ntroduced by Leroux [21], to be a vector space M wth three bnary operatons,, and that satsfy four relatons (see Eq. (16) below). As an analogue of the Rota-Baxter algebra case, t was shown [21] that, for a Njenhus algebra (N,P), the bnary operatons x P y := xp (y), x P y := P (x)y, x P y := P (xy), x, y R, defne an NS algebra on R. Consderng the adjont functor of the functor nduced by the above mentoned map from Rota-Baxter algebras to (tr-)dendrform algebras, the Rota-Baxter unversal envelopng algebra of a (tr-)dendrform algebra was constructed n [8]. For ths purpose, free Rota-Baxter algebras were frst constructed. In ths paper, we gve a smlar approach for Njenhus algebras, but we go beyond the case of Rota-Baxter algebras. Our frst goal s to gve an explct constructon of free Njenhus algebras n Secton 2. We consder both the cases when the free Njenhus algebra s generated by a set and by another algebra. Other than ts role n the theoretcal study of Njenhus algebras, ths constructon allows us to construct the unversal envelopng algebra of an NS algebra. We acheve ths n Secton 3. Knowng that a Njenhus algebra gves an NS algebra, t s natural to ask what other dendrform type algebras that Njenhus algebras can gve n a smlar way. As a second applcaton of our constructon of free Njenhus algebras, we determne all quadratc nonsymmetrc relatons that can be derved from Njenhus algebras and fnd that one can actually derve more relatons than gven by the NS algebra. Ths dscusson s presented n Secton 4. Notaton In ths paper, k s taken to be a feld. A k-algebra s taken to be nonuntary assocatve unless otherwse stated.

4 830 Peng LEI, L GUO 2 Free Njenhus algebra on an algebra We start wth the defnton of free Njenhus algebras. Defnton 1 Let A be a k-algebra. A free Njenhus algebra over A s a Njenhus algebra F N (A) wth a Njenhus operator P A and an algebra homomorphsm j A : A F N (A) such that, for any Njenhus algebra N and any algebra homomorphsm f : A N, there s a unque Njenhus algebra homomorphsm f : F N (A) N such that f j A = f : j A A F N (A) f f N For the constructon of free Njenhus algebras, we follow the constructon of free Rota-Baxter algebras [8,16] by bracketed words. Alternatvely, one can follow [9] to gve the constructon by rooted trees that s more n the sprt of operads [24]. One can also follow the approach of Gröbner-Shrshov bases [3]. Because of the lack of a unform approach (see [18,19] for some recent attempts n ths drecton) and to be notatonally self contaned, we gve some detals. We frst dsplay a k-bass of the free Njenhus algebra n terms of bracketed words n 2.1. The product on the free Njenhus algebra s gven n 2.2 and the unversal property of the free Njenhus algebra s proved n A bass of free Njenhus algebra Let A be a k-algebra wth a k-bass X. We frst dsplay a k-bass X of F N (A) n terms of bracketed words from the alphabet set X. Let and be symbols, called brackets, and let X = X {, }. Let M(X ) denote the free semgroup generated by X. Defnton 2 [8,16] Let Y,Z be two subsets of M(X ). Defne the alternatng product of Y and Z to be ( ( Λ(Y,Z)= Z ) r 1(Y r) r 0 ( r 1( Z Y ) r) ( r 0 ) (Y Z ) r Y ) ( Z Y ) r Z. (2) We construct a sequence X n of subsets of M(X ) by the followng recurson. Let X 0 = X and, for n 0, defne X n+1 =Λ(X, X n ). Furthermore, defne X = n 0 X n = lm X n. (3)

5 Njenhus algebras, NS algebras, and N-dendrform algebras 831 Here, the second equaton n Eq. (3) follows snce X 1 X n X n 1, we have X 0 and, assumng X n+1 =Λ(X, X n ) Λ(X, X n 1 ) X n. By [8,16], we have the dsjont unon ( X = ) r 1(X X r) ( r 0 ( ) (X X ) r X ( X) r 1( X r) r 0 Furthermore, every x X has a unque decomposton ) ( X X) r X. (4) x = x 1 x b, (5) where x, 1 b, s alternatvely n X or n X. Ths decomposton wll be called the standard decomposton of x. For x n X wth standard decomposton x 1 x b, we defne b to be the breadth b(x) ofx, and defne the head h(x) ofx to be 0 (resp. 1) f x 1 s n X (resp. n X ). Smlarly, defne the tal t(x) ofx to be 0 (resp. 1) f x b s n X (resp. n X ). 2.2 Product n a free Njenhus algebra Let F N (A) = x X kx. We now defne a product on F N (A) by defnng x x F N (A) forx, x X and then extendng blnearly. Roughly speakng, the product of x and x s defned to be the concatenaton whenever t(x) h(x ). When t(x) =h(x ), the product s defned by the product n A or by the Njenhus relaton n Equaton (1). To be precse, we use nducton on the sum n := d(x)+d(x ) of the depths of x and x. Then n 0. If n =0, then x, x are n X and so are n A and we defne x x = x x A F N (A). Here, s the product n A. Suppose that x x have been defned for all x, x X wth 0 n k and let x, x X wth n = k +1. Frst, assume the breadth b(x) =b(x )=1. Then x and x are n X or X. Snce n = k + 1 s at least one, x and x cannot be both n X. We accordngly defne xx, x (resp. x ) X, x (resp. x) X, x x = x x + x x x x, (6) x = x, x = x X.

6 832 Peng LEI, L GUO Here, the product n the frst case s by concatenaton and n the second case s by the nducton hypothess snce for the three products on the rght-hand sde, we have d( x )+d(x )=d( x )+d( x ) 1=d(x)+d(x ) 1, d(x)+d( x )=d( x )+d( x ) 1=d(x)+d(x ) 1, d(x)+d(x )=d( x ) 1+d( x ) 1=d(x)+d(x ) 2, whcharealllessthanorequaltok. Now, assume b(x) > 1orb(x ) > 1. Let x = x 1 x b, x = x 1 x b be the standard decompostons from Eq. (5). We then defne x x = x 1 x b 1 (x b x 1 )x 2 x b, (7) where x b x 1 s defned by Eq. (6) and the rest s gven by concatenaton. The concatenaton s well defned snce by Eq. (6), we have Therefore, h(x b )=h(x b x 1), t(x 1)=t(x b x 1). t(x b 1 ) h(x b x 1), h(x 2) t(x b x 1). We have the followng smple propertes of. Lemma 1 Let x, x X. Then we have the followng statements. () h(x) =h(x x ) and t(x )=t(x x ). () If t(x) h(x ), then x x = xx (concatenaton). () If t(x) h(x ), then for any x X, : (xx ) x = x(x x ), x (xx )=(x x)x. Extendng blnearly, we obtan a bnary operaton that we stll denote by For x X, defne F N (A) F N (A) F N (A). N A (x) = x. (8) Obvously, x s agan n X. Thus, N A extends to a lnear operator N A on F N (A). Let j X : X X F N (A) be the natural njecton whch extends to an algebra njecton j A : A F N (A). (9)

7 Njenhus algebras, NS algebras, and N-dendrform algebras 833 The followng s our frst man result whch wll be proved n the next subsecton. Theorem 1 Let A be a k-algebra wth a k-bass X. () The par (F N (A), ) s an algebra. () The trple (F N (A),,N A ) s a Njenhus algebra. () The quadruple (F N (A),,N A,j A ) s the free Njenhus algebra on the algebra A. The followng corollary of Theorem 1 wll be used later n the paper. Corollary 1 Let M be a k-module, and let T (M) = n 1 M n be the reduced tensor algebra over M. Then F N (T (M)), together wth the natural njecton M : M T (M) j T (M) F N (T (M)), s a free Njenhus algebra over M, n the sense that, for any Njenhus algebra N and k-module map f : M N, there s a unque Njenhus algebra homomorphsm ˆf : F N (T (M)) N such that ˆf k M = f. Proof Ths follows mmedately from Theorem 1 and the fact that the constructon of the free algebra on a module (resp. free Njenhus algebra on an algebra; free Njenhus on a module) s the left adjont functor of the forgetful functor from algebras to modules (resp. from Njenhus algebras to algebras; from Njenhus algebras to modules), and the fact that the composton of two left adjont functors s the left adjont functor of the composton. 2.3 Proof of Theorem 1 Proof of Theorem 1 () We just need to verfy the assocatvty. For ths we only need to verfy (x x ) x = x (x x ) (10) for x, x, x X. We wll do ths by nducton on the sum of the depths n := d(x )+d(x )+d(x ). If n =0, then all of x, x, x have depth zero and so are n X. In ths case, the product s gven by the product n A and so s assocatve. Assume that the assocatvty holds for n k and assume that x, x, x X have n = d(x )+d(x )+d(x )=k +1. If t(x ) h(x ), then by Lemma 1, we have (x x ) x =(x x ) x = x (x x )=x (x x ).

8 834 Peng LEI, L GUO A smlar argument holds when t(x ) h(x ). Thus, we only need to verfy the assocatvty when t(x )=h(x ), t(x )=h(x ). We next reduce the breadths of the words. Lemma 2 If the assocatvty (x x ) x = x (x x ) holds for all x, x, and x n X of breadth one, then t holds for all x, x, and x n X. Proof We use nducton on the sum of breadths m := b(x )+b(x )+b(x ). Then m 3. The case when m = 3 s the assumpton of the lemma. Assume the assocatvty holds for 3 m j and take x, x, x X wth m = j +1. Then j Therefore, at least one of x, x, x have breadth greater than or equal to 2. Frst, assume b(x ) 2. Then x = x 1 x 2 wth x 1, x 2 X and t(x 1 ) h(x 2 ). Thus, by Lemma 1, we obtan (x x ) x =((x 1x 2) x ) x =(x 1(x 2 x )) x = x 1((x 2 x ) x ). Smlarly, we have Thus, whenever x (x x )=(x 1x 2) (x x )=x 1(x 2 (x x )). (x x ) x = x (x x ) (x 2 x ) x = x 2 (x x ). The latter follows from the nducton hypothess. A smlar proof works f b(x ) 2. Fnally, f b(x ) 2, then x = x 1 x 2 wth x 1, x 2 X and t(x 1 ) h(x 2 ). By applyng Lemma 1 repeatedly, we obtan (x x ) x =(x (x 1x 2)) x =((x x 1)x 2) x =(x x 1)(x 2 x ). Inthesameway,wehave (x x 1)(x 2 x )=x (x x ). Ths agan proves the assocatvty.

9 Njenhus algebras, NS algebras, and N-dendrform algebras 835 To summarze, our proof of the assocatvty has been reduced to the specal case when x, x, x X are chosen so that (a) n := d(x )+d(x )+d(x )=k +1 1 wth the assumpton that the assocatvty holds when n k; (b) the elements have breadth one and (c) t(x )=h(x )andt(x )=h(x ). By (b), the head and tal of each of the elements are the same. Therefore, by (c), ether all the three elements are n X or they are all n X. If all of x, x, x are n X, then as already shown, the assocatvty follows from the assocatvty n A. Therefore, t remans to consder the case when x, x, x are all n X. Then x = x, x = x, x = x wth x, x, x X. Usng Eq. (6) and blnearty of the product, we have (x x ) x =( x x + x x x x ) x = x x x + x x x x x x = x x x + ( x x ) x ( x x ) x + x x x + (x x ) x (x x ) x x x x x x x + x x x =: I I 9. Applyng the nducton hypothess n n to I 5 and I 8, and then use Eq. (6) agan, we obtan (x x ) x = x x x + ( x x ) x ( x x ) x + x x x + x x x + x x x x x x (x x ) x x x x x x x (x x ) x + (x x ) x + x x x = x x x + ( x x ) x ( x x ) x + x x x + x x x + x x x x x x (x x ) x x x x (x x ) x + (x x ) x. By a smlar computaton, we obtan x (x x ) = x x x + x x x x x x + x x x x ( x x ) + x (x x )

10 836 Peng LEI, L GUO + x x x x (x x ) x (x x ) + x (x x ) x x x. Now, by nducton, the -th term n the expanson of (x x ) x matches wth the σ()-th term n the expanson of x (x x ). Here, the permutaton σ Σ 11 s gven by ( σ = ). (11) Ths completes the proof of Theorem 1 (). () The proof follows from the defnton N A (x) = x and Equaton (6). () Let (N,,P) be a Njenhus algebra wth multplcaton. Let f : A N be a k-algebra homomorphsm. We wll construct a k-lnear map f : F N (A) N by defnng f(x) forx X. We acheve ths by defnng f(x) forx X n, n 0, nductvely on n. For x X 0 := X, defne f(x) =f(x). Suppose that f(x) has been defned for x X n and consder x n X n+1 whch s, by defnton and Eq. (4), ( ) Λ(X, X n )= (X X n ) r X n ) r 1(X X r) ( r 0 ( r 0 X n (X X n ) r) ( r 0 ) X n (X X n ) r X. Let x be n the frst unon component r 1 (X X n ) r above. Then x = r (x 2 1 x 2 ) =1 for x 2 1 X and x 2 X n, 1 r. By the constructon of the multplcaton and the Njenhus operator N A, we have Defne x = r =1(x 2 1 x 2 )= r =1(x 2 1 N A (x 2 )). f(x) = r =1(f(x 2 1 ) N(f(x 2 ))), (12) where the rght-hand sde s well defned by the nducton hypothess. Smlarly, defne f(x) fx s n the other unon components. For any x X, we have P A (x) = x X, and by the defnton of f n (Eq. (12)), we have f( x ) =P (f(x)). (13)

11 Njenhus algebras, NS algebras, and N-dendrform algebras 837 Therefore, f commutes wth the Njenhus operators. Combnng ths equaton wth Eq. (12), we see that f x = x 1 x b s the standard decomposton of x, then f(x) =f(x 1 ) f(x b ). (14) Note that ths s the only possble way to defne f(x) norderforf to be a Njenhus algebra homomorphsm extendng f. It remans to prove that the map f defned n Eq. (12) s ndeed an algebra homomorphsm. For ths, we only need to check the multplcty f(x x )=f(x) f(x ) (15) for all x, x X. For ths, we use nducton on the sum of depths n := d(x)+d(x ). Then n 0. When n =0, we have x, x X. Then Eq. (15) follows from the multplcty of f. Assume the multplcty holds for x, x X wth n k and take x, x X wth n = k +1. Let x = x 1 x b and x = x 1 x b be the standard decompostons. Snce n = k +1 1, at least one of x b and x b s n X. Then, by Eq. (6), we have f(x b x 1 ), x b (resp. x 1 ) X, x 1 (resp. x b) X, f(x b x 1)= f( x b x 1 + x b x 1 x b x 1 ), x b = x b, x 1 = x 1 X. In the frst case, the rght-hand sde s f(x b ) f(x 1 ) by the defnton of f. In the second case, by Eq. (13), the nducton hypothess, and the Njenhus relaton of the operator P on N, we have f( x b x 1 + x b x 1 x b x 1 ) = f( x b x 1 )+f( x b x 1 ) f( x b x 1 ) = P (f( x b x 1 )) + P (f(x b x 1 )) P (f( x b x 1 )) = P (f( x b ) f(x 1 )) + P (f(x b) f( x 1 )) P (P (f(x b) f(x 1 ))) = P (P (f(x b )) f(x 1)) + P (f(x b ) P (f(x 1))) P (P ((f(x b ) f(x 1))) = P (f(x b )) P (f(x 1)) = f( x b ) f( x 1 ) = f(x b ) f(x 1). Therefore, f(x b x 1)=f(x b ) f(x 1).

12 838 Peng LEI, L GUO Then f(x x )=f(x 1 x b 1 (x b x 1 )x 2 x b ) = f(x 1 ) f(x b 1 ) f(x b x 1 ) f(x 2 ) f(x b ) = f(x 1 ) f(x b 1 ) f(x b ) f(x 1) f(x 2) f(x b ) = f(x) f(x ). Ths s what we need. 3 NS algebras and ther unversal envelopng algebras The concept of an NS algebra was ntroduced by Leroux [21] as an analogue of the dendrform algebra of Loday [22] and the trdendrform algebra of Loday and Ronco [23]. Defnton 3 An NS algebra s a module M wth three bnary operatons,, and that satsfy the followng four relatons: (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) (16) for x, y, z M. Here, denotes + +. NS algebras share smlar propertes as dendrform algebras. For example, the operaton defnes an assocatve operaton. Another smlarty s the followng theorem whch s an analogue of the results of Aguar [1] and Ebrahm- Fard [6] that a Rota-Baxter algebra gves a dendrform algebra or a trdendrform algebra. Theorem 2 [21] A Njenhus algebra (N,P) defnes an NS algebra (N, P, P, P ), where x P y = xp (y), x P y = P (x)y, x P y = P (xy). (17) Let NA denote the category of Njenhus algebras, and let NS denote the category of NS algebras. It s easy to see that the map from NA to NS n Theorem 2 s compatble wth the morphsms n the two categores. Thus, we obtan a functor E : NA NS. (18) We wll study ts left adjont functor. Motvated by the envelopng algebra of a Le algebra and the Rota-Baxter envelopng algebra of a trdendrform algebra [8], we are naturally led to the followng defnton.

13 Njenhus algebras, NS algebras, and N-dendrform algebras 839 Defnton 4 Let M be an NS-algebra. A unversal envelopng Njenhus algebra of M s a Njenhus algebra U N (M) NA wth a homomorphsm ρ: M U N (M)nNS such that for any N NA and homomorphsm f : M N n NS, there s a unque ˇf : U N (M) N n NA such that ˇf ρ = f. Let M := (M,,, ) NS. Let T (M) = M n n 1 be the tensor algebra. Then T (M) s the free algebra generated by the k-module M. By Corollary 1, F N (T (M)), wth the natural njecton M : M T (M) F N (T (M)), s the free Njenhus algebra over the vector space M. Let J M be the Njenhus deal of F N (T (M)) generated by the set {x y xp (y), x y P (x)y, x y P (x y) x, y M}. (19) Let π : F N (T (M)) F N (T (M))/J M be the quotent map. Theorem 3 Let (M,,, ) be an NS algebra. The quotent Njenhus algebra F N (T (M))/J M, together wth ρ := π M, s the unversal envelopng Njenhus algebra of M. Proof The proof s smlar to the case of trdendrform algebras and Rota- Baxter algebras [8]. Therefore, we skp some of the detals. Let (N,P) be a Njenhus algebra, and let f : M N be a homomorphsm n NS. More precsely, we have f :(M,,, ) (N, P, P, P ). We wll complete the followng commutatve dagram, usng notatons from Corollary 1: T (M) (20) k M j T (M) M M F N (T (M)) f f ˆf π N ˇf F N (T (M))/J M By the unversal property of the free algebra T (M)overM, there s a unque homomorphsm f : T (M) N such that f k M = f. Therefore, f(x 1 x n )=f(x 1 ) f(x n ).

14 840 Peng LEI, L GUO Here, s the product n N. Then by the unversal property of the free Njenhus algebra F N (T (M)) over T (M), there s a unque morphsm f : F N (T (M)) N n NA such that f j T (M) = f. By Corollary 1, f = ˆf. Then ˆf M = ˆf j T (M) k M = f k M = f. (21) Therefore, for any x, y M, we check that ˆf(x y xp (y)) = 0, ˆf(x y P (x)y) =0, ˆf(x y P (x y)) = 0. Thus, J M s n ker( ˆf) and there s a morphsm ˇf : F N (T (M))/J M N n NA such that ˆf = ˇf π. Then by the defnton of ρ = π M n the theorem and Eq. (21), we have ˇf ρ = ˇf π M = ˆf M = f. Ths proves the exstence of ˇf. Suppose that ˇf : F N (T (M))/J M N s also a homomorphsm n NA such that ˇf ρ = f. Then ( ˇf π) M = f =(ˇf π) M. By Corollary 1, the free Njenhus algebra F N (T (M)) over the algebra T (M) s also the free Njenhus algebra over the vector space M wth respect to the natural njecton M. Therefore, we have ˇf π = ˇf π n NA. Snce π s surjectve, we have ˇf = ˇf. Ths proves the unqueness of ˇf. 4 From Njenhus algebras to N-dendrform algebras In ths secton, we consder an nverse of Theorem 2 n the followng sense. Suppose that (N,P) s a Njenhus algebra and defne bnary operatons x P y = xp (y), x P y = P (x)y, x P y = P (xy). By Theorem 2, the three operatons satsfy the NS relatons n Eq. (16). Our nverse queston s, what other quadratc nonsymmetrc relatons could (N, P, P, P ) satsfy? We recall some background on bnary quadratc nonsymmetrc operads n order to make the queston precse. We then determne all the quadratc nonsymmetrc relatons that are consstent wth the Njenhus operator. 4.1 Background and statement of Theorem 4 For detals on bnary quadratc nonsymmetrc operads, see [16,24]. Defnton 5 Let k be a feld. ) A graded vector space s a sequence P := {P n } n 0 of k-vector spaces P n, n 0.

15 Njenhus algebras, NS algebras, and N-dendrform algebras 841 ) A nonsymmetrc (ns) operad s a graded vector space P = {P n } n 0 equpped wth partal compostons: := m,n, : P m P n P m+n 1, 1 m, (22) such that, for λ P l, μ P m, and ν P n, the followng relatons hold: () (λ μ) 1+j ν = λ (μ j ν), 1 l, 1 j m; () (λ μ) k 1+m ν =(λ k ν) μ, 1 <k l; () there s an element d P 1 such that d μ = μ and μ d = μ for μ P n, n 0. An ns operad P = {P n } s called bnary f P 1 = k.d and P n,n 3, are nduced from P 2 by composton. Then, n partcular, for the free operad, we have P 3 =(P 2 1 P 2 ) (P 2 2 P 2 ), (23) whch can be dentfed wth P2 2 P2 2. AbnarynsoperadP s called quadratc f all relatons among the bnary operatons n P 2 are derved from P 3. Thus, a bnary, quadratc, ns operad s determned by a par (V,R), where V = P 2, called the space of generators, andr s a subspace of V 2 V 2, called the space of relatons. Therefore, we can denote P = P(V )/(R). Note that a typcal element of V 2 s of the form k =1 (1) (2), (1), (2) V, 1 k. Thus, a typcal element of V 2 V 2 s of the form ( k =1 (1) (2), m j=1 ) (3) j (4) j, (1), (2), (3) j, (4) j V, 1 k, 1 j m. For a gven bnary quadratc ns operad P = P(V )/(R), a k-vector space A s called a P-algebra f A has bnary operatons (ndexed by) V and f, for ( k =1 (1) (2), m j=1 ) (3) j (4) j R V 2 V 2 wth (1), (2), (3) j, (4) j V, 1 k, 1 j m, we have k m (x (1) y) (2) z = x (3) j (y (4) j z), x, y, z A. (24) =1 j=1

16 842 Peng LEI, L GUO For example, from Eq. (16), the NS algebras are precsely the P-algebras, where P = P(V )/(R) wthr beng the subspace of V 2 V 2 spanned by the four elements where = + +. (, ), (, ), (, ), ( +, + ), Theorem 4 Let V = k{,, } be the vector space wth bass {,, }, and let P = P(V )/(R) be a bnary quadratc ns operad. The followng statements are equvalent. () For every Njenhus algebra (N,P), the quadruple (N, P, P, P ) s a P-algebra. () The relaton space R of P s contaned n the subspace of V 2 V 2 spanned by (, ), (, ), (, ), (, ), ( + +, + + ), (25) where = + +. More precsely, any P-algebra A satsfes the followng relatons: (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 +(x y) z = x (y z)+x (y z)+x (y z), x, y, z A. (26) Note that the relatons of the NS algebra n Eq. (16) are contaned n the space spanned by the relatons n Eq. (25). We call P defned by the relatons n Eq. (25) the N-dendrform operad and call a quadruple (A,,, ) satsfyng Eq. (26) an N-dendrform algebra. LetND denote the category of N-dendrform algebras. Then we have the followng mmedate corollary of Theorem 4. Corollary 2 () There s a natural functor F : NA ND, (N,P) (N, P, P, P ). (27) () There s a natural (ncluson) functor G : ND NS, (M,,, ) (M,,, ). (28)

17 Njenhus algebras, NS algebras, and N-dendrform algebras 843 () The functors F and G gve a refnement of the functor E : NA NS n Eq. (16) n the sense that the followng dagram commutes: F NA ND E G NS (29) 4.2 Proof of Theorem 4 Proof of Theorem 4 Wth V = k{,, }, we have V 2 V 2 = k( 1 2, 3 4 ). 1, 2, 3, 4 {,, } Thus, any element r of V 2 V 2 s of the form r := a 1 (, 0) + a 2 (, 0) + a 3 (, 0) + b 1 (, 0) + b 2 (, 0) + b 3 (, 0) + c 1 (, 0) + c 2 (, 0) + c 3 (, 0) + d 1 (0, )+d 2 (0, )+d 3 (0, ) + e 1 (0, )+e 3 (0, )+e 3 (0, ) + f 1 (0, )+f 2 (0, )+f 3 (0, ), where the coeffcents are n k. () () Let P = P(V )/(R) be an operad satsfyng the condton n (). Let r be n R expressed n the above form. Then for any Njenhus algebra (N,P), the quadruple (N, P, P, P )sap-algebra. Thus, x, y, z N, we have a 1 (x P y) P z + a 2 (x P y) P z + a 3 (x P y) P z + b 1 (x P y) P z + b 2 (x P y) P z + b 3 (x P y) P z + c 1 (x P y) P z + c 2 (x P y) P z + c 3 (x P y) P z + d 1 x P (y P z)+d 2 x P (y P z)+d 3 x P (y P z) + e 1 x P (y P z)+e 2 x P (y P z)+e 3 x P (y P z) + f 1 x P (y P z)+f 2 x P (y P z)+f 3 x P (y P z)=0, By the defntons of P, P, and P n Eq. (17), we have a 1 xp (y)p (z)+a 2 P (xp (y))z a 3 P (xp (y)z)+b 1 P (x)yp(z) + b 2 P (P (x)y)z b 3 P (P (x)yz) c 1 P (xy)p (z) c 2 P (P (xy))z + c 3 P (P (xy)z)+d 1 P (x)p (y)z + d 2 P (x)yp(z) d 3 P (x)p (yz) + e 1 xp (P (y)z)+e 2 xp (yp(z)) e 3 xp (P (yz)) f 1 P (xp (y)z) f 2 P (xyp (z)) + f 3 P (xp (yz)) = 0.

18 844 Peng LEI, L GUO Snce P s a Njenhus operator, we further have a 1 xp (yp(z)) + a 1 xp (P (y)z) a 1 xp 2 (yz)+a 2 P (xp (y))z a 3 P (xp (y)z)+b 1 P (x)yp(z)+b 2 P (P (x)y)z b 3 P (P (x)yz) c 1 P (xyp (z)) c 1 P (P (xy)z)+c 1 P 2 (xyz) c 2 P (P (xy))z + c 3 P (P (xy)z)+d 1 P (x)p (y)z + d 2 P (x)yp(z) d 3 P (xp (yz)) d 3 P (P (x)yz)) + d 3 P 2 (xyz)+e 1 xp (P (y)z)+e 2 xp (yp(z)) e 3 xp (P (yz)) f 1 P (xp (y)z) f 2 P (xyp (z)) + f 3 P (xp (yz)) = 0. Collectng the smlar terms, we obtan (a 1 + e 2 )xp (yp(z)) + (a 1 + e 1 )xp (P (y)z) (a 1 + e 3 )xp (P (yz) +(a 2 + d 1 )P (xp (y))z (a 3 + f 1 )P (xp (y)z)+(b 1 + d 2 )P (x)yp(z) +(b 2 + d 1 )P (P (x)y)z (b 3 + d 3 )P (P (x)yz) (c 1 + f 2 )P (xyp (z)) +(c 3 c 1 )P (P (xy)z)+(c 1 + d 3 )P 2 (xyz) (c 2 + d 1 )P (P (xy))z +(f 3 d 3 )P (xp (yz)) = 0. Now, we take the specal case when (N,P) s the free Njenhus algebra (F N (T (M)),P T (M) ) defned n Corollary 1 for our choce of M = k{x, y, z} and P T (M) (u) = u. Then the above equaton s just (a 1 + e 2 )x y z +(a 1 + e 1 )x y z (a 1 + e 3 )x yz +(a 2 + d 1 ) x y z (a 3 + f 1 ) x y z +(b 1 + d 2 ) x y z +(b 2 + d 1 ) x y z (b 3 + d 3 ) x yz (c 1 + f 2 ) xy z +(c 3 c 1 ) xy z +(c 1 + d 3 ) xyz (c 2 + d 1 ) xy z +(f 3 d 3 ) x yz =0. Note that the set of elements x y z, x y z, x yz, x y z, x y z, x y z, x y z, x yz, xy z, xy z, xyz, xy z, x yz s a subset of the bass X of the free Njenhus algebra F N (T (M)), and hence, s lnearly ndependent. Thus, the coeffcents must be zero, that s, a 1 = e 1 = e 2 = e 3, a 2 = b 2 = c 2 = d 1, a 3 = f 1, b 1 = d 2, b 3 = c 1 = c 3 = f 2 = f 3 = d 3. Substtutng these equatons nto the general relaton r, we fnd that the any relaton r that can be satsfed by P, P, and P for all Njenhus algebras (N,P) softheform r = a 1 ((x y) z x (y z) x (y z) x (y z)) + b 1 ((x y) z x (y z)) + d 1 (x (y z) (x y) z (x y) z (x y) z)+a 3 ((x y) z x (y z)) + b 3 ((x y) z +(x y) z +(x y) z x (y z) x (y z) x (y z)),

19 Njenhus algebras, NS algebras, and N-dendrform algebras 845 where a 1,b 1,d 1,a 3,b 3 k can be arbtrary. Thus, r s n the subspace prescrbed n (), as needed. () () We check drectly that all the relatons n Eq. (26) are satsfed by (N, P, P, P ) for every Njenhus algebra (N,P). Frst of all, (x P y) P z = xp (y)p (z) = xp (yp(z)) + xp (P (y)z) xp 2 (yz) = x P (y P z)+x P (y P z)+x P (y P z), provng the frst equaton n Eq. (26). The proofs of the second and thrd equatons are smlar. For the fourth equaton, we have (x P y) P z = P ((xp (y))z) = P (x(p (y)z)) = x P (y P z). Fnally, for the last equaton, we verfy and (x P y) P z +(x P y) P z +(x P y) P z = P ((P (x)y)z) P (xy)p (z)+p(p(xy)z) = P (P (x)yz) P (xyp (z)) P (P (xy)z)+p 2 (xyz)+p(p(xy)z) = P (P (x)yz) P (xyp (z)) + P 2 (xyz), x P (y P z)+x P (y P z)+x P (y P z) = P (x)p (yz) P (x(yp(z))) + P (xp (yz)) = P (xp (yz)) P (P (x)yz)+p 2 (xyz) P (xyp (z)) + P (xp (yz)) = P (P (x)yz)+p 2 (xyz) P (xyp (z)). Therefore, the two sdes of the last equaton agree. Thus, f the relaton space R of an operad P = P(V )/(R) scontaned n the subspace spanned by the vectors n Eq. (25), then the correspondng relatons are lnear combnatons of the equatons n Eq. (26), and hence, are satsfed by (N, P, P, P ) for each Njenhus algebra (N,P). Therefore, (N, P, P, P )sap-algebra. Ths completes the proof of Theorem 4. Acknowledgements The authors thank the referee for helpful comments. L. Guo thanks NSF grant DMS for support. References 1. Aguar M. On the assocatve analog of Le balgebras. J Algebra, 2001, 244: Baxter G. An analytc problem whose soluton follows from a smple algebrac dentty. Pacfc J Math, 1960, 10:

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