Input/output coloring and Gröbner basis for dioperads

Input/output coloring and Gr¨ obner basis for diop erads An ton Khoroshkin ∗ F ebruary 24, 2026 Abstract By selecting a sp eciﬁc input or output of a diop eradic tree, we transform it into a ro oted tree and induce a corresp onding colored operadic structure. This fundamen tal pictorial construction demonstrates ho w the machinery of Gr¨ obner bases and the theory of Hilb ert series (w ell-established for (colored) op erads) can b e adapted to the diop eradic setting. W e illustrate this framew ork by providing several examples and applications: (1) we compute the dimensions of the spaces of op erations for the dioperad of Lie bialgebras; (2) w e describ e a Gr¨ obner basis and a minimal resolution for the dioperad of triangular Lie bialgebras; (3) w e provide computations for the diop erad of “algebraic string op erations”; (4) w e present a graphical construction that establishes the existence of quadratic Gr¨ obner bases and the Koszul prop ert y for a broad class of diop erads originating from cyclic operads. Con ten ts 0 In tro duction 2 0.1 Structure of the pap er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Diop erads – recollection 3 2 F rom Diop erads to colored op erads 4 2.1 The pictorial map Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Koszul prop ert y and Hilb ert series . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Colored shuﬄe op erads and Gr¨ obner bases for diop erads . . . . . . . . . . . . . . 7 2.3.1 Sh uﬄe (colored) op erads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.2 Rewriting systems for Shuﬄe op erads . . . . . . . . . . . . . . . . . . . . 10 2.4 Koszulness and Anick-t ype resolution . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Coloring of inputs/outputs in a (cyclic) op erad 12 3.1 The pictorial maps Θ with examples Θ( Lie ) and Θ( A ss ) . . . . . . . . . . . . . . 12 3.2 Deﬁning relations and Gr¨ obner basis for diop erads Θ c ( P ) . . . . . . . . . . . . . 13 4 Examples 17 4.1 The F rob enius diop erad F r ob . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 The diop erad L ieb of Lie bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Quasi-Lie bialgebras and Pseudo-Lie bialgebras . . . . . . . . . . . . . . . . . . . 20 4.4 The diop erad L ieb △ of triangular Lie bialgebras . . . . . . . . . . . . . . . . . . 21 4.5 The Diop erad V ( d ) of T radler and Ze inalian . . . . . . . . . . . . . . . . . . . . . 25 4.6 The diop erad Q p ois of quadratic P oisson structures . . . . . . . . . . . . . . . . 28 ∗ Departmen t of Mathematics, Universit y of Haifa, Moun t Carmel, 3498838, Haifa, Israel 1 0 In tro duction While n -ary operations on a v ector space V are traditionally view ed as elemen ts of Hom( V ⊗ n , V ), man y natural algebraic structures (lik e Lie bialgebras) require m ultilinear op erations of the form Hom( V ⊗ m , V ⊗ n ). Such op erations, featuring m inputs and n outputs, are geometrically repre- sen ted b y v ertices with m ultiple incoming and outgoing edges. Dep ending on the class of graphs allo w ed for their comp osition, one arriv es at diﬀerent categorical framew orks: PROPs [7] and prop erads [53] corresp ond to (connected) directed acyclic graphs, wheeled prop erads [42] allow for cycles, and mo dular op erads [22] describ e structures asso ciated with surfaces. Despite their imp ortance, these framew orks are known to b e diﬃcult for explicit compu- tation. The lack of a general computational theory often necessitates case-by-case analyses. F or many fundamental examples – including Lie bialgebras [41], double Poisson brac k ets [6], cob oundary Lie bialgebras [44], balanced inﬁnitesimal bialgebras [48] —- the presentations in- v olv e only graphs of genus zero. This observ ation motiv ates the study of diop erads: structures that encode multilinear op erations while restricting comp ositions to directed trees. Muc h of the com binatorial and homological machinery developed for op erads extends naturally to the diop eradic setting, including the Koszul dualit y theory established b y Gan in [21]. Therefore, en umerativ e and homological questions regarding the prop eradic env elopes of the diop erads should b e ﬁrst examined at the level of the underlying diop erads. A key technical insight of this pap er is as follo ws (see Picture (2.1.1)): Fixing a r o ot in a diop er adic tr e e le ads to a c onsistent 2-c oloring of its e dges. This coloring dep ends on whether the orientation induced b y the diop eradic structure agrees with the orientation to ward the root. In Prop osition 2.1.2 we pro ve that this coloring deﬁnes an exact functor Ψ from the category of diop erads to the category of 2-colored op erads. Crucially , this functor commutes with bar-cobar constructions, pro viding a p o w erful to ol for verifying the Koszul prop ert y . This approac h yields tw o immediate applications. First, it allows us to deriv e a functional equation relating the generating series of Koszul-dual diop erads (Corol- lary 2.2.2), which w e illustrate by computing the dimensions for the Lie bialgebra diop erad (Prop osition 4.2.2): dim L ieb ( m, n ) = ( m + n − 2)! 2 ( m − 1)!( n − 1)! . Second, by applying the Gr¨ obner basis techniques for colored op erads developed in [28], we obtain a corresponding Gr¨ obner basis theory for diop erads , what is the primary goal of this pap er. As an illustration we compute explicit Gr¨ obner bases for several foundational examples. W e demonstrate that the diop erads gov erning Lie bialgebras ( § 4.2), quadratic P oisson structures ( § 4.6), and the diop erad of T radler and Zeinalian gov erning ”algebraic string op erations” ( § 4.5) admit quadratic Gr¨ obner bases (rewriting system). F urthermore, w e sho w that the spanning relations for triangular Lie bialgebras ( L ieb △ ) form a Gr¨ obner basis (Theorem 4.4.5), conﬁrming that the conjectural resolution prop osed b y Merkulo v in [44] coincides with the Anic k resolution (Theorem 4.4.10). Additionally , b y partitioning the legs of a cyclic op erad into inputs and outputs, w e deﬁne a pictorial functor Θ c from cyclic op erads to diop erads. In Corollary 3.2.12, we prov e that if a cyclic op erad P admits a quadratic rewriting system with caterpillar normal forms, then Θ c ( P ) also admits a quadratic rewriting system. This facilitates the veriﬁcation of Koszulness for a broad family of diop erads (see e.g., Examples 3.1.3 and 3.1.4). Finally , it is worth men tioning the computational feasibilit y . While Gr¨ obner basis machin- ery for op erads has b een implemented (e.g., the softw are dev elop ed under the supervision of Vladimir Dotsenk o), such to ols are b est suited for a small num b er of generators. In the colored 2 setting arising from diop erads, the num b er of generators grows dramatically , rendering existing soft w are diﬃcult to apply . Nev ertheless, we exp ect that sp ecialized implemen tations for v eri- fying conﬂuence in diop erads will signiﬁcan tly simplify the veriﬁcation of the Koszul prop erty , whic h has typically b een a nontrivial problem. 0.1 Structure of the pap er F ollowing a brief review of diop erads in § 1, we introduce the pictorial functor Ψ from diop erads to colored op erads in § 2.1. Its applications to generating series and the Koszul property are discussed in § 2.2. In § 2.3, we recall the theory of shuﬄe op erads and Gr¨ obner bases, adapting the framework to the 2-colored op erads arising from Ψ. § 2.4 then outlines the homological applications of Gr¨ obner bases. Section 3 is dev oted to the functor Θ, which constructs a diop erad from a cyclic op erad. Our main result, Theorem 3.2.11, establishes that a rewriting system for a diop erad can b e deriv ed from the Gr¨ obner basis of the underlying cyclic op erad under sp eciﬁc assumptions. Finally , Section 4 pro vides v arious illustrative examples. Notable applications include com- puting the dimensions of operations for Lie bialgebras ( § 4.2), as w ell as conﬂuence computations and the construction of Anic k resolutions for triangular Lie bialgebras ( § 4.4) and the diop erad of T radler and Ze inalian ( § 4.5). 1 Diop erads – recollection W e hop e that the reader is familiar with the notion of an algebraic op erad and Koszul duality theory for them (see e.g. [23], [37] for details). Recall that op erad consist of collections of op erations with multiple inputs and one output. This pap er is devoted to suggesting some computational metho ds dealing with algebraic structure of op eradic nature, but those who has v arious n um b ers of inputs and outputs. While working with algebraic structures with several inputs and several outputs one typically dra w an element of P ( m, n ) b y a corolla with m incoming edges and n outgoing. While working with p ossible univ erses (universal operations one can write down out of this data) p eople end up with muc h more sophisticated comp ositions enco ding the notions of: a PROP ([7]) and that of a prop erad ([53]) corresp ond to directed (connected) graphs without directed cycles, wheeled prop erads ([42, 43]) allow graphs with cycles, and mo dular operads ([22]) describ e structures enco ded b y graphs embedded on surfaces. All these frameworks admit interesting examples motiv ated b y their representations, yet they are notoriously diﬃcult for explicit computations. A t present, no general computational metho ds are kno wn: eac h example requires a detailed, case-b y-case analysis and often has its own p eculiarities. The diop erads deal with the simplest part of the op erations one can write down – the underlying graphs are trees. In particular, the diop erads is a part of the all univ erses we men tioned ab o ve. One of the main goals of this pap er is to suggest an ”arythmetic” of monomials that will help to answer on basis enumerativ e questions for dioperads. Let us give a formal deﬁnition of a diop erad, (w e also refer to more details in [21]): Deﬁnition 1.0.1. A diop er ad P c onsists of • a c ol le ction of op er ations P ( m, n ) with m ⩾ 1 inputs and n ⩾ 1 outputs e quipp e d with S m action on inputs and S n action on outputs. • inﬁnitesimal c omp osition rules: i ◦ j : P ( m, n ) ⊗ P ( m ′ , n ′ ) → P ( m + m ′ − 1 , n + n ′ − 1) that inserts i ’th ouput of P ( m, n ) into the j ’th input of P ( m ′ , n ′ ) 3 such that c omp osition rules ar e asso ciative and S -e quivariant with r esp e ct to the action of S × S op on inputs and outputs. While looking for an iterated inﬁnitesimal comp ositions one has to assign a composition rule ◦ T to any diop eradic tree T : ◦ T : O v ∈ V ( T ) P ( in ( v ) , out ( v )) → P ( in ( T ) , out ( T )) . Here by a diop eradic tree T w e mean a tree with a direction on all edges and with the la- b elling/n umbering of inputs in ( T ) and outputs out ( T ). The asso ciativit y constraints tells that the result of iterated comp osition do es not dep end on the w ay w e contract edges in a diop eradic graph. F or example, F or T := w e hav e ◦ T : P (3 , 3) ⊗ P (2 , 2) ⊗ P (2 , 3) → P (5 , 6) . It is w orth mentioning that the free diop erad F ree(Υ) ge nerated by an S × S op -collection Υ( m, n ) is spanned b y diop eradic trees T whose vertex v is indexed b y elemen t of Υ( in ( v ) , out ( v )). Deﬁnition 1.0.2. A structur e of a P -algebr a (also c al le d a r epr esentation of a diop er ad P ) on a ve ctor sp ac e V is a morphism of diop er ads P → End V , wher e the diop er ad End V is an endomorphism diop er ad: End V ( m, n ) := Hom( V ⊗ m , V ⊗ n ) with obvious c omp osition rules. Similarly to the case of other algebraic structures like op erads one can deﬁne the notion of the bar and cobar construction functors. The bar-construction functor B maps a diop erad P to the free dg-co diop erad F ( s P ) generated by P and the cobar construction functor Ω asso ciates to a co-diop erad Υ a quasi-free dg-diop erad F ( s Υ). As alw a ys one has an quasiisomorphism of op erads Ω( B ( P )) → P for any diop erad P . Ho w ev er, such a free resolution is not a minimal one. F or quadratic diop erads one extends the standard Koszul duality theory , that helps to ﬁnd minimal resolutions for some go o d quadratic dioperads. Koszul dualit y theory is w ell known for algebras (see [47], [46], generalized for quadratic op erads in [23] and can b e extended word b y word to man y diﬀerent op eradic-t yp e structures. W e are mainly interested in the case of coloured operads (w orked out in [34]), and the case of diop erads describ ed in [21]. Our notations are sligh tly diﬀerent and more follow the one for op erads ([37]) and we use one of equiv alen t deﬁnitions of koszulness. Deﬁnition 1.0.3. We say that a quadr atic diop er ad P := F ree(Υ | R ) gener ate d by Υ subje ct to quadr atic r elations R is Koszul if the c anonic al pr oje ction fr om the c ob ar c onstruction of the quadr atic dual c o diop er ad P ¡ := coF ree( s Υ | sR ⊥ ) Ω diop ( P ¡ ) quis − → P is a quasiisomorphism. 2 F rom Diop erads to colored op erads This section details the central pictorial intuition that allo ws us to derive a theory of Gr¨ obner bases for diop erads from the established mac hinery for colored op erads. W e omit a formal 4 review of colored op erads here and instead refer the reader to [39], as well as our preceding pap er [28] and the references therein, for a comprehensive treatment of Gr¨ obner bases in the colored setting. 2.1 The pictorial map Ψ Supp ose T is a diop eradic tree. All edges of T are directed; we refer to this as the inner orien tation. Cho ose a leaf r of T and declare it the global ro ot. This c hoice deﬁnes a unique path from eac h vertex to r . W e then impose a new orien tation on the edges suc h that r b ecomes the unique sink, which we call the outer orientation. The outer orientation transforms T into an op eradic tree T r , where each vertex has exactly one outgoing edge and all other incident edges are incoming. This resulting op eradic tree carries a tw o-coloring of its edges: an edge is drawn as a straight line if its outer and inner orien tations coincide, and as a dotted line if they are opposite. Note that the outer orien tation is entirely determined by the choice of the ro ot, and the inner orientation can b e reco v ered from the coloring. The following pictorial example illustrates the pro cedure for transforming a diop eradic tree in to a tw o-colored op eradic tree, with the outgoing edge ¯ 6 selected as the ro ot: 1 2 3 4 5 ¯ 1 ¯ 2 ¯ 3 ¯ 4 ¯ 5 ¯ 6 7→ 1 2 3 4 5 ¯ 1 ¯ 2 ¯ 3 ¯ 4 ¯ 5 ¯ 6 = ¯ 2 1 2 ¯ 1 3 4 ¯ 3 5 ¯ 4 ¯ 5 a diop eradic tree with a chosen root edge coloring the corresp onding colored op eradic tree (2.1.1) The result dep ends strongly on whic h input or output is selected as the ro ot. Sp eciﬁcally , to eac h diop eradic corolla with m inputs and n outputs, w e asso ciate t w o t yp es of op eradic corollas: one with m straight inputs and n − 1 dotted inputs, and another with m − 1 straigh t inputs and n dotted inputs. The op eradic comp osition of these colored trees corresp onds exactly to the diop eradic comp osition in the original diop erad, leading to the follo wing observ ation: Prop osition 2.1.2. The c ol le ction of al l p ossible r o ot choic es in a diop er ad deﬁnes a faithful exact functor Ψ : Diop er ads − → 2 -c olor e d Op er ads . The sp ac e of op er ations Ψ( P ) | ( m, n − 1) with a str aight output, m str aight inputs, and n − 1 dotte d inputs, and the sp ac e Ψ( P ) . . . ( m − 1 , n ) with a dotte d output, m − 1 str aight inputs, and n dotte d inputs, ar e b oth c anonic al ly isomorphic to the sp ac e P ( m, n ) of op er ations in the underlying diop er ad P : Ψ( P ) | ( m, n − 1) = Ψ( P ) . . . ( m − 1 , n ) := P ( m, n ) . Mor e over, op er adic c omp osition in the c olor e d op er ad agr e es with c omp osition in the diop er ad. Pr o of. This follows directly from the pictorial description. Example 2.1.3. L et F diop ( µ ) b e the fr e e diop er ad gener ate d by a single element µ ∈ F (2 , 2) , symmetric with r esp e ct to inputs and skew-symmetric with r esp e ct to outputs: µ := 1 2 1 2 = − 2 1 1 2 = 1 2 2 1 = − 2 1 2 1 . 5 Then the 2 -c olor e d op er ad Ψ( F diop ( µ )) is the fr e e 2 -c olor e d op er ad gener ate d by the fol lowing two op er ations, which satisfy the fol lowing symmetry c onditions: µ + := 1 2 ¯ 1 = 2 1 ¯ 1 ; µ − := 1 ¯ 1 ¯ 2 = − 1 ¯ 2 ¯ 1 . The functor Ψ generalizes straigh tforw ardly to co-diop erads and colored co-op erads, leading to the following observ ation: Theorem 2.1.4. The functor Ψ is exact, maps fr e e diop er ads to fr e e c olor e d op er ads, and c ommutes with the b ar and c ob ar c onstructions B , Ω : Ψ ◦ Ω Dioperads = Ω Operads ◦ Ψ , Ψ ◦ B Dioperads = B Operads ◦ Ψ . (2.1.5) Pr o of. The pro of relies on a direct comparison of diop eradic and op eradic trees and their col- oring. Each diop eradic tree is uniquely iden tiﬁed by replacing the inner and outer directions with coloring information, establishing a bijection b et ween these classes of trees. Consequently , Ψ maps a free diop erad to a free 2-colored op erad. The (co)bar construction Ω Dioperads assigns to a co dioperad P the free dg-diop erad generated b y s P , where s denotes the appropriate shift of the collection (cf. [37]). Similarly , the construction Ω Operads assigns to a colored coop erad Ψ( P ) the free colored dg-op erad generated by s Ψ( P ) with the same shift. The equiv alences in (2.1.5) then follow from the isomorphism of the underlying free ob jects. 2.2 Koszul prop ert y and Hilb ert series Corollary 2.2.1. The diop er ad P is quadr atic (r esp e ctively Koszul) if and only if the asso ciate d 2 -c olor e d op er ad Ψ( P ) is quadr atic (r esp e ctively Koszul). Pr o of. The functor Ψ sends a free diop erad generated b y a S × S op -collection Υ to the free colored op erad generated by the collection Ψ(Υ). More generally , a presentation of a diop erad b y generators and relations is mapp ed by Ψ to a presentation of a colored operad whose gen- erators and relations are the images under Ψ of the original ones. In particular, Ψ preserv es quadratic presen tations. F urthermore, thanks to Theorem 2.1.4, a morphism of dg-diop erads is a quasiisomorphism if and only if its image under Ψ is a quasiisomorphism. Therefore, the canonical morphism deﬁning Koszulness satisﬁes Ω diop ( P ¡ ) quis − → P ⇔ Ω Operad  Ψ( P ¡ )  = Ψ  Ω diop ( P ¡ )  quis − → Ψ( P ) , whic h prov es the claim. In particular, one can apply the theory of Gr¨ obner bases and rewriting systems dev elop ed for colored operads in [28] in order to prov e the Koszulness of quadratic diop erads. W e brieﬂy review this theory in the next section. W e also state a simple corollary concerning the generating series of Koszul dual diop erads. Supp ose that the diop erad P is graded, P = ∞ M r =0 P ( r ) . W e then deﬁne its graded generating series by χ q P ( x, y ) := X m,n ∞ X r =0 q r dim P ( m, n ) ( r ) ! x m m ! y n n ! = X m,n dim q P ( m, n ) x m m ! y n n ! . 6 F or example, if the diop erad P = F ree(Υ , R ) is quadratic, then P (0) = P (1 , 1) = Id , P (1) = Υ, and P (2) = Υ ◦ ′ Υ R , where ◦ ′ denotes the inﬁnitesimal comp osition. Moreov er, in all examples considered here, the total n umber of inputs and outputs of the generators Υ is ﬁxed. As a result, the parameter q is uniquely determined by the x - and y -gradings, since the sum of the n umbers of inputs and outputs is constant on each graded comp onen t. Corollary 2.2.2. If the diop er ad P is Koszul, then the gr ade d gener ating series of P and its Koszul dual P ! satisfy the fol lowing algebr aic identity in the gr oup of automorphisms of a 2 -dimensional ve ctor sp ac e: ∂ χ − q P ! ∂ y ; ∂ χ − q P ! ∂ x ! ◦  ∂ χ q P ∂ y ; ∂ χ q P ∂ x  = Id . (2.2.3) Pr o of. The argument follo ws the standard pro of for op erads (see, for example, [37, § 7.5]). The Koszulness of the diop erad P is equiv alen t to the Koszulness of the colored op erad Ψ( P ). In turn, this is equiv alen t to the acyclicit y of the Koszul complex K (Ψ( P )), whic h, as a graded v ector space, is isomorphic to Ψ( P ¡ ) ◦ ′ Ψ( P ). The homological grading is determined by the degree of P ¡ . It therefore remains to compute the generating series of Ψ( P ), as is done for colored op erads (see, for example, [28]). The generating series of a colored op erad is given b y a collection of generating series corresp onding to operations with a ﬁxed output color. In the present situation, there are tw o such series, corresp onding to the “ | ”(straigh t) and “ . . . ”(dotted) colors: F | P ( x, y ) := X m,n dim q Ψ( P ) | ( m, n ) x m m ! y n n ! = X m,n dim q P ( m, n + 1) x m m ! y n n ! = ∂ χ q P ∂ y , F . . . P ( x, y ) := X m,n dim q Ψ( P ) . . . ( m, n ) x m m ! y n n ! = X m,n dim q P ( m + 1 , n ) x m m ! y n n ! = ∂ χ q P ∂ x . Finally , the generating series of the op erad Ψ( P ! ) coincides with that of the coop erad Ψ( P ¡ ), and replacing q by − q accounts for the homological grading, which con tributes signs to the Euler characteristic. W e illustrate this theory with the computation of the generating series for the diop erad of Lie bialgebras ( § 4.2). Notably , one may also deﬁne generating series for colored op erads that accoun t for the action of the symmetric group (see e.g. [28, § 1.6]). The corresp onding functional equation is expressed via plethystic substitutions of symmetric functions, a framework that generalizes readily to the diop eradic setting. 2.3 Colored shuﬄe op erads and Gr¨ obner bases for diop erads Gr¨ obner basis theory originates from the study of monomials, their divisibilit y prop erties, and the existence of a monomial ordering compatible with multiplication (see, for example, [1] for a general introduction in the commutativ e setting). It w as in tro duced for Lie algebras b y Shirsho v [49], for noncommutativ e asso ciativ e algebras by Bokut [8], and indep enden tly for commutativ e algebras by Buc h b erger [9]. Such an approach cannot b e directly applied to symmetric op erads, since there is no monomial ordering compatible with the action of the 7 symmetric group. This diﬃculty is resolved b y the notion of sh uﬄe op erads, introduced in [13], where part of the symmetric group action is forgotten. In [28], this idea was extended to the colored setting by introducing colored sh uﬄe op erads. F rom a pictorial p oin t of view, the passage from diop erads to colored shuﬄe op erads is straigh tforw ard. How ever, it leads to a signiﬁcan t increase in the n um b er of generators, since one must consider all p ossible c hoices of colorings. In what follows, w e recall the main deﬁnitions from [28] and [13], adapting them to the case of 2-colored op erads. Remark 2.3.1. During the pr ep ar ation of this manuscript, we note d a description of PBW b ases for diop er ads in the do ctor al thesis [27]. Unfortunately, the notion of shuﬄe c omp osition mentione d in that p ap er is not asso ciative and one c annot describ e a shuﬄe diop er ad, so we have some doubts whether the ar guments of the thesis [27] ar e c orr e ct. 2.3.1 Sh uﬄe (colored) op erads Deﬁnition 2.3.2. A shuﬄe 2 -c olor e d tr e e T is a planar r o ote d tr e e with e dges oriente d fr om le aves towar d the r o ot, satisfying: • Each e dge is assigne d one of two sp e ciﬁe d c olors. • The set of le aves of T is e quipp e d with a total (line ar) or der. • F or every internal vertex v ∈ T , the r elative or der of the minimal lab els of its inc oming subtr e es is c omp atible with the planar structur e. Notation 2.3.3. • A shuﬄe c or ol la is a shuﬄe tr e e p ossessing exactly one internal vertex. • F or e ach internal vertex v of a shuﬄe tr e e T , we denote by ar( v ) the c orr esp onding c or ol la that pr eserves the c oloring of the inputs and output of v . • F or a shuﬄe tr e e T , we denote by ar( T ) the total arity and c oloring of its le aves and r o ot. W e represent sh uﬄe trees in the plane with the ro ot at the b ottom and all edges orien ted do wn w ard. The planar structure is ﬁxed by the lo cal order of m inima, whic h increases from left to right. Consequen tly , in a shuﬄe corolla, the leav es are arranged from left to righ t according to their lab els. The following ﬁgure displays three sh uﬄe corollas, follo wed by three colored planar trees with t w o in ternal v ertices. The ﬁrst tw o are v alid shuﬄe trees; the third is not, as the compatibility condition is violated at one v ertex. The indices causing the violation are circled. 1 2 3 , 1 2 3 , 1 2 3 ; 1 4 2 3 , 1 3 2 4 ; 3 ○ 4 1 ○ 2 . Sh uﬄe corollas Sh uﬄe trees Non-sh uﬄe tree Deﬁnition 2.3.4. A (c olor e d) shuﬄe op er ad P in a monoidal c ate gory C c onsists of: • A n assignment of an obje ct P (ar( v )) in C to e ach c olor e d shuﬄe c or ol la v . • F or e ach c olor e d shuﬄe tr e e T , an op er adic c omp osition map ◦ T : O v ∈ T P (ar( v )) − → P  ar( T )  , wher e the tensor pr o duct is taken over the internal vertic es of T . 8 The maps ◦ T must b e asso ciative: for any shuﬄe subtr e e T ′ ⊂ T , p erforming c omp osition within T ′ fol lowe d by c omp osition in the c ontr acte d tr e e T /T ′ must c oincide with the dir e ct c omp osition over T : ◦ T ≃ ◦ T /T ′ ◦ ◦ T ′ . The deﬁning c haracteristic of shuﬄe operads is that free shuﬄe op erads p ossess a monomial basis and admit an ordering compatible with op eradic comp ositions. Prop osition 2.3.5. L et { α i } i ∈ I b e an alphab et of gener ators with pr escrib e d arities and c olors. L et B { α i } SH ( a r ( w )) b e the set of al l shuﬄe tr e es or arity ar ( w ) whose vertic es ar e lab ele d by elements of the alphab et such that the lab el at vertex v matches ar( v ) . Then n B { α i } SH ( a r ( w )) : w – c or ol la o forms the b asis of the fr e e c olor e d shuﬄe op er ad gener ate d by { α i } . Deﬁnition 2.3.6. In a fr e e c olor e d shuﬄe op er ad F r ee ( { α i } ) , a total or der ⩽ on the tr e e monomials of B { α i } SH ( a r ( w )) of the same (c olor e d) arity ar ( w ) is admissible if it is c omp atible with c omp osition: ∀ α ⩽ α ′ ∈ B { α i } SH ( a r ( v )) , ∀ β ⩽ β ′ ∈ B { α i } SH ( a r ( w )) ⇒ α ◦ α ′ ⩽ β ◦ β ′ ∈ B { α i } SH ( a r ( v ◦ w )) . It is well known that admissible orderings for (colored) shuﬄe op erads exist (see [28, § 4.1]). The most widely used is the path-lexicographical ordering in tro duced b y E. Hoﬀb ec k in [25] (see also [13, § 3.2.1]); for our computations, w e ﬁnd the generalizations suggested by V. Dotsenko in [12] particularly useful. W e do not provide the technical details of these orderings here, as w e will primarily employ the framew ork of rewriting systems in most of our examples. Deﬁnition 2.3.7. L et M b e an op er adic ide al in a fr e e c olor e d shuﬄe op er ad F . Given an admissible or dering of monomials, a set G of gener ators of M is a Gr¨ obner b asis if for every f ∈ M , the le ading term of f is divisible by the le ading term of some element in G . The following result ensures that Gr¨ obner basis theory for sh uﬄe operads is applicable to symmetric op erads: Theorem 2.3.8 ([13]) . Ther e exists an exact for getful functor SH fr om the c ate gory of (c olor e d) symmetric op er ads to the c ate gory of (c olor e d) shuﬄe op er ads. This functor for gets p arts of the c omp osition and the symmetric gr oup action, and it maps fr e e (c olor e d) symmetric op er ads to fr e e (c olor e d) shuﬄe op er ads. Remark 2.3.9. We intend to apply this the ory to 2 -c olor e d op er ads derive d fr om diop er ads. Sp e ciﬁc al ly, we se ek a Gr¨ obner b asis for the shuﬄe 2 -c olor e d op er ad SH (Ψ( P )) , wher e P is a diop er ad in the c ate gory of ve ctor sp ac es. The primary dr awb ack of this appr o ach is the signiﬁc ant incr e ase in the numb er of gener ators and deﬁning r elations within SH (Ψ( P )) . F or instance, consider P as the free diop erad from Example 2.1.3 generated by a single elemen t µ with tw o inputs and t w o outputs: µ := 1 2 1 2 = − 2 1 1 2 = 1 2 2 1 = − 2 1 2 1 . The corresp onding colored shuﬄe op erad con tains the following six generators, whic h are dis- tinguished by the p erm utation of their colorings (which w e also refer to as their arity): µ 1 + := 1 2 3 ; µ 2 + := 1 2 3 , µ 3 + := 1 2 3 , µ 1 − := 1 2 3 , µ 2 − := 1 2 3 , µ 3 − := 1 2 3 . 9 2.3.2 Rewriting systems for Sh uﬄe op erads As observ ed, the functor Ψ signiﬁcantly increases the num b er of generators, occasionally making the standard Gr¨ obner bases machinery to o restrictive. Consequently , we prop ose a generaliza- tion via the notion of a rewriting system. Unlike Gr¨ obner bases, whic h require a global total ordering on all monomials, a rewriting system only requires a sp eciﬁc c hoice of a leading term for each relation to deﬁne a directed reduction. F or the general theory of rewriting systems, w e refer to [4], and for their adaptation to Gr¨ obner bases for algebras and op erads, we refer to [10, § 2.6]. Below, we pro vide the deﬁnitions tailored to shuﬄe op erads. Deﬁnition 2.3.10. A r ewriting system S for a shuﬄe op er ad P = F ( V ) /J , gener ate d by a shuﬄe c ol le ction V subje ct to an ide al of r elations J , is a set of rules S = { ( τ i , f i ) } i ∈ I wher e: • τ i is a shuﬄe tr e e (designate d as the le ading monomial). • f i ∈ F ( V ) is a line ar c ombination of shuﬄe tr e es. Each p air deﬁnes a r e duction rule τ i − → f i , such that the r elations { τ i − f i } i ∈ I gener ate the ide al J . A sh uﬄe tree T is called reducible if it con tains a subtree isomorphic to some τ i . Speciﬁcally , if T = a ◦ τ i ◦ ( b 1 , . . . , b k ), where ◦ denotes op eradic comp osition, the reduction step is deﬁned as: T − → a ◦ f i ◦ ( b 1 , . . . , b k ) . This op eration extends linearly to the free shuﬄe op erad F ( V ). W e write g → S h if h is obtained from g b y a ﬁnite sequence of such reductions. Deﬁnition 2.3.11. A shuﬄe tr e e is c al le d S -irr e ducible if it do es not c ontain any τ i as a subtr e e. The ve ctor sp ac e of normal forms is the subsp ac e sp anne d by the set of al l irr e ducible shuﬄe tr e es: B S irr = { T ∈ Shuﬄe T r e es : ∀ i ∈ I , τ i is not a subtr e e of T } . Deﬁnition 2.3.12. The system S is c al le d terminating if ther e is no inﬁnite se quenc e of r e- ductions m → S m 1 → S m 2 → S . . . for any monomial m . In the shuﬄe op erad setting, assuming termination, we deﬁne the follo wing: Deﬁnition 2.3.13. • A mbiguities (overlaps) o c cur when two le ading tr e es τ i and τ j ar e r e alize d as subtr e es of a shuﬄe tr e e m such that: (1) they shar e at le ast one c ommon internal vertex, and (2) al l internal vertic es of m b elong to the union of these two subtr e es. • F or e ach such ambiguity, we form the S -p olynomial (the diﬀer enc e of p ossible r e ductions): S ( i, j ) = ( r e duction of m using rule i ) − ( r e duction of m using rule j ) . • The system is c onﬂuent if for every ambiguity, the c orr esp onding S -p olynomial r e duc es to zer o: S ( i, j ) → S . . . → S 0 . The following theorem is the analogue of Bergman’s Diamond Lemma for sh uﬄe op erads: Theorem 2.3.14. A terminating r ewriting system S is c onﬂuent if and only if the set B S irr forms a ve ctor sp ac e b asis for the shuﬄe op er ad P . In this c ase, every element in P p ossesses a unique normal form in Span ( B S irr ) . It is imp ortan t to note that the rewriting system { (lm( f ) , f ) : f ∈ G } asso ciated with a Gr¨ obner basis G is alwa ys terminating and conﬂuen t. How ever, there exist rewriting systems that do not originate from a Gr¨ obner basis. 10 2.4 Koszulness and Anic k-type resolution One of the most prominent homological applications of noncomm utative Gr¨ obner bases is the Anic k resolution of the trivial mo dule. This construction is based on deforming the diﬀeren tial in the minimal resolutions of monomial algebras, as in tro duced in [3] (for a detailed combinatorial treatmen t of the Anick resolution and Anic k chains, we recommend the exp osition in [52, § 3]). Unfortunately , a similar description for the minimal resolution of a sh uﬄe op erad with monomial relations is currently unknown. How ever, in [14, § 2.1], we deﬁned a dg-op erad I q Q , termed the ”inclusion-exclusion” op erad, which pro vides a quasi-free resolution of a monomial sh uﬄe op erad; this resolution prov es to b e minimal in many sp eciﬁc cases. W e recall this construction b elo w, as w e are interested in its applications (see § 4.4). Deﬁnition 2.4.1. The diﬀer ential-gr ade d inclusion-exclusion op er ad ( I q Q , d I ) asso ciate d with a shuﬄe op er ad Q = F ( E ; R ) – gener ate d by the set E subje ct to an ide al of r elations gener ate d by the set of monomials R – is the dg-op er ad whose monomial b asis is given by:  m ⊗ g 1 ∧ . . . ∧ g k deg( m ⊗ g 1 ∧ . . . ∧ g k ) = k     m − a shuﬄe monomial in F ( E ) , g i − a divisor of m isomorphic to a monomial r elation in R  The diﬀer ential in I q Q omits one of the divisors while le aving the underlying monomial m un- change d: d I ( m ⊗ g 1 ∧ . . . ∧ g k ) := k X i =1 ( − 1) k m ⊗ g 1 ∧ . . . ∧ b g i ∧ . . . ∧ g k The gener ators of the fr e e shuﬄe op er ad I q Q ar e monomials with divisors that c annot b e factor e d as a pr o duct of two such monomials; in p articular, the divisors { g q } must form a non-split c overing of the underlying op er adic tr e e. Starting from a sh uﬄe operad P with a chosen Gr¨ obner basis (or a terminating conﬂuen t rewriting system), we consider the inclusion-exclusion quasi-free op erad I q gr P asso ciated with the corresp onding monomial shuﬄe operad gr P . By deforming the diﬀerential, we obtain a resolution of P , denoted by I q P (see [14] for details). Corollary 2.4.2. If the 2 -c olor e d op er ad Ψ( D ) asso ciate d with a diop er ad D admits a quadr atic terminating c onﬂuent r ewriting system, then: • the shuﬄe op er ad Ψ( D ! ) asso ciate d with the quadr atic dual diop er ad D ! also admits a quadr atic terminating c onﬂuent r ewriting system; • the diop er ads D and D ! ar e Koszul. Pr o of. Consider a (colored) sh uﬄe op erad Q with quadratic monomial relations. Its quadratic dual shuﬄe op erad Q ! p ossesses complementary quadratic monomial relations (i.e., a quadratic monomial is zero in Q ! if and only if it is non-zero in Q ). F urthermore, there is a one-to-one corresp ondence b etw een the monomial generators of the inclusion-exclusion op erad I q Q and the non-zero sh uﬄe monomials in Q ! . F or grading reasons, this implies that Q is Koszul and that the inclusion-exclusion op erad I q Q is indeed the minimal resolution of Q . No w, supp ose that the (colored) shuﬄe op erad P admits a quadratic terminating conﬂuent rewriting system such that the associated monomial quadratic shuﬄe operad is isomorphic to Q . By similar degree considerations (where the cohomology grading coincides with the syzygy grading), the corresp onding inclusion-exclusion resolution I q P with the deformed diﬀerential remains minimal. Consequen tly , P is Koszul, and the monomials of Q ! deﬁne a basis for P ! . Th us, for an y quadratic monomial g ∈ F ( E ) that v anishes in Q ! , one can assign a rewriting rule g 7→ f , where f is the linear combination of non-zero monomials in Q ! represen ting the 11 class of g in P ! . The resulting rewriting system is clearly terminating and conﬂuent b ecause the monomials from Q ! constitute a basis. W e refer to § 4.4 for an explicit example of the Anick resolution applied to a diop erad with non-quadratic relations. 3 Coloring of inputs/outputs in a (cyclic) op erad 3.1 The pictorial maps Θ with examples Θ( Lie ) and Θ( A ss ) In this section we suggest more pictorial maps gov erened by coloring of inputs and outputs. As an application we provide a wa y of pro ducing rewriting systems for certain diop erads. Deﬁnition 3.1.1. A cyclic op er ad P c onsists of • a c ol le ction of m -ary op er ations P ( m + 1) with m ⩾ 1 e quipp e d with S m +1 action; • inﬁnitesimal c omp osition rules: i ◦ j : P ( m + 1) ⊗ P ( n + 1) → P ( m + n ) that c onne cts i ’th and j ’th inputs c orr esp ondingly. such that c omp osition rules ar e asso ciative and S -e quivariant. In particular, after ﬁxing an output in n -ary op erations P b ecame an ordinary symmetric op erad. Deﬁnition 3.1.2. • We c al l the subset c ∈ Z ⩾ 0 × Z ⩾ 0 to b e the c oloring rule if it is close d under the fol lowing op er ation: if ( m, n ) , ( m ′ , n ′ ) ∈ c then ( m + m ′ − 1 , n + n ′ − 1) ∈ c . • The c -c oloring of the inputs/outputs by the rule: Θ c ( P )( m, n ) := ( P ( m + n ) , if ( m, n ) ∈ c , 0 , otherwise. deﬁnes an exact functor fr om the c ate gory of Cyclic op er ads to the c ate gory of diop- er ads . Wher e the c omp osition rules in a diop er ad c oincide with the r estriction of the c omp osition rule in the cyclic op er ad P . The examples of the coloring rules include: Z > 0 × { 1 } ; Z > 0 × Z > 0 ; Z ⩾ 0 × Z > 0 ; { ( n, n ) : n ∈ Z > 0 } ; { ( kn + 1 , l n + 1) : n ∈ Z > 0 } . In particular, • The coloring with resp ect to the c = Z > 0 × { 1 } deﬁnes an ordinary symmetric op erad by forgetting the cyclic structure; • The coloring Θ Z > 0 × Z > 0 applied to the commutativ e op erad Comm gives the F robenius diop erad (see § 4.1); • The coloring Θ c with resp ect to the set Z ⩾ 0 × Z > 0 applied to the same op erad Comm of comm utativ e algebra gives the diop erad that is Koszul dual to the diop erad of quasi-Lie bialgres and to the coloring Z ⩾ 0 × Z ⩾ 0 is isomorphic to the diop erad Koszul dual to the diop erad of pseudo Lie bialgebras (see § 4.3); • the coloring with an equal n umber of inputs/outputs of the same commutativ e op erad pro vides the dioperad Koszul dual to the diop erad of quadratic P oisson structures (see § 4.6 for details). 12 Note that the functor Θ c ma y signiﬁcantly alter the sets of generators and relations. Ho w- ev er, Theorem 3.2.11 demonstrates that, under certain assumptions, this functor enables the construction of a Gr¨ obner basis for the image and facilitates pro ofs of Koszulness for v arious natural diop erads. Example 3.1.3. The diop er ad Θ Z > 0 × Z > 0 ( Lie ) is gener ate d by a p air of skew-symmetric gener- ators (Lie br acket and Lie c obr acket): Θ Z > 0 × Z > 0 ( Lie )(2 , 1) := sp an  2 1 = − 1 2  , Θ Z > 0 × Z > 0 ( Lie )(1 , 2) := sp an  2 1 = − 1 2  by the S op × S -invariant ide al gener ate d by the fol lowing line ar c ombinations                3 2 1 + 1 3 2 + 2 1 3 ; 3 2 1 + 1 3 2 + 2 1 3 ; 2 1 1 2 − 1 2 1 2 + 2 1 1 2 ; 1 2 1 2 − 2 1 2 1 .                Example 3.1.4. The diop er ad Θ Z > 0 × Z > 0 ( A ss ) assigne d to the cyclic op er ad of asso ciative al- gebr as is gener ate d by a p air of nonsymmetric multiplic ation and c omultiplic ation: Θ Z > 0 × Z > 0 ( A ss )(2 , 1) := sp an  2 1 , 1 2  , Θ Z > 0 × Z > 0 ( A ss )(1 , 2) := sp an  2 1 , 1 2  Subje ct to the fol lowing list of r elations (and al l their symmetries):      3 2 1 = 1 2 3 ; 3 2 1 = 1 2 3 ; 2 1 1 2 = 1 2 1 2 = 2 1 2 1      . 3.2 Deﬁning relations and Gr¨ obner basis for diop erads Θ c ( P ) The results in this section can b e readily adapted to v arious other settings; how ever, to a void unnecessary technical complications, w e restrict our attention to the sp eciﬁc case c := Z > 0 × Z > 0 . In this context, to each cyclic generator γ ∈ P ( n + 1), we assign n generators of diﬀerent arities γ i,j ∈ Θ c ( P )( i, j ), where i + j = n + 1 and i, j > 0. In particular, one m ust exclude all op erations of arities ( n + 1 , 0) and (0 , n + 1). Prop osition 3.2.1. Supp ose that the cyclic op er ad P admits a pr esentation F ( E ; R ) as a symmetric op er ad. In p articular, assume that E and R p ossess cyclic symmetry. Then, the diop er ad Θ c ( P ) is gener ate d by Θ c ( E ) – obtaine d by c oloring the gener ators as inputs/outputs –subje ct to the union of the r elations Θ c ( R ) (the set of al l p ossible c olorings of inputs, outputs, and internal e dges of the r elations in R ) and the fol lowing quadr atic r elations, which admit the pictorial description b elow: ∀ α, β ∈ E α ... ... β ... ... = α ... ... β ... ... (3.2.2) 13 Pr o of. Directly from the deﬁnitions of the pictorial functors Θ c and Ψ, it follows that any elemen t in the 2-colored op erad Ψ(Θ c ( P )) can b e represen ted as a linear com bination of operadic trees underlying P , equipp ed with a coloring of all edges in one of t wo colors. This coloring is sub ject to a mild constraint: no v ertex exists where all incoming edges are of one color and the outgoing edge is of a diﬀeren t color. In particular, Θ c ( P ) is clearly generated b y Θ c ( E ), since eac h corolla is obtained b y assigning an appropriate coloring to the adjacent edges – eﬀectively determining whic h are inputs and which are outputs. Relation (3.2.2) implies that the coloring of in ternal edges in the op eradic tree is immaterial. This ensures that all aforemen tioned relations are satisﬁed in Θ c ( P ). Con v ersely , one can easily sho w by induction on the size of a coloured monomial M that an y t w o diﬀerent admissible colorings of in ternal edges of M can b e connected by a sequence of c hanging the color in one edge. Therefore, the size of the diop erad generated by Θ c ( E ), sub ject to the coloring of relations Θ c ( R ) and the inv ariance of internal colors, cannot exceed the size of the diop erad Θ c ( P ), which concludes the pro of. In particular, we observ e that the functor Θ c maps quadratic cyclic op erads to quadratic di- op erads. It is not currently kno wn whether Θ c maps Koszul cyclic op erads to Koszul diop erads, though we susp ect this is not generally the case. How ever, we provide b elo w suﬃcien t condi- tions on the Gr¨ obner basis of a cyclic op erad P that ensure the Koszulness of the corresp onding diop erad Θ c ( P ). T o this end, we must analyze the eﬀect of the comp osition of functors Ψ ◦ Θ c on shuﬄe monomials. Indeed, supp ose we choose a set of shuﬄe generators { γ i : i ∈ I } for the free cyclic op erad F generated b y a cyclic collection E . In other w ords, { γ i : i ∈ I } deﬁnes a basis for the S -collection E . Then the 2-colored sh uﬄe op erad SH (Ψ(Θ c ( F ))) is generated by the follo wing set: { γ i, ¯ c : i ∈ I , ¯ c ∈ F m i +1 2 \ { (0; 1 , . . . , 1) , (1; 0 , . . . , 0) } , where ar ( γ i ) = m i } . (3.2.3) Sp eciﬁcally , to each generator γ i of cyclic arity m i + 1, we assign a (2 m i +1 − 2)-tuple of m i -ary generators in the 2-colored shuﬄe op erad. The v ector ¯ c denotes the coloring of the inputs and outputs: c 0 = 0 if the output is a straight edge, while for j = 1 , . . . , m , the j -th co ordinate c j equals 0 (resp. 1) if the j -th input of γ i, ¯ c is colored as a straight (resp. dotted) edge. F urthermore, the quadratic relation in Equation (3.2.2) leads to the following set of quadratic equations in the shuﬄe op erad: γ i, ( ..., 1 ,... ) ◦ a γ j (1; ... ) = γ i, ( ..., 0 ,... ) ◦ a γ j, (0; ... ) . (3.2.4) That we call the ”recoloring relation” b ecause they c hange the color of an inner edge in the sh uﬄe colored op erad SH (Ψ(Θ c ( F ))). The corresp onding rewriting rule, when applied to an y colored sh uﬄe monomial, do es not c hange the underlying uncolored monomial. Thus, to an y colored shuﬄe monomial T in the free 2-colored shuﬄe op erad generated by (3.2.3), we assign the set: [ T ] c := { T ′ ∈ F ( γ i, ¯ c ) : Shap e ( T ′ ) = Shap e ( T ) } , where Shape ( T ) denotes the underlying shuﬄe monomial obtained b y forgetting the colors of all internal edges while preserving the colors of the inputs and outputs. Let us deﬁne a partial order ≺ on this set by declaring the co v ering relations to b e given b y recoloring a single dotted in ternal edge to a straight one. As illustrated in the following diagrammatic Example (3.2.5), this p oset may lack a supremum, and instances without an inﬁm um also exist.       c :=      ≻ ≺      . (3.2.5) 14 Ho w ev er, such cases do not o ccur for caterpillar monomials, whic h we deﬁne as follows: Deﬁnition 3.2.6. • A n op er adic tr e e is c al le d a c aterpil lar tr e e if it c ontains no vertex with mor e than one inc oming internal e dge. Equivalently, the subtr e e sp anne d by the internal vertic es is a p ath (also r eferr e d to as the spine or trunk), and al l le aves ar e attache d dir e ctly to this p ath. Pictorial ly, the fol lowing subtr e e is forbidden: • A shuﬄe tr e e monomial T is c al le d a c aterpil lar monomial if its underlying op er adic tr e e is a c aterpil lar tr e e. Lemma 3.2.7. F or any c oloring of the inputs and outputs of a c aterpil lar shuﬄe monomial T , the p oset ([ T ] c , ≺ ) of al l c olor e d shuﬄe monomials of a given shap e p ossesses a unique supr emum T max . This supr emum is char acterize d by having the maximal p ossible numb er of internal e dges c olor e d as str aight. Pr o of. Recall that for a caterpillar monomial, the set of in ternal edges forms a spine (a central path). F or a given coloring of the leav es (inputs and outputs), certain internal edges may hav e forced colors, while others ma y admit diﬀerent colorings. W e claim that there exists a unique maximal monomial T max in which all such ”ﬂexible” internal edges are colored as straight. Consider any monomial T ∈ [ T ] c . If the color of the edge adjacent to the ro ot is ﬁxed, we can cut this edge and pro ceed b y induction on the size of the caterpillar monomial. So, without loss of generality , we assume that the ﬁrst edge is ﬂexible. This may happ en whenev er the color of the output coincides with the color of one of the non-spine inputs of the ro ot vertex. If T is not T max , there must exist at least one internal edge e colored as dotted that can b e recolored as straigh t without violating the consistency relations at the v ertices. T o see this, consider the set of nonro oted vertices where one of the inputs is colored as straight but the output is colored as dotted. Among these, let v b e a vertex that is closest to the ro ot, and let v ′ b e the adjacent v ertex immediately closer to the ro ot. By the minimality of v , the v ertex v ′ m ust either hav e a straigh t output, or all of its inputs must b e dotted, or v ′ is a root (otherwise, v ′ w ould ha ve been c hosen instead of v ). Consequently , the conﬁguration allo ws for the internal edge connecting v and v ′ to b e changed from dotted to straight. By iterativ ely applying this recoloring process, w e mo v e upw ard in the p oset ([ T ] c , ≺ ) un til all p ossible in ternal edges are straight. This pro cess terminates at a unique element T max , which serves as the supremum of the p oset. Deﬁnition 3.2.8. A terminating and c onﬂuent r ewriting system S for a (c olor e d) shuﬄe op er ad is c al le d c aterpil lar if every irr e ducible shuﬄe monomial in B S irr is a c aterpil lar monomial. Example 3.2.9. • The standar d quadr atic Gr¨ obner b asis of the c ommutative op er ad Comm (with r esp e ct to the p ath-lexic o gr aphic al or dering) deﬁnes a c aterpil lar r ewriting system. This is b e c ause al l irr e ducible monomials ar e right-normalize d c ombs, wher e the internal vertic es gr ow exclusively to the right. • The quadr atic Gr¨ obner b asis of the Lie op er ad Lie (with r esp e ct to the r everse p ath- lexic o gr aphic al or dering) is c aterpil lar, as its normal forms c onsist of left-normalize d c ombs, wher e the internal vertic es gr ow exclusively to the left. • Ther e exists a quadr atic Gr¨ obner b asis for the asso ciative op er ad Asso c such that al l normal forms ar e left-normalize d c ombs (se e e.g. [29, Example 2.14]) and thus Asso c admits a quadr atic c aterpil lar r ewriting system. 15 • F ol lowing r esults on gener ating series (se e [29, Example 5.11]), one c an show that the Koszul-self dual cyclic op er ad Pois of Poisson algebr as do es not admit a (quadr atic) c ater- pil lar r ewriting system. Supp ose that the shuﬄe op erad SH ( P ) asso ciated with a given cyclic op erad P admits a caterpillar rewriting system. That is, w e are given a set of generators { γ i : i ∈ I } of arit y m i and a terminating, conﬂuen t rewriting system S = { ( τ j , f j ) } j ∈ J suc h that all normal forms are caterpillar. In particular, eac h f j is spanned by caterpillar monomials. Then, b y Lemma 3.2.7, for each j ∈ J , any coloring ¯ c of the inputs/outputs of τ j , and an y coloring ¯ b of its internal edges, w e can assign the following rewriting rule: ( τ j ) ¯ c, ¯ b → [ f j ] max ¯ c . (3.2.10) In other words, we rewrite every colored sh uﬄe monomial ( τ j ) ¯ c, ¯ b as a sum of caterpillar mono- mials p ossessing the maximal num b er of straigh t internal edges. W e denote the union of the ”recoloring” rules from (3.2.4) and the rules deﬁned in (3.2.10) as Θ c ( S ). W e are now ready to state the main result of this section. Theorem 3.2.11. Supp ose that a terminating and c onﬂuent r ewriting system S for the shuf- ﬂe op er ad asso ciate d with a cyclic op er ad P is c aterpil lar. Then Θ c ( S ) is a terminating and c onﬂuent c aterpil lar r ewriting system for the diop er ad Θ c ( P ) . Pr o of. First, by Prop osition 3.2.1, the set of generators describ ed in (3.2.3) generates the sh uﬄe op erad SH (Ψ(Θ c ( F ( E )))). Second, it is straightforw ard to verify that these rewriting rules generate the ideal of relations for the diop erad. Third, T o pro v e termination, we observe that each recoloring rule (3.2.4) increases the n um b er of straigh t internal edges. Since the n umber of in ternal edges is ﬁnite for any given monomial, the recoloring process m ust terminate. The ov erall termination of Θ c ( S ) then follo ws from the termination of the underlying system S . F orth, similarly , for conﬂuence, it suﬃces to notice that after applying a suﬃcien t num b er of recoloring rules (3.2.4), w e only need to v erify conﬂuence for caterpillar monomials with the maximal n umber of straight edges. This conﬂuence is equiv alent to the conﬂuence of the original rewriting system S . Corollary 3.2.12. If the shuﬄe op er ad asso ciate d with a cyclic op er ad P admits a quadr atic Gr¨ obner b asis with c aterpil lar normal forms (or, e quivalently, a quadr atic terminating and c on- ﬂuent c aterpil lar r ewriting system), then the c orr esp onding 2 -c olor e d shuﬄe op er ad SH (Ψ(Θ c ( P ))) also admits a quadr atic terminating and c onﬂuent r ewriting system. Conse quently, the diop er ad Θ c ( P ) is Koszul. Pr o of. By Theorem 3.2.11, the 2-colored op erad Ψ(Θ c ( P )) admits a terminating and conﬂuent rewriting system that gov erns the quadratic relations, pro vided the cyclic op erad p ossesses a quadratic Gr¨ obner basis. The Koszulness of the diop erad Θ c ( P ) then follo ws directly from Corollary 2.4.2. Corollary 3.2.13. The diop er ads Θ c ( Lie ) and Θ c ( A ss ) describ e d in Examples 3.1.3 and 3.1.4 admits a quadr atic terminating c onﬂuent r ewriting systems and, ther efor e, ar e Koszul diop er ads. Remark 3.2.14. It is worth noting that the statements of Pr op osition 3.2.1 and The or em 3.2.11 c an b e r e adily gener alize d to the various c ases of c discusse d ab ove. In p articular, for most standar d diop er adic c olorings c , the Koszulness of the diop er ad Θ c ( P ) is ensur e d by the existenc e of a quadr atic (c aterpil lar) Gr¨ obner b asis for the underlying cyclic op er ad P . 16 In the subsequent section, we discuss v arious applications of this result. Our examples include the dioperads of F rob enius algebras ( § 4.1), Lie bialgebras ( § 4.2), quasi-Lie bialgebras, and pseudo-Lie bialgebras ( § 4.3). Notably , this framework pro vides a straightforw ard pro of for the Koszulness of the diop erad of quadratic P oisson structures. 4 Examples In this section, w e describe Gr¨ obner bases and Hilb ert series of dimensions for diﬀerent diop- erads that app ear in the literature and whose description is motiv ated by the category of their represen tations. 4.1 The F rob enius diop erad F r ob Recall that a non-unital F rob enius algebra is a triple ( V , · , ∆), where · deﬁnes a comm utative asso ciativ e multiplication and ∆ a co comm utative coasso ciativ e comultiplication, such that for all a, b ∈ V : ∆( a · b ) = a (1) ⊗ ( a (2) · b ) = ( a · b (1) ) ⊗ b (2) . Here, we employ Sweedler’s notation ∆( a ) := a (1) ⊗ a (2) for the comultiplication map. The corresp onding diop erad (prop erad) of F rob enius algebras F r ob is generated by a sym- metric multiplication of arity (2 , 1) and a symmetric com ultiplication of arity (1 , 2). These are depicted as trees with black v ertices: 2 1 = 1 2 ; 2 1 = 1 2 . The quadratic relations imply that the space of quadratic comp ositions is one-dimensional for all arities where a comp osition is p ossible:              3 2 1 = 1 3 2 = 2 1 3 ; 3 2 1 = 1 3 2 = 2 1 3 ; 1 2 1 2 = 1 2 2 1 = 2 1 1 2 = 2 1 2 1 = 2 1 1 2              Consequen tly , it follows that dim F r ob ( m, n ) = 1 for all m, n ⩾ 1. Remark 4.1.1. The F r ob enius diop er ad is isomorphic to the c oloring Θ c ( Comm ) for c := Z > 0 × Z > 0 . In p articular, sinc e the c ommutative op er ad admits a quadr atic Gr¨ obner b asis with c aterpil lar normal forms, the F r ob enius diop er ad likewise admits a quadr atic, terminating, and c onﬂuent r ewriting system. Below, we explain that this r ewriting system is originate d fr om an appr opriate quadr atic Gr¨ obner b asis. The 2-colored shuﬄe op erad SH (Ψ( F r ob )) p ossesses six binary generators with distinct input/output colorings, which we order as follo ws: α 1 2 ⩽ β 1 2 ⩽ γ 1 2 ⩽ δ 1 2 ⩽ ε 1 2 ⩽ ζ 1 2 The relations reﬂect the prop ert y that a comp osition of tw o op erations, if it exists, is uniquely determined b y the coloring of the inputs and outputs. F or the path-lexicographical ordering, the set of normal forms of degree 2 is similarly determined by the input/output coloring. These 17 normal forms consist of right-gro wing trees (righ t com bs), where in ternal edges are straight whenev er such a coloring is admissible: α α 1 2 3 , α ϵ 1 2 3 , α γ 1 2 3 , γ α 1 2 3 , γ ε 1 2 3 , γ γ 1 2 3 , ε ζ 1 2 3 ; δ δ 1 2 3 , δ β 1 2 3 , β α 1 2 3 , δ ζ 1 2 3 , β ε 1 2 3 , β γ 1 2 3 , ζ ζ 1 2 3 . (4.1.2) Note that while there are 8 p ossible w ays to color the inputs and output of a binary corolla, and 16 for a corolla with 3 inputs, we observe only 6 binary generators and 14 quadratic com- p ositions. The ”forbidden” op erations corresp ond to cases where all inputs share one color but the output is colored diﬀerently . (A more comprehensive discussion of the remaining generators and op erations is pro vided in § 4.3, within the con text of quasi- and pseudo-Lie bialgebras.) It follows that for any p ermitted coloring, there exists exactly one normal form (a tree monomial that is not divisible b y the leading term of an y relation). This normal form is a tree gro wing strictly to the righ t, in which all in ternal edges are colored straight whenever p ossible. This normal form is non-zero as it represents the unique non-zero op eration in F r ob ( m, n ). 4.2 The diop erad L ieb of Lie bialgebras Recall that a Lie-bialgebr a is a vector space V together with t wo op erations [ , ] : V ∧ V → V and δ : V → V ∧ V . Where [ , ] is a Lie-brack et, δ is a Lie co-brac ket and the op erations satisfy the relation δ ([ a, b ]) = ( ad a ⊗ 1 + 1 ⊗ ad a ) δ ( b ) − ( ad b ⊗ 1 + 1 ⊗ ad b ) δ ( a ) for any a, b ∈ V and ad a ( b ) = [ a, b ]. The corresponding diop erad L ieb of Lie bialgebras has the follo wing pictorial description. L ieb is generated by an S × S op -bimo dule E = { E ( m, n ) } m,n ⩾ 1 with all E ( m, n ) = 0 except E (2 , 1) := span  2 1 = ( − 1) 1 2  , E (1 , 2) := span  2 1 = ( − 1) 1 2  b y the ideal generated b y the follo wing relations                3 2 1 + 1 3 2 + 2 1 3 = 3 2 1 + 1 3 2 + 2 1 3 = 0; − 2 1 1 2 − 1 2 1 2 − 2 1 1 2 + 2 1 2 1 − 1 2 2 1 = 0                (4.2.1) The diop erad L ieb is Koszul dual to the diop erad F r ob of F rob enius algebras (see e.g. [41]). This follows that L ieb admits a quadratic Gr¨ obner basis with resp ect to the opp osite ordering of monomials. In particular, the quadratic righ t-comb monomials (4.1.2) assemble the set of leading terms of this quadratic Gr¨ obner basis. 18 Prop osition 4.2.2. The gener ating series of the diop er ad of Lie bialgebr as is given by: χ L ieb ( u, v ) := X m,n ⩾ 1 dim L ieb ( m, n ) u m m ! v n n ! = − Z ln 1 − u + v + p 1 − 2( u + v ) + ( u − v ) 2 2 ! d v . In p articular, the dimensions of the sp ac es L ieb ( m, n ) ar e: dim L ieb ( m, n ) = ( n + m − 2)!  n + m − 2 m − 1  = ( m + n − 2)! 2 ( m − 1)!( n − 1)! . Pr o of. The dioperad of Lie bialgebras is Koszul, and its Koszul dual is the diop erad of F rob enius algebras F r ob . The generating series for F r ob is particularly simple because for ev ery m ⩾ 1 and n ⩾ 1, there exists exactly one operation (up to a scalar) with m inputs and n outputs. Consequen tly , the generating series is: χ F r ob ( x, y ) := X m,n ⩾ 1 dim F r ob ( m, n ) x m m ! y n n ! = X m,n ⩾ 1 x m m ! y n n ! = ( e x − 1)( e y − 1) . The generating series of the asso ciated colored op erad Ψ( F r ob ) consists of tw o comp onents: χ Ψ( F r ob ) =  ∂ χ F r ob ( x, y ) ∂ y , ∂ χ F r ob ( x, y ) ∂ x  =  e x + y − e y , e x + y − e x  . Since all generators of Ψ( F r ob ) are binary , the q -grading is uniquely determined by the arit y of the op erations. Thus, we ha ve the follo wing functional equation for the generating series of the Koszul dual: χ Ψ( L ieb ) ( − x, − y ) ◦ χ Ψ( F r ob ) ( − x, − y ) = ( x, y ) . Setting u = ∂ χ L ieb ∂ v and v = ∂ χ L ieb ∂ u , we solve the functional equation system: ( u, v ) = − χ Ψ( F r ob ) ( − x, − y ) ⇔ ( u = e − y − e − x − y v = e − x − e − x − y ⇔ ( u − v = e − y − e − x v = e − x − e − x e − y Substituting e − y = u − v + e − x in to the second equation: v = e − x − e − x ( u − v + e − x ) = ⇒ e − 2 x − e − x (1 − u + v ) + v = 0 . By the quadratic formula: e − x = 1 − u + v ± p (1 − u + v ) 2 − 4 v 2 . Since w e are working with p o wer series where x → 0 as u, v → 0, w e must hav e e − x → 1. Only the p ositiv e root satisﬁes this b oundary condition. Thus, we obtain: x = − ln 1 − u + v + p 1 − 2( u + v ) + ( u − v ) 2 2 ! . F rom the functional e quation, we identify x = ∂ χ L ieb ( u,v ) ∂ v . Let z = 1 − e − x . Then we ha v e the following equality: z u = 1 − e − x u = e y = e − x e − x − y = 1 − z 1 − v − z =  1 − v 1 − z  − 1 . 19 Let H ( z ) := − ln(1 − z ) = x . W e apply the Lagrange Inv ersion Theorem (LIT) with resp ect to z , treating v as a parametr: [ u m ] ∂ χ L ieb ( u, v ) ∂ v = [ u m ] x ( u, v ) = [ u m ] H ( z ) LIT = LIT = 1 m [ z m − 1 ]  H ′ ( z )  z u  m  = 1 m [ z m − 1 ] 1 1 − z ·  1 − v 1 − z  − m ! . No w we can recall that v is also a formal v ariable and compute the co eﬃcient near v n : [ v n ][ u m ] x ( u, v ) = [ v n ] 1 m [ z m − 1 ] 1 1 − z ·  1 − v 1 − z  − m !! = = 1 m [ z m − 1 ]  1 1 − z · [ v n ]  1 − v 1 − z  − m  = 1 m [ z m − 1 ]  1 1 − z ·  (1 − z ) − n  n + m − 1 m − 1  = = 1 m  n + m − 1 m  [ z m − 1 ](1 − z ) − n − 1 = 1 m  n + m − 1 m − 1  n + m − 1 m − 1  = ( n + m − 1)!  n + m − 1 m − 1  m ! n ! Consequen tly , we hav e: dim L ieb ( m, n ) = dim Ψ( L ieb ) | ( m, n − 1) = ( m + n − 2)!  m + n − 2 m − 1  . W e refer to [50, § 6.2] for details on the Lagrange In version Theorem. 4.3 Quasi-Lie bialgebras and Pseudo-Lie bialgebras While considering diﬀeren t algebraic structures related to quan tum groups, Drinfeld also deﬁned diﬀeren t generalizations of the notion of Lie bialgebras by adding extra structures to them (see e.g. [18]). Let us recall some of them Deﬁnition 4.3.1. • A pseudo-Lie bialgebr a is a 5-tuple ( V , [ − , − ] , δ, ϕ, η ) wher e V is a Z gr ade d ve ctor sp ac e and δ ∈ Hom( V , Λ 2 V ) , [ − , − ] ∈ Hom(Λ 2 V , V ) and ϕ ∈ Hom( k , Λ 3 V ) , η ∈ Hom(Λ 3 V , k ) such that δ [ a, b ] = ( ad a ⊗ 1 + 1 ⊗ ad a ) δ ( b ) − ( ad b ⊗ 1 + 1 ⊗ ad b ) δ ( a ) + ϕ (1) ⊗ ϕ (2) ( η ( ϕ (3) ⊗ a ⊗ b )); 1 2 Alt 3 ( δ ⊗ id)( δ ( a )) = [ a ⊗ 1 ⊗ 1 + 1 ⊗ 1 ⊗ 1 + 1 ⊗ 1 ⊗ a, ϕ ]; [[ a, b ] , c ] + [[ c, a ] , b ] + [[ b, c ] , a ] = = η ( a, b, δ (1) ( c )) δ (2) ( c )+ η ( c, a, δ (1) ( b )) δ (2) ( b ) + η ( b, c, δ (1) ( a )) δ (2) ( a ); Alt 4 ( δ ⊗ id ⊗ id)( ϕ ) = 0; η ([ x, y ] , z , w ) + η ([ x, z ] , y , w ) + η ([ x, w ] , y , z )+ + η ([ y , z ] , x, w ) + η ([ y , w ] , x, z ) + η ([ z , w ] , x, y ) = 0 . wher e Alt k : V ⊗ k → V ⊗ k denotes the op er ator P σ ∈ S k sgn( σ ) σ . • A quasi-Lie bialgebr a is a pseudo-Lie algebr a wher e η = 0 . • When η = 0 and ϕ = 0 , the deﬁnition r e duc es to that of a Lie bi-algebr a ( V , [ - , - ] , δ ) . 20 The corresp onding diop erad P L ieb n is generated by the following sk ew-symmetric opera- tions: E ( m, n ) =            ( m, n ) = (0 , 3) ( m, n ) = (1 , 2) ( m, n ) = (2 , 1) ( m, n ) = (3 , 0) mo dulo the ideal generated b y R ( m, n ) =                                        1 2 3 4 + 1 3 2 4 + 1 4 2 3 + 2 3 1 4 + 2 4 1 3 + 3 4 1 2 , ( m, n ) = (0 , 4) 1 2 3 + 2 3 1 + 3 1 3 + 1 2 3 + 2 3 1 + 3 1 2 , ( m, n ) = (1 , 3) 1 2 1 2 + 1 2 1 2 + 2 1 1 2 + 1 2 2 1 + 2 1 2 1 + 1 2 1 2 , ( m, n ) = (2 , 2) 1 2 3 + 2 3 1 + 3 1 2 + 1 2 3 + 2 3 1 + 3 1 2 , ( m, n ) = (3 , 1) 1 2 3 4 + 1 3 2 4 + 1 4 2 3 + 2 3 1 4 + 2 4 1 3 + 3 4 1 2 , ( m, n ) = (4 , 0) Similarly , one ma y deﬁne the diop erad (or properad) QL ieb of Quasi-Lie bialgebras. The Koszulness of the corresp onding prop erad (and consequently the diop erads) was established in [24], follo wing the metho dology used for the Koszul prop ert y of the prop erad of Lie bialgebras. In particular, it can b e sho wn that each space of ( m, n )-ary op erations in the Koszul-dual diop erad is one-dimensional whenev er it is non-empty . Let us adapt the framework dev elop ed in this pap er to these examples. Prop osition 4.3.2. • The fol lowing isomorphisms of diop er ads hold: P L ieb ! ≃ Θ Z ⩾ 0 × Z ⩾ 0 ( Comm ) , QL ieb ! = Θ Z ⩾ 0 × Z > 0 ( Comm ) . (4.3.3) In p articular, b oth diop er ads admit a quadr atic, terminating, c onﬂuent, c aterpil lar r ewrit- ing system. • Sp e ciﬁc al ly, the 2 -c olor e d shuﬄe op er ad Ψ( P L ieb ) is gener ate d by eight binary gener ators (one for e ach p ossible c oloring), subje ct to sixte en quadr atic r elations (one for e ach p ossible input/output c oloring). • The r ewriting system S := ( τ i → f i ) , wher e τ i is a right-gr owing shuﬄe ternary tr e e with an internal e dge c olor e d str aight, is b oth terminating and c onﬂuent. Pr o of. First, a direct comparison of the generators and relations for P L ieb ! and QL ieb ! with those arising from the diﬀeren t colorings of the comm utativ e op erad sho ws Isomorphism (4.3.3). Second, we generalize the arguments we pro vided for Lie bialgebras; it is suﬃcient to construct the Gr¨ obner basis for the Koszul-dual 2-colored shuﬄe op erad, which is a straightforw ard veri- ﬁcation. 4.4 The diop erad L ieb △ of triangular Lie bialgebras The follo wing algebraic concepts were introduced by V. Drinfeld in [17] as a foundational com- p onen t of his seminal w ork on quantum groups. 21 Deﬁnition 4.4.1. A Lie bialgebr a ( g , [ - , - ] , δ ) is c al le d a c ob oundary Lie bialgebr a if ther e exists an element r ∈ g ⊗ g such that the Lie c obr acket δ : g → g ⊗ g is deﬁne d by the fol lowing c ob oundary c ondition: ∀ x ∈ g δ ( x ) = ad x ( r ) = [ x ⊗ 1 + 1 ⊗ x, r ] . Deﬁnition 4.4.2. A c ob oundary Lie bialgebr a ( g , [ - , - ] , δ, r ) is said to b e triangular if the deﬁning r -matrix is skew-symmetric (i.e., r ∈ Λ 2 g ) and satisﬁes the Classic al Y ang-Baxter Equation (CYBE): [[ r , r ]] = [ r 12 , r 13 ] + [ r 12 , r 23 ] + [ r 13 , r 23 ] = 0 (4.4.3) wher e [[ · , · ]] denotes the algebr aic Schouten br acket on g ⊗ g ⊗ g . The diop erad L ieb △ , whose represen tations corresp ond to triangular Lie bialgebras, is gener- ated by tw o sk ew-symmetric operations in arities (2 , 1) and (0 , 2), resp ectiv ely . These generators represen t the Lie brack et and the r -matrix: 2 1 = ( − 1) 1 2 := [- , -]; 2 1 = ( − 1) 1 2 := r . The structure is go verned by the follo wing relations, represen ting the Jacobi identit y and the CYBE:    3 2 1 + 1 3 2 + 2 1 3 = 0; 1 2 3 + 2 3 1 + 3 1 2 = 0    (4.4.4) F urthermore, the asso ciated 2-colored sh uﬄe op erad Ψ( L ieb △ ) is generated b y one unary generator and three binary generators, the latter of whic h arise from the v arious colorings of the Lie brack et: 1 := 1 0 = ( − 1) 0 1 ; 1 2 0 = ( − 1) 2 1 0 , 1 2 0 := 0 2 1 = ( − 1) 2 0 1 , 1 2 0 := ( − 1) 0 1 2 = 1 0 2 . Theorem 4.4.5. The shuﬄe r elations 1 2 3 = 3 2 1 − 2 3 1 , 1 2 3 = 3 2 1 − 2 3 1 , 1 2 3 = 3 2 1 − 2 3 1 , 1 2 3 = 3 2 1 − 2 3 1 . (4.4.6) 2 1 = 1 2 − 1 2 (4.4.7) 22 asso ciate d with the r elations in (4.4.4) c onstitute a Gr¨ obner b asis with r esp e ct to the r everse p ath-lexic o gr aphic al or dering of monomials induc e d by the fol lowing or dering of gener ators: 1 2 > 1 2 > 1 2 > 1 (In p articular, the left-hand sides of the r elations in (4.4.6) – (4.4.7) ar e the le ading terms, which we have expr esse d as a r ewriting system). Pr o of. T o v erify that these relations form a Gr¨ obner basis, we m ust sho w that all S -p olynomials reduce to zero. Note that the only nontrivial comp ositions occur b et ween the t wo leading terms of the (colored) Jacobi identities (4.4.6) and a unique comp osition arising from the tw o Y ang- Baxter equation (4.4.7). The v anishing of the S -p olynomials for the comp ositions of the (colored) Jacobi iden tities follo ws from Theorem 2.3.14. The reduction of the remaining S -p olynomial is illustrated in the pictorial computation b elo w. Namely , we compare t wo diﬀeren t reduction paths for the same monomial. F or the reader’s con venience, after each reduction, w e encircle (in dotted red) the divisor to b e reduced in the subsequent step, and we denote the corresp onding reduction in brac k ets after the equality: 1 2 3 = 1 2 3 − 3 2 1 =  1 2 3 − 1 3 2  −  1 3 2 − 3 2 1  = =  3 1 2 − 1 2 3  −  2 1 3 − 1 3 2  − 1 3 2 +  3 2 1 − 1 3 2  = =  3 2 1 − 3 2 1  −  3 2 1 − 3 2 1  −  2 3 1 − 2 3 1  +  2 3 1 − 2 3 1  − 1 3 2 + + 3 2 1 −  2 3 1 − 2 3 1  = 3 2 1 − 3 2 1 + 3 2 1 − 2 3 1 + 2 3 1 − 1 3 2 . (4.4.8) 23 1 2 3 = 1 2 3 − 1 2 3 =  1 2 3 − 1 3 2  −  1 2 3 − 2 3 1  = =  3 2 1 − 2 3 1  − 1 3 2 −  3 2 1 − 3 2 1  + 2 3 1 . (4.4.9) One can easely see that the Right Hand sides of Reductions (4.4.8) and (4.4.9) coincide, what follo ws that the corresp onding S -p olynom is equal to zero. Theorem 4.4.10. L et ( Q q , d ) b e the quasi-fr e e diop er ad gener ate d by the fol lowing two c ol le c- tions of skew-symmetric gener ators: ∀ σ ∈ S m , m ⩾ 2 ℓ m := 1 2 ... m = ( − 1) | σ | σ (1) ... σ ( m ) ∈ Q ( m, 1) , ∀ τ ∈ S n , n ⩾ 2 r n := 1 2 ... n = ( − 1) | τ | τ (1) ... τ ( n ) ∈ Q (0 , n ) (with de gr e es 2 − m and 2 − n , r esp e ctively), wher e the diﬀer ential is given by: d   1 2 ... m   = P A ⊊ [ m ] # A ⩾ 2 ± ... ... A [ m ] \ A , d 1 2 ... n ! = P k ⩾ 2 , [ n ]= ⊔ [ n q ] , n 0 =1 ,n 1 ,...,n k ⩾ 1 ± . . . . . . . . . . . . k z }| { |{z} n 0 |{z} n 1 |{z} n 2 |{z} n k . Then the diop er ad morphism Q q → L ieb △ that sends ℓ 2 to the Lie br acket, r 2 to the c op airing, and al l other gener ators to zer o is the minimal mo del of the diop er ad L ieb △ . Pr o of. T o prov e the theorem, w e establish the following three conditions: First , we verify that the aforementioned map d is indeed a diﬀerential ( d 2 = 0). This v eriﬁcation is straightforw ard to chec k pictorially; w e refer to [44] for the detailed veriﬁcation. Second , we sho w that the map Q q → L ieb △ is a surjective morphism of dioperads. Note that all generators of Q q are negatively graded except for ℓ 2 and r 2 . The zero-th cohomology of Q q is the free diop erad generated by ℓ 2 and r 2 sub ject to the relations arising from the diﬀerential of ℓ 3 (the Jacobi identit y) and the diﬀeren tial of r 3 (the classical Y ang-Baxter equation); th us, it coincides with L ieb △ . Third , we demonstrate that all negatively graded cohomology of Q q v anishes. T o this end, we apply the Anick-t yp e resolutions discussed in Deﬁnition 2.4.1. It suﬃces to sho w that 24 the higher cohomology v anishes for the 2-colored quasi-free sh uﬄe op erad SH (Ψ( Q q )), as both functors Ψ and SH are exact and do not aﬀect the underlying vector spaces. F or the shuﬄe op erad, we consider the decreasing ﬁltration F arising from the Gr¨ obner theory of monomials. Sp eciﬁcally , w e deﬁne: F k (Ψ( Q q )) := Span D sh uﬄe tree monomials with at least k internal edges growing to the left E . The asso ciated graded diﬀeren tial d gr F ma y only create edges that grow to the righ t in the shuﬄe description. This yields the following pictorial description of the asso ciated graded diﬀerential d gr F : ∀ m ⩾ 2 , d gr F (Ψ( ℓ m )) = d gr F   1 2 ... m   = P m − 2 j =1 ± m ... j +1 j ... 1 , ∀ n ⩾ 1 , d gr F (Ψ( r n +1 )) = d gr F   1 2 ... n   = P n − 1 j =0 ± n ... j +1 j j − 1 ... 1 . Note that Ψ( ℓ m ) yields m + 1 distinct generators dep ending on which input or output is c hosen as the root. W e illustrate only the case corresp onding to the ”straight” color; the others diﬀer b y assigning the ”dotted” color to one of the inputs and the output. The right-hand side of the diﬀeren tial also v aries according to the corresp onding coloring. Finally , one observ es that these generators coincide with the generators of the corresp onding inclusion-exclusion op erad asso ciated with the following sh uﬄe monomials (where corresp onding divisors are encircled in red): ... 2 m − 1 1 m , for j = 1 . . . m, ... ... m 1 2 j m − 1 , 1 2 ... n − 1 n . Here, the ﬁrst tw o diagrams corresp ond to diﬀerent colorings of the inputs/output in the gener- ator ℓ m , while the ﬁnal diagram corresponds to the generator Ψ( r n +1 ). Consequen tly , the higher homology v anishes for the asso ciated graded diﬀerential d gr F , which completes the pro of. 4.5 The Diop erad V ( d ) of T radler and Zeinalian The representations of the diop erad V ( d ) , introduced b y T radler and Zeinalian in [51], consist of an associative algebra ( A, · ) equipp ed with a symmetric and in v arian t elemen t c ∈ S 2 A of degree d . W riting c = c (1) ⊗ c (2) in Sw eedler’s notations, the symmetry and inv ariance conditions are expressed as: c (1) ⊗ c (2) = ( − 1) | c (1) || c (2) | c (2) ⊗ c (1) , ( a · c (1) ) ⊗ c (2) = ( − 1) | a | ( | c (1) | + | c (2) | ) c (1) ⊗ ( c (2) · a ) for all a ∈ A, resp ectiv ely . 25 F rom a diagrammatic persp ectiv e, the dioperad V ( d ) is generated by the follo wing subspaces: V ( d ) (2 , 1) := k [ S 2 ] = span  2 1 , 1 2  ; V ( d ) (0 , 2) := 1 2 [ d ] = span D 2 1 = 1 2 E , sub ject to the quadratic relations:    3 2 1 = 1 2 3 ; 1 2 = 2 1    . (4.5.1) The Koszul prop ert y of V ( d ) w as originally veriﬁed in [45]. Belo w, w e pro vide a signiﬁcan tly simpler pro of of this fact using the Gr¨ obner basis theory introduced earlier. Notation 4.5.2. The 2 -c olor e d non-symmetric op er ad V ( d ) is gener ate d by thr e e binary op er a- tions of de gr e e 0 and a single unary op er ation of de gr e e d : , , , subje ct to the fol lowing r elations: = , = , = , = ; (4.5.3) = , = . (4.5.4) Prop osition 4.5.5. The 2 -c olor e d symmetric op er ad Ψ( V ( d ) ) and the symmetrization Sym ( V ( d ) ) of the non-symmetric 2 -c olor e d op er ad V ( d ) ar e isomorphic. Pr o of. Note that the relations in (4.5.1) are planar (i.e., they do not permute the order of inputs or outputs as dra wn in the plane). A straigh tforw ard comparison of these relations completes the pro of. Sp eciﬁcally , relations (4.5.3) arise from selecting the input in the asso ciativit y relation, while relations (4.5.4) corresp ond to the second relation in (4.5.1). Theorem 4.5.6. The r ewriting system that maps the left-hand sides to the right-hand sides in r elations (4.5.3) and (4.5.4) is terminating and c onﬂuent. Pr o of. F ollowing the framework of quan tum-monomial orderings introduced in [12], w e deﬁne an admissible monomial ordering suc h that the left-hand sides of the relations are the leading monomials. This choice automatically ensures the termination of the rewriting system. Conse- quen tly , these relations constitute a Gr¨ obner basis with resp ect to this ordering (see [16] for a detailed treatment of Gr¨ obner bases for non-symmetric op erads). Recall that monomials in the ring of quantum p olynomials k ⟨ x, y , q | xq = q x, y q = q y , y x = q xy ⟩ admit a multiplication-compatible ordering deﬁned by: x k 1 y l 1 q m 1 > x k 2 y l 2 q m 2 ⇔      k 1 > k 2 , k 1 = k 2 and l 1 > l 2 , k 1 = k 2 , l 1 = l 2 , and m 1 > m 2 . 26 T o each tree monomial T with n lea v es, w e assign a collection of n quan tum monomials. The i -th monomial is the noncomm utativ e w ord in v ariables x and y corresp onding to the path from the ro ot to the i -th input, where x represents asso ciativ e m ultiplication and y represen ts the unary operation. T o compare tree monomials T 1 and T 2 , we compare their corresponding quan tum monomials lexicographically . W e omit the proof that this is an admissible ordering, as it follows the construction in [12]. Conﬂuence for comp ositions of the asso ciativit y relations follows from the standard conﬂu- ence of asso ciativit y . Conﬂuence for the remaining comp ositions is veriﬁed b elo w for tw o of the three p ossible colorings of inputs (the third case b eing entirely analogous): ; . (4.5.7) Th us, relation 4.5.4 deﬁnes a distributiv e law: λ :   ◦ Ψ( Asso c ) − → Ψ( Asso c ) ◦   b et ween the 2-colored op erad asso ciated with the non-symmetric asso ciativ e op erad (viewed as a dioperad) and the 2-colored op erad consisting of a single unary op eration. W e refer to [40] for the original deﬁnition of distributiv e la ws and to [37, § 8.6] for a reﬁned exp osition. Corollary 4.5.8. The non-symmetric 2 -c olor e d op er ad V is Koszul. F urthermor e, for e ach c oloring of inputs/outputs, ther e exists a unique normal monomial in which al l multiplic ations pr e c e de the c op airing and gr ow to the right: ... , ... ... (4.5.9) Conse quently, the diop er ad V ( d ) is Koszul and satisﬁes dim V ( d ) ( m, n ) = ( m + n − 1)! . Pr o of. First, the existence of a quadratic Gr¨ obner basis for V (Theorem 4.5.6) implies the Koszulness of the non-symmetric op erad V (b y Corollary 2.4.2). Second, the symmetrization functor Sym is an exact functor that preserves the Koszul prop ert y (see, e.g., [38] for the uncolored case). Th us, the 2-colored op erad Ψ( V ( d ) ) is Koszul, and b y Corollary 2.2.1, we conclude that the diop erad V ( d ) is Koszul. 27 The description of normal forms is straightforw ard: for each p ossible coloring of inputs and outputs, there is exactly one normal form. Since the n um b er of such colorings is giv en by a binomial co eﬃcien t, w e ha ve: dim V ( d ) ( m, n ) = dim  Ψ( V ( d ) ) | ( m, n − 1)  = dim  Sym ( V | ( m, n − 1)  = = m ! · ( n − 1)! · dim  V | ( m, n − 1)  = m ! · ( n − 1)!  m + n − 1 m  = ( m + n − 1)! . It is also ask ed in [45] whether the diop erad W ( d ) , obtained by replacing the symmetric elemen t c with a skew-symmetric one, is Koszul. W e claim that in this case, one must add a sign in relation (4.5.4), whic h fundamentally c hanges the structure. Sp eciﬁcally , the rewriting system is no longer conﬂuent; in the rewriting rule corresp onding to (4.5.7), the num b er of resulting signs is three, whic h leads to dim W (2 , 2) = 0. It follo ws that dim W ( m, n ) = 0 for all m, n ⩾ 2. A computation of the Hilb ert series conﬁrms that the resulting dioperad is not Koszul. 4.6 The diop erad Q p ois of quadratic Poisson structures Recall that a Poisson structure on a ﬂat space V ∼ = R d is deﬁned by a bivector ﬁeld π := P i,j π ij ∂ ∂ x i ∧ ∂ ∂ x j , where x 1 , . . . , x d are co ordinates on V , satisfying the Poisson identit y: 0 = [ π , π ] S ch = X i,j,k,l  π ij ∂ π kl ∂ x i ∂ ∂ x j ∧ ∂ ∂ x k ∧ ∂ ∂ x l − π ij ∂ π kl ∂ x j ∂ ∂ x i ∧ ∂ ∂ x k ∧ ∂ ∂ x l  . Here, [ · , · ] S ch denotes the Schouten-Nijenh uis brack et on p olyv ector ﬁelds. A P oisson structure is called quadratic if the coeﬃcients π ij are quadratic functions of the coordinates. In this case, the Poisson bivector π ma y b e viewed as an elemen t of S 2 V ⊗ Λ 2 V ∗ . The corresp onding prop erad Q p ois is generated by a single element that is symmetric with resp ect to its inputs and skew-symmetric with resp ect to its outputs: 1 2 1 2 = − 2 1 1 2 = 1 2 2 1 = − 2 1 2 1 . This generator satisﬁes the following quadratic diop eradic relations: 1 2 3 1 2 3 + 2 3 1 1 2 3 + 3 1 2 1 2 3 + 1 2 3 2 3 1 + 2 3 1 2 1 3 + 3 1 2 2 1 3 + 1 2 3 3 1 2 + 2 3 1 3 1 2 + 3 1 2 3 1 2 = 0 . (4.6.1) The deformation theory of the prop erad Q p ois and its relationship with the Kontsevic h graph complex w ere discussed in [32]. The Koszul prop ert y for this prop erad was recently pro v en in [31]. In what follows, w e provide an alternative p erspective b y showing that this diop erad admits a quadratic Gr¨ obner basis. The Koszul-dual diop erad Q p ois ! also p ossesses a single generator of arit y (2 , 2), and all of its non-trivial quadratic comp ositions are prop ortional to one another. Sp eciﬁcally: Q p ois ! := * 1 2 1 2 = ( − 1) τ σ (1) σ (2) τ (1) τ (2)        1 2 3 1 2 3 = ( − 1) τ σ (1) σ (2) σ (3) τ (1) τ (2) τ (3) + 28 where σ, τ ∈ S 3 . Consequently , w e hav e: dim( Q p ois ! )( m, n ) = ( 1 , if m = n, 0 , otherwise. Let Comm 2 b e the sub operad of the comm utative op erad Comm whose n -ary op erations are non-zero only for o dd n : Comm 2 ( n ) := ( Comm ( n ) , if n = 2 k + 1; 0 , if n = 2 k . The op erad Comm 2 is a cyclic op erad generated by a single symmetric ternary operation. W e refer to [30] for a description of its Koszul dual and to [20] for the connection b et ween this op erad and the real lo cus of the Deligne-Mumford compactiﬁcation of the mo duli space of mark ed p oin ts. Notably , Comm 2 is known to admit a quadratic Gr¨ obner basis with caterpillar normal forms. It follows directly from the deﬁnition that Q p ois ! diﬀers from the diop erad Θ c ( Comm 2 ) only b y the twisting of outputs b y the sign representation: Q p ois ! ( n, n ) ≃ S n × S op n 1 n ⊗ Sgn n ≃ S n × S op n Θ c ( Comm 2 )( n, n ) ⊗ Sgn n . Prop osition 4.6.2. The diop er ad Q p ois admits a quadr atic Gr¨ obner b asis. Pr o of. By a straightforw ard generalization of Theorem 3.2.11 and Corollary 3.2.12 to the col- oring c = { ( n, n ) : n ∈ Z > 0 } , we conclude that Θ c ( Comm 2 ) admits a terminating and conﬂuent quadratic rewriting system. This system is induced b y the coloring of the quadratic Gr¨ obner basis for the cyclic operad Comm 2 . It is easily v eriﬁed that this rewriting system corresp onds to a Gr¨ obner basis under the path-lexicographical ordering for any c hoice of ordering on the generators. The resulting set of normal forms consists of righ t-com bs, (the coloring of the inputs and outputs uniquely determines the coloring of the internal edges). 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