Quadratic Algebras arising from Hopf operads generated by a single element
aa r X i v : . [ m a t h . QA ] J u l Quadratic Algebras arising from Hopf operads generated by asingle element
Anton Khoroshkin ∗ July 15, 2019
Abstract
The operads of Poisson and Gerstenhaber algebras are generated by a single binary element ifwe consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). Inthis note we discuss in details the Hopf operads generated by a single element of arbitrary arity. Weexplain why the dual space to the space of n -ary operations in this operads are quadratic and Koszulalgebras. We give the detailed description of generators, relations and a certain monomial basis inthese algebras. Contents
Pois kd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Cofibrant dgca model of Pois kd via oriented graphs and koszulness of t kd ( n ) . . . . . . . . . 103.5 Generalizing Orlik-Solomon algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Nondegenerate pairing between operad Pois kd and cooperad OS kd . . . . . . . . . . . . . . . 133.7 A basis in generalized Orlik-Solomon algebras OS kd ( n ) . . . . . . . . . . . . . . . . . . . . 14 Pois kd and graph complexes 15A Combinatorial computations with graphs 17 A.1 Computing homology of
Graphs k ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17A.2 Computing homology of ICG k ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 ∗ International Laboratory of Representation Theory and Mathematical Physics, National Research University HigherSchool of Economics, 20 Myasnitskaya street, Moscow 101000, Russia & Institute for Theoretical and Experimental Physics,Moscow 117259, Russia; Introduction
Recall that the homology of the topological operad is an operad in the category of commutative coalgebraswhich is nowdays called a Hopf operad. Many modern results in deformation theory and homotopy theoryof operads deal with Hopf operads (see e.g.[28],[8]). On the other hand very few examples of Hopf operads,that has a rigorously simple description is known so far. One of the main goals of this paper is to givedifferent detailed descriptions of the simplest algebraic Hopf operads with multi-ary generators one canever encounter. Namely, for each k ∈ N and d ∈ Z we define the Hopf operad called Pois kd , such that for k = 1, d > E d and Pois is equal to the ordinary operad of Poisson algebras. We define operads Pois kd in Section § k + 1) arguments, yielding the Leibnizrule (3.9) and generalized Jacobi identity (3.6). It turns out, that Pois kd admits a Hopf structure and isgenerated by a single primitive operation ν k ∈ Pois kd ( k + 1) of homological degree (1 − d ) that is supposedto be primitive with respect to the Hopf structure (see Definition 2.7). This is the reason why we saythat the operads Pois kd are the simplest Hopf operads. The cheap statement is that the operads Pois kd areKoszul as operads in the category of graded vector spaces (Proposition 3.15). However, the proof thatfor each n the space of n -ary operations Pois kd ( n ) is a quadratic Koszul coalgebra is much more involved(see Theorem 3.39). We call the corresponding algebras generalized Orlik-Solomon algebras and denotethem by OS kd ( n ). We present their definition already in the introduction because of the beauty of thisdescription: OS kd ( n ) := k ω S , S ⊂ { , . . . , n } , | S | = k + 1 , deg( ω S ) = kd − ,ω { i σ (1) ,...,i σ ( k +1) } = ( − d | σ | ω { i ,...,i k +1 } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω S ω T , if | S ∩ T | > ,ω { i ...i k +1 } ω { i k +1 ...i k +1 } ++ ω { i ...i k +2 } ω { i k +2 ...i k +1 i } ++ . . . + ω { i k +1 ...i k +1 } ω { i k +1 i ...i k } . . It is worth mentioning that for the case k = 2, d = 1 these algebras is known to coincide with thehomology of the real locus of the moduli space of stable rational curves ([7]), the koszulness of OS kd ( n )was proven just recently in [12]. We also describe the similar generalizations of Drinfeld-Kohno Liealgebras t kd ( n ) that happens to be the Koszul dual to Orlik-Solomon algebras OS kd ( n ): t kd ( n ) := Lie ν I , I ⊂ [1 n ] , | I | = k + 1; ν σ (1) ...σ ( k +1) = ( − d | σ | ν ...k +1 deg( ν I ) = 2 − kd. (cid:12)(cid:12)(cid:12)(cid:12) [ ν I , ν J ] , if I ∩ J = ∅ , (cid:2) ν I , P s ∈ I ν J \{ t }∪{ s } (cid:3) . if I ∩ J = { t } The (co)operadic composition rules for the generalized Orlik-Solomon algebras and Drinfeld-Kohno Liealgebras are natural generalizations of the known classical one (see (3.32) and § Pois kd ( n ) are quadratic uses the sequence of intermediate equivalent models of these dg-Hopfoperads. Let us draw the diagram of all equivalences we are going to prove in the paper: hoPois kd ( n ) Graphs k ↓ d +1 ( n ) ICG k ↓ d +1 ( n ) Pois kd ( n ) OS kd ( n ) t kd ( n ) § . , Theorem 3 . § . H ( − ) C CE q ( − ) § . , Theorem 3 . H ( − ) h− ; −i§ . . As a byproduct of our theorems we construct a monomial basis of OS kd ( n ) in Section § n ) of the Hilbert series of dimensions ofgraded components of the algebras OS kd ( n ).The bad news about Hopf operads is that it seems to be a rare situation that the Hopf operadis quadratic (and Koszul) as an operad and all spaces of n -ary operations are quadratic (and Koszul)algebras at the same time. For example, if one considers the Hopf operad generated by a pair of(skew)symmetric primitive elements of the same arity k then the corresponding algebras will be quadraticonly for k = 1, but even for k = 1 the corresponding algebras does not satisfy the Koszul property asshown in [5]. 2 cknowledgement I would like to thank Vladimir Dotsenko, Nikita Markarian, Sergei Merkulov, Dmitri Piontkovski andThomas Willwacher for stimulating discussions. I would like to acknowledge Yurii Ivanovich Manin andBruno Vallette for the correspondence concerning [18] which forses me to finish this paper.My research was carried out within the HSE University Basic Research Program and funded (jointly)by the Russian Academic Excellence Project ’5-100’, Results of Section § We refer to the textbooks [17, 21] for different standard definitions of operads. We typically use capitalletters for the notations of operads. E.g. we denote by
Comm the operad of commutative algebras andby
Lie the operad of Lie algebras. The space of n -ary operations of the operad P is denoted by P ( n ) andin the case we want to mark the inputs by a finite set I of cardinality n we will write P ( I ).It is often convenient for applications to deal with cooperads, for example a Hopf cooperad is acooperad in the category of commutative algebras. We will typically use the additional symbol ∗ beforethe name of the corresponding operad and denote the cooperad ∗ P for the dual operad especially in case P ( n ) are finite-dimensional, for all n . We denote partial cocompositions in the cooperad ∗ P by the greekletter ϕ : ϕ I,J ∗ P : ∗ P ( I ⊔ J ) → ∗ P ( I ⊔ {∗} ) ⊗ ∗ P ( J ) (2.1)and the usual dot-sign is used for the algebra multiplication. We deal a lot with Koszul duality for operads and would like to fix certain conventions. Let P be analgebraic operad. We say that a structure of an algebra over a homologically shifted ( k -suspended)operad P{ k } on a chain complex V q is in one-to-one correspondence with the structure of a P -algebraon a shifted complex V q [ k ]. In particular, the homological shift (also called the suspension) increasesthe homological degree of the space of n -ary operations P ( n ) by ( n −
1) and multiplies with a signrepresentation − : P{ } ( n ) := P ( n )[1 − n ] ⊗ − Following the same ideology the homological shift of a cooperad shifts the degrees of cogenerators in theother direction. In particular, the cooperad ∗ Comm { k } is cogenerated by a single element of degree − k which is skew-symmetric for k odd.In order to preserve the standard conventions suggested by [11] that predicts the Koszul dualitybetween Comm and
Lie operads we pose the following conventions:Let P be an algebraic operad and ∗ P be the corresponding cooperad. Then as a chain complex the cobarconstruction Ω( ∗ P ) is isomorphic to the homological shift of the free operad generated by the shiftedsymmetric collection ∗ P [ −
1] := ∪ ∗ P ( n )[ − ∗ P ) := F ( ∗ P [ − {− } Respectively, we use the following degree conventions for the bar-construction: B ( P ) := F c ( P [1]) { } and we say that a Koszul dual operad P ! is the homology of the cobar construction Ω( ∗ P ). For example,if the Koszul operad P is generated by a single symmetric ternary operation of degree 0 then the operad P ! is generated by a single skew-symmetric ternary element of degree 1 − − .3 Hopf operads An operad P in a symmetric monoidal category ( C , ⊗ ) is called Hopf if there exists a coassociativecomultiplication ∆ P : P → P ⊗ P . Example 2.2.
An operad P in the category T op of topological spaces is a Hopf operad whose comulti-plication is given by the diagonal map. The homology (and the chains) of a topological operad provide examples of Hopf operads in (differ-ential) graded vector spaces. We say that an operad in the category of differential graded commutativecoalgebras (dgca for short) is an algebraic Hopf operad .Suppose that P is an operad in T op such that the space of n -ary operations P ( n ) is a connected CWcomplex. Then the rational homotopy type π q ( P Q ) form an operad in the category of L ∞ -algebras. Lemma 2.3.
The Hopf operad P and the Hopf cooperad ∗ P are dual to each other iff there exists acollection of nondegenerate pairings h - , - i : ∗ P ( n ) ⊗ P ( n ) → k , ∀ n ∈ N (2.4) that is compatible with the (co)operadic structure: h f, α ◦ I,J β i = h ( ϕ I,J ∗ P ) ( f ) , α ih ( ϕ I,J ∗ P ) ( f ) , β i and is compatible with the (co)multiplication (Hopf structure): h f · g, α i = h f, ∆ ( α ) ih g, ∆ ( α ) i . Proof.
Directly follows from the definitions.One of the main viewpoint of this article is to insist the reader that the structure of a Hopf operadis very rigid. Namely, as we will see from the examples, if the Hopf operad P is finitely presented asan operad and the algebras ∗ P are finitely presented algebras then the pairing(2.4) requires to have abig kernel both on algebra and operadic level, in particular. We will try to explain the meaning of thismeta-statement for the case of Hopf operads generated by a single element. This section can be omitted because it contains definitions that are not used in the core of the paper.However, these definitions explains the nature of the examples of the Hopf operads we are working with.
Definition 2.5.
A Hopf operad P is called • unital if for all n the coalgebras P ( n ) are counital and the collection of counits ε : P ( n ) ։ k defines a surjective morphism onto operad of commutative algebras: ε : P ։ Comm • augmented if for all n the coalgebras P ( n ) admits a coaugmentation ǫ : k ֒ → P ( n ) that assemblesa morphism of the operad ǫ : Comm ֒ → P such that the composition ε ◦ ǫ is an identity automorphism of the commutative operad. We reservethe letter µ for the image of the commutative associative multiplication in P (2) . • connected if there exists a unique -ary operation κ providing maps of coalgebras ◦ i ( κ ) : P ( n + 1) → P ( n ) . In particular, if P is connected and augmented then µ ◦ κ = 1 ∈ P (1) . graded if P ( n ) = ⊕ i ∈ Z P ( n ) i is a graded commutative coalgebra such that the operadic compositionspreserves grading. Definition 2.6.
An element γ ∈ P ( n ) of a connected Hopf operad P is called • nilpotent if γ ◦ i κ = 0 for all i = 1 ,. . . , n . • (skew)symmetric if γ ( x σ (1) , . . . , x σ ( n ) ) = ± γ ( x , . . . , x n ) . Definition 2.7. • An element γ ∈ P ( n ) of a Hopf augmented operad P is called primitive iff ∆ P ( γ ) = µ ◦ µ ◦ . . . ◦ µ | {z } n ⊗ γ + γ ⊗ µ ◦ µ ◦ . . . ◦ µ | {z } n . In other words, γ is a primitive element of the coalgebra P ( n ) , since the itereted composition ofcommutative multiplication defines a counit in P ( n ) . • A collection of primitive elements { γ s | s ∈ S } of a Hopf augmented operad P is called primitivegenerators of P if the set { µ } ⊔ { γ s | s ∈ S } generates the operad P . • A Hopf operad is called generated by primitives iff there exists a collection of primitive gener-ators which is minimal as a set of generators.
Remark 2.8.
Note that when we say that a connected operad P is generated by the set S we meanthat P (0) = k κ , P (1) ∋ Id and ⊕ n > P ( n ) is generated by S . In particular, if P (1) = k Id , one cansay that the element γ ∈ P ( n ) of arity n > is indecomposable if it may not be presented as a sum ofcompositions of elements of lower arities that are greater than . The key motivation for Definition 2.5 is the following
Proposition 2.9.
Let M be an operad in the category of topological spaces then • H q ( M ; k ) is a graded Hopf operad and respectively H q ( M ; k ) is the corresponding dual graded Hopfcooperad. • the map M → point makes an operad H q ( M ; k ) to be unital. • If, moreover, for all n > the spaces of n -ary operations M ( n ) are connected then the Hopf operad H q ( M , k ) is augmented with µ = H ( M (2)) . • The operad H q ( M , k ) is connected if M (0) is a contractible space. Remark 2.10.
There exists other (co)monoidal functors from the category of topological spaces to thecategory of commutative (co)algebras that produce other examples of connected, graded, unital Hopf op-erads. However, the augmentation is not given by default even if the spaces of operations of any givenarity is connected. One has to deal with pointed spaces.
Example 2.11.
The cohomology of the little cubes operad E d is a unital connected Hopf operad with aunique primitive generator ν ∈ H d − ( E d (2); Q ) = H d − ( S d − ; Q ) . Conjecture 2.12.
If the Hopf operad P is generated by primitive nilpotent elements which generatesa (Koszul) suboperad Q then there exists a ditributive law λ : Q ◦
Comm → Comm ◦ Q that implies theisomorphism of symmetric collections
P ≃
Comm ◦ Q . The category of Lie algebras admits a simple monoidal structure given by direct sum. Consequently, thecomposition rules in the operad g ( n ) in the category of Lie algebras are given by partial compositions: ◦ i : g ( m ) ⊕ g ( n ) → g ( m + n − g ( m ) and from g ( n ) correspondingly: ◦ i := ” ◦ i := ◦ i | g ( m ) ” + ” ◦ i := ◦ i | g ( n ) ” g ( m ) g ( n ) · · · = g ( m ) · · · + g ( n ) · · · Note that the Chevalley-Eilenberg complex (as well as its homology) are monoidal functors from thecategory of Lie algebras to the category of cocommutative coalgebras: C CE q ( g ⊕ g ′ ) ≃ C CE q ( g ) ⊗ C CE q ( g )Consequently, if ∪ g ( n ) is an operad in Lie algebras then the Chevalley-Eilenberg complexes ∪ C CE q ( g ( n ))and ∪ H CE q ( g ( n )) are Hopf (dg)operads. Remark 2.13.
If the operad g in Lie has no operations in arity then the corresponding Hopf operads C CE q ( g ) , H CE q ( g ) are graded, unital, connected operads. Unfortunately, as we will see below, if the operad g in Lie is finitely generated the space of n -aryoperations g ( n ) might be infinite-dimensional. Consequently, the corresponding Chevalley-Eilenbergcomplex will be infinite-dimensional as well, however, the (co)homology might be finite-dimensional andassemble a finitely generated Hopf operad. If in addition the algebras g ( n ) are (pronilpotent) then wehave the following splitting: Comm = H CE ( g ) → H CE q ( g ) → Comm
Consider an operad t k in Lie generated by a single element t [1 k +1] ∈ t k ( k + 1) (with symmetry relation σ · t [1 k +1] = ( − k | σ | t [1 k +1] ) and yielding the Leibniz rule: t [1 k +1] ◦ k (0) k +1 ,k +2 = (0) ,k +2 ◦ t [1 k +1] + (0) ,k +1 ◦ t [1 k +2] \{ k +1 } (2.14)and this is the only condition we impose. Proposition 2.15.
The space of n -ary operations in the operad t k is the quadratic Lie algebra: Lie ν I , I ⊂ [1 n ] , | I | = k + 1; ∀ σ ∈ S k +1 ν σ (1) ...σ ( k +1) = ( − | σ | k ν ...k +1 ∀ τ ∈ A n ⊂ S n τ ( ν I ) = ν τ ( I ) (cid:12)(cid:12)(cid:12)(cid:12) [ ν I , ν J ] , if I ∩ J = ∅ , (cid:2) ν I , P s ∈ I ν J \{ t }∪{ s } (cid:3) , if I ∩ J = { t } Proof.
The generator ν I is the image of the generator t [1 k +1] with respect to appropriate operadic com-position with all other inputs being zero: ν I = t [1 k +1] I The first quadratic relations follows from the commutativity of the monoidal structure in the operad ofLie algebras and the second relation is the application of the Leibniz rule (2.14).It is easy to check that the aforementioned quadratic Lie algebras assemble an operad and conse-quently this operad coincides with t k which is defined by universal properties of single generation thatis obviously satisfied. 6 Single-generated Hopf operads
The operad
Comm is the first example of the quadratic Koszul operad generated by a single binarysymmetric operation µ ( x , x ) = µ ( x , x ). The space of n -ary operations Comm ( n ) is a one-dimensionaltrivial S n -representation with a chosen basic element µ n − = µ ◦ µ ◦ . . . ◦ µ | {z } n − = µ ( x σ (1) , µ ( x σ (2) , . . . µ ( x σ ( n − , x σ ( n ) ) , ∀ σ ∈ S n The homologically shifted operad
Comm { d } is a Koszul operad generated by a single binary operation µ ( x , x ) = ( − d µ ( x , x ) yielding the (super)commutativity relation. Remark 3.1.
The operad generated by a single skew-symmetric binary operation µ ( x , x ) = − µ ( x , x ) and yielding the associativity relations µ ( µ ( x , x ) , x ) = µ ( µ ( x , x ) , x ) = µ ( µ ( x , x ) , x ) is called Comm − and is not Koszul and even has no nontrivial operations in arity because: µ ( µ ( µ ( x , x ) , x ) , x ) = − µ ( µ ( x , µ ( x , x )) , x ) = µ ( x , µ ( µ ( x , x ) , x )) == − µ ( x , µ ( x , µ ( x , x ))) = µ ( µ ( x , x ) , µ ( x , x )) = − µ ( µ ( µ ( x , x ) , x ) , x )Consider the natural analogues of the commutative operad denoted by Comm k . Definition 3.2.
The operad
Comm k is generated by a single S k +1 -symmetric operation µ k +1 of arity k + 1 and of -homological degree subject to the following quadratic relations: ∀ σ ∈ S k +1 µ k +1 ( µ k +1 ( x σ (1) , . . . , x σ ( k +1) ) , x σ ( k +2) , . . . , x σ (2 k +1) ) == µ k +1 ( µ k +1 ( x , . . . , x k +1 ) , x k +2 , . . . , x k +1 ) (3.3) Proposition 3.4.
The quadratic operad
Comm k is Koszul and one has an isomorphism of S n -modules Comm k ( n ) = the trivial S n representation , if ( n − ... k, , if ( n − ... k . The corresponding generating series of dimensions has the following presentation χ Comm k ( t ) := X n > dim Comm k ( n ) n ! t n = X n > t kn +1 ( kn + 1)! = 1 k k − X j =0 ξ − j e ξ j t where ξ is the k ’th primitive root of unity, e.g. ξ = exp( π √− k ) . The symmetric function given bygenerating series of the corresponding S n -characters admits the following description in the basis ofNewton power sums p m := P x mi : χ Comm k ( x , x , . . . ) = X n > h nk +1 = 1 k k − X j =0 ξ − j exp X n > ξ jn p n n (3.5) Proof.
The operad
Comm k admits a quadratic Gr¨obner basis with respect to the path-lexicographicalordering such that the only normal quadratic monomial is µ ( x , . . . , x k , µ ( x k +1 , . . . , x k +1 )), all otherquadratic monomials are the leading monomials of the given Gr¨obner basis. Consequently, Comm k is aKoszul operad thanks to the results of [3]. See also § n -ary operations. 7he homologically shifted operads Comm k { d } are Koszul as well. Note that Comm k { d } is generatedby a single k + 1-ary operation µ k +1 of homological degree dk and for σ ∈ S k +1 we have σ · µ k +1 =( − | σ | d µ k +1 . The Koszul-dual operad Lie k {− d } := ( Comm k { d } ) ! is generated by a single element ν k +1 = ( − | σ | d σ · ν k +1 of homological degree 1 − dk yielding the following quadratic relation: X σ ∈ A k +1 ∩ S k +1 / S k +1 × S k ν ( ν ( x σ (1) , . . . , x σ ( k +1) ) , x σ ( k +2) , . . . x σ (2 k +1) ) = 0 (3.6)The corresponding generating series of Lie k is the inverse of the generating series of Comm k and itscoefficients can be computed using, for example the Lagrange inverse formula:dim Lie k ( n ) = X m > ( − t ) km ( km + 1)! − n ( n − . (3.7)Here [ f ( t )] ( n ) denotes the n -th coefficient f n in the Taylor expansion of f ( t ) = P n > f n n ! t n . Moreover,one can use the presentation (3.5) in terms of Newtons sums to get the inverse with respect to plethysticsubstitution that remembers the S -character of Lie k . Recall that for k = 1 Lie k = Lie and the generatingseries χ Lie ( t ) = − ln(1 − x ), respectively χ Lie ( t ) = arcsinh ( t ). However, we were not able to recognise thegenerating series χ Lie k ( t ) as a known function for k > Remark 3.8.
The operad
Comm k − (as well as its homological shifts Comm k − { d } ) that are generated bya single skew-symmetric operation are obviously not Koszul and thus they are out of our consideration.We refer to [20] for the corresponding nonsymmetric analogues of these operads. Let us define a higher-dimensional analogue of the operad of Poisson algebras. Let
Pois k be an algebraicoperad generated by a commutative associative binary multiplication µ and a skew-symmetric k + 1-aryoperation ν k +1 of homological degree 1 − k that generates the suboperad Lie k := ( Comm k ) ! subject tothe following Leibniz identity: ν k +1 ( x , . . . , x k , µ ( x k +1 , x k +2 )) == µ ( ν k +1 ( x , . . . , x k , x k +2 ) , x k +1 ) + ( − ǫ µ ( ν k +1 ( x , . . . , x k , x k +1 ) , x k +2 ) . (3.9)Here ǫ = | x k +1 | · (1 − k + | x | + . . . + | x k | ). In other words, the Leibniz identity says that the higherLie bracket with all arguments fixed (except one) defines a derivation of the commutative associativeproduct: Let D (-) := ν ( x , . . . , x k , -) then D ( µ ( a, b )) = µ ( D ( a ) , b ) + ( − | D || a | µ ( a, D ( b )) . Let
Pois kd be the operad that generalizes the homology of the little discs operad. That is, Pois kd isgenerated by degree 0 binary commutative multiplication µ and k + 1-ary (skew)-symmetric operation ν that generates the shifted operad Lie k { − d } yielding the Leibniz identity (3.9). Lemma 3.10.
The following map of generators: ∆( µ ) = µ ⊗ µ , ∆( ν k ) = µ ◦ k ⊗ ν k + ν k ⊗ µ ◦ k (3.11) uniquelly extents to a Hopf structure ∆ : Pois kd → Pois kd ⊗ Pois kd on the operad Pois kd .Proof. In order to show that ∆ :
Pois k → Pois k ⊗ Pois k defines a morphism of operads one has to checkthat the comultiplication of the relations of Pois k is zero. Let us show that this indeed happens for the8eneralized Jacobi identity (for all other relations it is obvious):∆ X I ⊔ J =[12 k +1] , | I | +1= | J | = k +1 ± ν ( x I , ν ( x J )) = X I ⊔ J =[12 k +1] , | I | +1= | J | = k +1 (cid:20) ± ν ( x I , ν ( x J )) ⊗ µ ◦ k +1 + ± (cid:0) ( ν ( x I , -) ⊗ µ ◦ k +1 ) ◦ ( µ ◦ k +1 ⊗ ν ( x J )) (cid:1) + (cid:0) ( µ ◦ k +1 ⊗ ν ( x I , -)) ◦ ( ν ( x J ) ⊗ µ ◦ k +1 ) (cid:1) + ± µ ◦ k +1 ⊗ ν ( x I , ν ( x J )) (cid:21) . (3.12)The first and the last summands in (3.12) dissapear because of the Jacobi identity in each tensor multipleof Pois k ⊗ Pois k . Let us expand middle terms using the Leibniz rule. Each summand in the expansionwill be a tensor product of shuffle monomials of the form: (cid:0) µ ◦ k +1 ◦ ν ( x I ) (cid:1) ⊗ (cid:0) µ ◦ k +1 ◦ ν ( x J ) (cid:1) with | I ∩ J | = 1 , | I | = | J | = k + 1 . (3.13)For all pairs of subsets I and J the term (3.13) appears in the expansion of (3.12) twice and thanks to thesymmetry of the group S k +1 we can check that corresponding coefficients has opposite signs for a chosen I = { , . . . , k + 1 } and J = { k + 1 , . . . , k + 1 } . Indeed, one monomial comes out from the expansion ofthe second term in the summand of (3.12) for I = { , . . . , k } , J = { k +1 , . . . , k +1 } and the of expansionof the third term of the summand of (3.12) for I = { k + 1 , . . . , k } and J = { k + 1 , , . . . , k } gives thesecond nontrivial contribution. The sign is affected by the Koszul sign rule while interchanging the tensorproduct (the degree of ν ) and by the sign of the long cycle (12 . . . k k +1) and we get ( − − k ( − k = − µ as well as its iterated compositions µ ◦ N plays a role of a counit in Pois k ( n ). Remark 3.14.
The operad
Pois kd is generated by a single primitive element ν and admits a aryoperation κ such that µ ◦ κ = Id and ν ◦ κ = 0 . Proposition 3.15. • The Leibniz identity (3.9) defines a distributive law in the sence of [19]:
Lie k { − d } ◦ Comm → Comm ◦ Lie k { − d } between Koszul operads Comm and
Lie k { − d } . • The quadratic operad
Pois kd is Koszul. • There exists an isomorphism of symmetric collections
Pois kd and Comm ◦ Lie k { − d } . In particular,we have the following description of generating series: χ Pois k ( t, q ) := X n > dim q ( Pois k ( n )) n ! t n = exp (cid:18) χ Lie k ( qt ) q (cid:19) − Proof.
The proof is standard and repeats the one known for Poisson operad
Pois (see e.g. [19]). Moreover,one can define a quadratic Gr¨obner basis in
Pois k using the methods of [2]. Pois kd Consider the following Hopf dg-operad hoPois kd generated by the commutative associative product µ anda collection of operations ν nk +1 , n = 1 , , . . . each yielding the Leibniz rule (3.9) with the multiplication µ . The differential acts nontrivially only on operations ν q : d ( ν nk +1 ) := X i + j = n ± ν ik +1 ◦ ν jk +1 , d ( µ ) = 0 . The dg-suboperad generated by all ν q is isomorphic to the Koszul resolution Ω( ∗ Comm kd ) ։ Lie k { − d } and is denoted by L k ∞ { − d } . 9 heorem 3.16. The surjection hoPois kd ։ Pois kd is a quasiisomorphism.Proof. First, It is easy to show that the Leibniz rule defines a distributive law between the operad
Comm and the (dg) operad L k ∞ { − d } . Generalizing the known basis and ordering of shuffle monomials for Pois suggested in [2] one can easily find a quadratic Gr¨obner basis for the operad hoPois kd . Second, wealready know from the previous section that Leibniz rule defines a distributive law on Pois kd . Third,thanks to Koszulness of Lie k we know that there is a quasi-isomorphism L k ∞ ։ Lie k . what implies thequasi-isomorphism: Comm ◦ ( L k ∞ { − d } ) ։ Comm ◦ Lie kd (3.17)Note that the source and the target of the quasiiso (3.17) are the associated graded with respect tothe appropriate filtrations on the operads hoPois kd and Pois kd that are predicted by distributive law.Consequently, they are quasiisomorphic. Pois kd via oriented graphs and koszulness of t kd ( n ) As mentioned in the title we suggest a combinatorial models of
Pois kd that are cofibrant meaning thateach space of n -ary operations is a free dg cocommutative coalgebra. These models are used to show thekoszulness of Pois kd ( n ) and compare them with the Chevalley-Eilenberg complex of t kd ( n ).Let us start from combinatorial definitions: Definition 3.18.
We say that a graph Γ is k -mod oriented iff Γ is a graph whose edges are oriented,such that in addition • Γ has n > external nubered (white) vertices drawn as rounded numbers s ; • there are no edges starting in an external vertex; • with arbitrary amount of unordered (black) vertices; • for all internal vertex v the remainder modulo k of the number of outgoing edges from v is equalto ; • there are no directed cycles; • we do not allow vertices with one output and at most one input.We say that the homological degree of • an edge is equal to − d • an internal vertex is equal to d ; • an external vertex is equal to .The homological degree affects the symmetry group and the full homological degree of a graph. The linear span of k -mod oriented graphs is denoted by Graphs k ↓ d ( n ). The linear dual space (withhomological degree reversed) has the same basis given by k -mod oriented graphs and will be denotedby ∗ Graphs k ↓ d ( n ). Suppose Γ is a k -mod oriented subgraph on m vertices in a k -mod oriented graphΓ on n vertices such that if the vertex v belongs to Γ then all its outgoing edges also belongs to Γ ,then if we contract a subgraph Γ and replace it by an external vertex the resulting graph Γ / Γ is alsoa k -mod oriented graph with n − m + 1 external vertices. This operation defines a cooperad structureon ∗ Graphs kd . The dual operation defines an operad structure on Graphs kd .The edge contraction operation defines a differential on the cooperad ∗ Graphs kd . The operad Graphs kd is a dg-operad with the linear dual differential given pictorially by vertex splitting differential as itis in all graph complexes and operads of Graphs used in [15, 16, 28, 29]. Moreover, every pair ofgraphs in ∗ Graphs kd can be glued through the external vertices and we end up with the Hopf (co)operadstructure on ( ∗ ) Graphs kd . As suggested in [27] let us denote by ICG k ↓ the subspace of internally connected k -mod oriented graphs. The collection ICG k ↓ ( n ) is an operad in the category of L ∞ -algebras, wherethe L ∞ structure is prescribed by the isomorphism of Graphs kd ( n ) and the Chevalley-Eilenberg complex C CE ( ICG k ↓ ( n )). Theorem 3.19.
The following assignment of a graph to each generator of hoPois kd hoPois kd (2) ∋ µ , hoPois d ( mk + 1) ∋ ν mk +1 mk+1 · · · mk (3.20) We use index d because the graphs corresponds to the differential forms in the upper-half space of R d what helps toremember the gradings, see e.g. [29, 12]. xtends to a quasiisomorphism of Hopf operads hoPois kd → Graphs k ↓ d +1 .Proof. The proof of Theorem 3.19 generalizes the one given in [29] and is postponed to the Appendix § A.1.Let us suppose that k > k = 1 leads to the classical story of the little discs operad ([29]).The case k = 2 was covered in [12] and we are going to suggest the direct generalization of all argumentssuggested in that paper.First of all, let us recall that all inner vertices of a graph Γ ∈ ICG k ↓ as well as Γ ∈ Graphs k ↓ maynot have univalent and bivalent vertices. Moreover, if an internal vertex has more than 1 outgoing edgesthen it has at least k + 1 outgoing edges. If the vertex has exactly one outgoing edge than it has at leasttwo incoming edges. Moreover, in addition to the homological grading there exists a grading on ICG k ↓ given by the loop order divided by k :deg(Γ) := − k that respects the L ∞ -structure. The same grading exists for internally disconnected graphs Graphs k ↓ d as well and Theorem 3.19 predicts that the graded component ( ICG k ↓ d +1 ) m of degree m belongs to thehomological degree m (1 − kd ). Notice that the homological and internal gradings of the Lie algebra t kd ( n )differ by analogous linear transformation and we are ready to state one of the main theorems of thisnote: Theorem 3.21.
1. The assignment ψ : t [1 k +1] k+1 · · · k (3.22) extends uniquely to the isomorphism of the operads ψ : t kd → H q ( ICG k ↓ d +1 ) in the category of graded L ∞ -algebras.2. The Quadratic Lie algebras t kd ( n ) are Koszul. The core of the proof is hidden in the following technical Lemma 3.23 whose proof is postponed tothe Appendix § A.2.
Lemma 3.23.
All non-trivial cohomology classes in
ICG k ↓ can be represented by linear combinations ofinternally trees whose all inner vertices are either has one output and two inputs or has k + 1 outgoingedges and no incoming edges. In particular, for each loop degree m there exists a unique nonvanishinghomological component of H ( ICG k ↓ ) m . Let us explain how Lemma 3.23 implies Theorem 3.21.
Proof.
First, let us notice that the gradings bounds implies vanishing of all higher Lie brackets on thehomology H q ( ICG k ↓ ( n )). Second, we see from the recursive description of cycles representing cohomology H q ( ICG k ↓ ( n )) that the corresponding Lie algebra is one generated and the generators are given by theimage of generators of t ( n ) described in the assignment 3.22. Third, the relations in the Lie algebra t k ( n )are easily verified in H q ( ICG k ↓ ( n )) as a boundary of a simplest graph with one inner edge. Let us givethe pictorial presentation for k = 2: d i j k p q = i j k p q + j k i p q + k i j p q = (3.24)= [ ψ ( ν ijk ) , ψ ( ν kpq )] + [ ψ ( ν ijk ) , ψ ( ν ipq )] + [ ψ ( ν ijk ) , ψ ( ν jpq )] . (3.25)Consequently, the map ψ : t kd ( n ) → H q ( ICG k ↓ ( n )) is a surjective map of Lie algebras. Fourth, thanks toTheorem 3.19 we already know that H CE ( ICG k ↓ ( n )) = H ( Graphs k ↓ ( n )) = Pois kd ( n ). In particular, the m -th cohomology differs from zero only in the m -th inner (=loop) degree. Therefore, the Lie algebra11 CG k ↓ ( n ) is Koszul and, hence, quadratic. It remains to show that the set of quadratic relations coincidewith the image of the known relations for t k ( n ). It is easy to find a bijection between quadratic shufflemonomials in Lie k (2 k + 1) and the set of quadratic relations in t kd (2 k + 1) which is drawn by the pictureon the left hand side of (3.24). The generalized Jacobi identity in Lie k (2 k + 1) corresponds to the uniquelinear dependence for the aforementioned quadratic relations in H q ( ICG k ↓ (2 k + 1)) : d · · · = X σ ∈ S k +1 ± σ σk +1 σ · · · σk +2 σ k +1 · · · (3.26)What follows, that the quadratic relations in the Lie algebra t kd spans the whole set of quadratic rela-tions for those image. Therefore, H ( ICG k ↓ )( n ) and t kd ( n ) are isomorphic as graded Lie algebras and, inparticular, t kd ( n ) is a quadratic Koszul Lie algebra. Let us define a collection of quadratic algebras that generalizes the Orlik-Solomon algebras ([23]). Fora given positive integer k and a finite set I let OS kd ( I ) be a graded algebra generated by elements ω S indexed by linearly ordered subsets S ⊂ I of cardinality k + 1 subject to the following quadratic relations: ω S ω T = 0 if | S ∩ T | > , (3.27) ω { i ...i k +1 } ω { i k +1 ...i k +1 } + Z k +1 -cyclic permutations = 0 (3.28)We pose the homological degree of ω S to be equal kd − ω σ ( S ) = ( − d | σ | ω S for σ ∈ S k +1 . Remark 3.29.
The difference of two relations (3.28) with two consecutive indices interchanged leadesto the following identity for any decomposition I ⊔ J ⊔ { a, b, c } of a set of cardinality k + 1 : ω a,I,c ω c,J,b − ( − d ω b,I,c ω c,J,a + ω c,J,b ω b,I,a − ( − d ω c,J,a ω a,I,b = 0 (3.30) Proposition 3.31.
For a given k, d the collection OS kd ([1 n ]) assembles a structure of the Hopf cooperadsubject to the cocomposition ϕ I,J OS : OS ( I ⊔ J ) → OS ( I ⊔ {∗} ) ⊗ OS ( J ) defined on the generators by thefollowing rule: ϕ I,J OS : ω S ω S ⊗ if S ⊂ I, ⊗ ω S if S ⊂ J,ω S \{ s }⊔{∗} if S ∩ J = { s } , , if | S ∩ I | > | S ∩ J | > . (3.32) Proof.
It is a straightforward check that cocomposition preserves the relations. Let us illustrate it forthe most interesting case I = { , . . . , k } , J = { k + 1 , . . . , k } and the relation (3.28): ϕ I,J OS k X m =0 ω [ m +1] k +1 ,..., [ m + k +1] k +1 ω [ m + k +1] k +1 ,..., [ m +2 k +1] k +1 ! == ( ω ,...,k, ∗ ⊗ · (1 ⊗ ω k +1 ,..., k +1 ) + (1 ⊗ ω k +1 ,..., k +1 ) · ( ω ∗ , ,...,k ⊗
1) == ω ,...,k, ∗ ⊗ ω k +1 ,..., k +1 + ( − kd − d ( k ) ω ,...,k, ∗ ⊗ ω k +1 ,..., k +1 = 0Here [ l ] k +1 denotes the remainder of l modulo 2 k + 1. Example 3.33. • The algebra OS kd ( I ) is isomorphic to the ground field k if the cardinality of I isless then k + 1 . • The algebra OS kd ( k + 1) is two-dimensional and has a basis { , ω ,...,k +1 } . Lemma 3.34.
The following set of quadratic monomials B (2 k + 1) := (cid:26) ω { }⊔ I ω J (cid:12)(cid:12)(cid:12)(cid:12) I ∩ J = { s } , I ∪ J ∪ { } = { , . . . , k + 1 } s = min( J ) or s = max( I ) . (cid:27) (3.35) spans the second graded component of the algebra OS k (2 k + 1) . roof. Thanks to the relation (3.27) we know that the second graded component is spanned by monomials ω I ω J with | I ∩ J | = 1 and | I ∪ J | = 2 k + 1. Using the relation (3.28) we can even suppose that 1 / ∈ I ∩ J .Let us call the intersection I ∩ J by the repeated element and it remains to use the relation (3.30) tomake this repeated element either the maxima of I or the minima of J . We can do it inductively, becausefor a monomial ω ⊔ I ω J / ∈ B (2 k + 1) there exists a triple a ∈ I ∩ J , b ∈ J , c ∈ I such that c > b > a : ω ⊔ I ω J = ± ω ⊔ I ω { c }⊔ J \{ a } ± ω ⊔{ b }⊔ I \{ c } ω J ± ω ⊔{ b }⊔ I \{ c } ω { c }⊔ J \{ b } . and we see that in the first and in the third summands of the right hand side we increased (in the localorder of I ) the number of the repeated element, and in the second summand we decreased (in the localorder of J ) the number of the repeated element. Corollary 3.36.
There is an upper bound on the dimensions of the component of degree in the algebra OS kd : dim OS kd (2 k + 1) (2) dim Lie kd (2 k + 1) Proof.
It is straightforward to verify that the map ϕ below between shuffle monomials in Lie (2 k + 1) thatare not divisible by ν ( ν ( x , . . . , x k +1 ) , x k +2 , . . . , x k +1 ) and the elements of the set B (2 k + 1) is bijective: ϕ : ν ( ν ( x , x i , . . . , x i k ) , x j , . . . , x j k ) ω { ,i ,...,i k } ω { i k ,j ,...,j k } (3.37) ϕ : ν ( x , x i , . . . , x i s − , ν ( x i s , x j , . . . , x j k ) , x i s +1 , . . . , x i k ) ω { ,i ,...,i k } ω { i s ,j ,...,j k } . (3.38) Pois kd and cooperad OS kd The goal of this section is to recognize the Hopf structure on
Pois kd , namely, we want to prove theisomorphism ∗ Pois kd ( n ) ≃ OS kd ( n ). Theorem 3.39.
The Hopf cooperad OS kd is dual to the Hopf operad Pois kd . In particular,1. The pairing of the generators of the the Hopf cooperad OS and the Hopf operad Pois kd : h , µ i = h ω S , ν ( x S ) i = 1 , h ω S , µ ◦ k i = h , ν i = 0 extends uniquely to a nondegenerate pairing between the Hopf (co)operads that is compatible withthe Hopf (co)operads structure in the sense of Lemma 2.3.2. Moreover, the pairing between an operadic monomial γ ∈ Pois kd ( I ) and a monomial f := ω S · . . . ω S m in the algebra OS dk ( I ) differs from zero if and only if there exists a bijection ψ between vertices of γ labelled by ν and the generators ω S t dividing f such that for all t = 1 . . . m there exists an orderingof the elements of the subset S t := { s < . . . < s k +1 } such that the leaf s i of γ belongs to the subtreegrowing from the i -th input of the vertex ψ ( ω S t ) .Proof. The second part of the theorem follows directly from the compatibility of the pairing and the Hopfoperad structure. On the other hand these vanishing properties of monomials explains the straightforwardcomputation that shows that quadratic relations in the operad
Pois kd and quadratic relations in algebra OS kd belongs to the kernel of this pairing what follows that the pairing is well defined. E.g. there areeither 0 or two monomials (with opposite signs) from the l.h.s. of (3.28) that have nonzero pairing witha shuffle monomial ν ( x I , ν ( x J )). Similarly, there are either 0 or two shuffle monomials (with oppositesigns) in the Jacobi identity, that has nonzero pairing with a monomial ω S ω T .Thanks to Theorem 3.19 and Theorem 3.21 we already know that the coalgebra Pois kd ( n ) is a quadraticKoszul algebra whose Koszul dual is isomorphic to t kd ( n ). Thus, it remains to show that the aforemen-tioned pairing is nondegenerate on the level of quadratic components where one can easely see that thematrix of pairing between monomial shuffle basis in Lie kd (2 k + 1) and the spanning set B (2 k + 1) of OS kd (2 k + 1) is diagonal: h γ, ϕ ( γ ′ ) i = δ γ,γ ′ The bijection ϕ was defined in 3.37,3.38. 13hile forgetting the operad structure in the isomorphism ∗ Pois kd ( n ) ≃ OS kd ( n ) we end up with one ofthe main applications: Corollary 3.40.
The quadratic (super)commutative generalized Orlik-Solomon algebra OS kd ( n ) and thequadratic generalized Drinfeld-Kohno Lie algebra t kd ( n ) are quadratic dual to each other and satisfy theKoszul property. OS kd ( n ) Let us use the nondegeneracy of the pairing discovered in 3.39 and present a basis in OS kd ( n ) that is dualto the operadic basis in Pois kd ( n ). Lemma 3.41.
The set
Shuf - Lie k ( n ) consisting of shuffle monomials generated by a single k + 1 -aryoperation ν that are not divisible by the shuffle monomial ν ( ν ( x , . . . , x k +1 ) , x k +2 , . . . , x k +1 ) form abasis of the space of n -ary operations of the operad Lie k .Proof. The Koszul-dual operad
Comm k admits a quadratic Gr¨obner basis with more or less any compati-ble ordering of shuffle monomials. Let us consider, for example, the reverse path-lexicographical ordering.The only nontrivial operation in Comm k (2 k +1) is given by the monomial ν ! ( ν ! ( x , . . . , x k +1 ) , x k +2 , . . . , x k +1 )that has to be the leading term of the Koszul dual quadratic Gr¨obner basis of Lie k . Definition 3.42.
Let
Grph kd ( n ) denotes the subset of oriented graphs Graphs kd ( n ) with n white numberedvertices and with arbitrary amount of unordered black vertices such that each internal (black) vertex hasno incoming edges and exactly ( k + 1) outgoing edges. Multiple edges are not allowed. Note that each edge in
Grph k starts in an internal vertex and ends in an external one. Hence, theorientation of a graph is prescribed by its shape and can be omitted for simplicity. Moreover, thereis an obvious bijection between monomials in the free (super)commutative algebra generated by theset of elements { ω S | S ⊂ [1 n ] , | S | = k + 1 } and the aforementioned set of graphs Grph k ( n ), such thateach multiple ω S produces an internal black vertex connected with the subset of external white verticesindexed by the elements of S . Lemma 3.43.
If the graph Γ ∈ Grph kd +1 ( n ) contains a loop then the corresponding monomial in OS kd ( n ) is equal to zero. Hence, the set of monomials assigned to the subset of trees Trees k ( n ) ⊂ Grph k ( n ) spansthe algebra OS kd ( n ) .Proof. Let us do the induction on the length of a cycle. For the base of induction we notice that if agraph Γ ∈ Grph k ( n ) has a cycle of length 4 (i.e. containing two internal and two external vertices) thenthe corresponding monomial is divisible by ω S ω T with | S ∩ T | > m . Let a be an external vertex in this cycle and let b and c be theadjacent external vertices in this loop. Then the corresponding monomial ω (Γ) has a factor ω a,b,I ω a,c,J for appropriate subsets I = { i , . . . , i k − } and J = { j , . . . , j k − } of cardinality k −
1. If we apply therelation (3.27) to the following cyclically ordered set of indices: { b, i , . . . , i k − , a, j , . . . , j k − , c } we can rewrite ω a,b,I ω a,c,J as a sum of quadratic monomials, such that each summand has a factor ω b,c,S for appropriate S . Notice that the corresponding graphs will have cycles with the vertex a erased and b and c become external vertices that are adjacent in a given cycle. Thus the length of the correspondingcycles becomes less and equal to 2 m − ϕ (given for quadratic terms in 3.37,3.38) and define (recursively) thecombinatorial map ϕ that assings to each shuffle monomial γ ∈ Shuf - Lie k ( S ) a monomial ϕ ( γ ) ∈ OS kd ( S ).Note that S is supposed to be a linearly ordered set that labels leaves of shuffle monomials:1. We pose ϕ ( ν ( x s , . . . , x s k +1 )) := ω { s ,...,s k +1 } ;2. Suppose that v is a vertex of γ ∈ Shuf - Lie k ( S ) whose all incoming edges are leaves marked by s < . . . < s k +1 . (In other words v corresponds to a single operation ν ( x s , . . . , x s k ).) Suppose14hat, moreover, the edge outgoing from v is not the leftmost income of the corresponding innervertex of γ . Then we set ϕ ( γ ) := ω { s ,...,s k +1 } ϕ ( γ v ↔ s )where γ v ↔ s is the shuffle monomial with the branch growing from v replaced by a leaf indexed bya minimal leaf of this branch s .3. Let v be a vertex of γ ∈ whose all incoming edges are leaves marked by s < . . . < s k +1 and theedge outgoing from v is the leftmost income of the corresponding inner vertex of γ . Then we set ϕ ( γ ) := ω { s ,...,s k +1 } ϕ ( γ v ↔ s k +1 )where γ v ↔ s k +1 ∈ Shuf - Lie k ( S \ { s , . . . , s k } ) is the shuffle monomial with the branch growing from v replaced by a leaf indexed by a maximal leaf of this branch s k +1 . But we pose a new order onthe new set of leaves S ′ := S \ { s , . . . , s k } saying that s k +1 > s (resp. s k +1 < s ) iff s > s (resp. s < s ). In other words we erase all elements s , . . . , s k +1 and replace s with s k +1 .For example, ϕ ( ν ( ν ( x , ν ( x , x , x ) , x ) , x , ν ( x , x , x )) = ω ω ω ω . The set of admissible shuffle monomials in
Pois kd represents the iterated µ -multiplication of shuffle mono-mials in Lie k depending on different subset of indices. Thus, we can easely assign to each shuffle monomial α = µ ( γ , µ ( γ , . . . , µ ( γ m , γ m +1 ))) ∈ Comm ( m + 1) ◦ Lie kd ⊂ Pois kd the monomial ϕ ( γ ) . . . ϕ ( γ m +1 ) ∈ OS kd . Lemma 3.44.
The image under assignment ϕ of admissible shuffle monomials of Shuf - Lie k ( nk + 1) constitute a basis of the n -th graded component of the algebra OS k ( nk + 1) and the image of admissibleshuffle monomials of Pois kd ( S ) constitute a basis of the algebra OS kd ( S ) .Proof. Suppose that for a pair of
Shuf - Lie k ( nk + 1)-monomials γ, γ ′ we have the nonvanishing of thepairing h γ, ϕ ( γ ′ ) i 6 = 0. Let us show by induction on n that γ = γ ′ . Since we are dealing with trees onecan find a multiple ω i ,...,i k +1 of ϕ ( γ ′ ) that has at most one index i s that appears in another multipleof ϕ ( γ ′ ). Note, that ϕ is constructed in the way that all external vertices has at most two edges. Thatis i s appears as an index in exaclty one another generator. Now if i s is a minima of I := { i , . . . , i k +1 } then there exists a unique vertex v of γ connected directly with leaves indexed by i , . . . , i k +1 . Erasingthe vertex v from γ and the corresponding multiple ω i ,...,i k +1 from ϕ ( γ ′ ) we can proceed with induction.Similarly, if i s is not a minima of I , and this happens for all multiples of ϕ ( γ ′ ) then there exists a multiple ω J yielding the property that there exists exaclty one index j s ∈ J that appears in other multiples of ϕ ( γ ′ ), and j s has to be the maxima of J and the same induction procedure makes sence.Thus, we conclude that matrix pairing between Shuf - Lie k ( nk +1) and ϕ ( Shuf - Lie k ( nk +1)) is diagonal.Therefore, the matrix pairing between Pois kd ( S ) and their images under ϕ is also given by diagonal matrix.On the other hand we know that Pois kd ( S ) and OS kd ( S ) are dual vector spaces. Hence, both sets of elementsare linearly independent and spans Pois kd ( S ) and OS kd ( S ) respectively. Remark 3.45.
As mentioned in [18] (see the sketch of the proof of 5.20) one should expect the directimplication of the relations (3.28) , (3.27) that the set of monomials ϕ ( Shuf - Lie k ) spans the correspondinggraded component but we are a bit lazy to go through the combersome combinatorics involved since wecan use the Koszulness instead. Pois kd and graph complexes We do not recall here the precise definition of the deformation complex of a (Hopf) operad and referto [22, 28, 8] for precise definition. The key point is that in order to define a biderivation (deformation)complex of the map of Hopf operads P f → Q one has to find a fibrant resolution of P (that is a quasi-freeoperad) and a cofibrant replacement of Q (that has to be quasifree as an algebra). Note that we do havefound both fibrant and cofibrant replacement of the Hopf operad Pois kd . However, the full deformationcomplex is huge and the goal of this section is to repeat the arguments of [29] in order to compare the15orresponding deformation complexes with the Kontsevich’s graph complexes. Since this is also out ofthe main strem of this note we do not recall the precise definitions of the graph complexes GC d , directedgraph complexes dGC d and oriented graph complexes GC ↓ d and refer to the papers [28, 29]. The roughidea is that the graph complex consists of connected graphs that do not have external vertices and thedifferential is given by vertex splitting. As always the dual differential is given by edge contraction. Theconvention on grading is the same. Vertices has degree d and edges has degree 1 − d . The ordinary graphcomplex GC d consists of graphs whose edges are not oriented. Respectively the directed graph complex dGC d consists of graphs with a chosen orientation of each edge and the oriented graph complex GC ↓ d is asubset of dGC d containing graphs with no directed cycles. Finally, c GC ↓ d consists of directed graphs with hairs and no directed cycles. Where by a hair we mean an edge that starts in a vertex of a graph and hasno end. The degree of a hair is set to be 0. The number of loops defines a grading in all aforementionedgraph complexes. Let dGC dk be the graded Lie subalgebra of the directed graph complex spanned bygraphs whose number of loops is divisible by k . Note that as a subcomplex dGC dk is a direct summandof dGC d . Respectively, let GC ↓ ,kd be the corresponding subcomplex of the oriented graph complex. Thestandard action of the (oriented) graph complexes on the operad of (oriented) graphs([28, 29]) restrictsto the natural action of GC ↓ ,kd on the operad Graphs ↓ ,kd by homotopy derivations as a Hopf operad. Theorem 4.1.
The map GC ↓ ,kd → Def(Ω( ∗ ( Pois kd − ) ! ) → Graphs k ↓ d )[1] ≃ BiDer h ( Pois kd − ) is a quasiisomorphism of complexes.Proof. The proof repeats the one presented in [29] and is conducted in several steps:Note that the Koszul resolution Ω( ∗ ( Pois kd − ) ! ) ։ Pois kd − contains, in particular, the dg-suboperad L k ∞ { − d } ⊂ Ω( ∗ ( Pois kd − ) ! ) is generated by operations ν nk +1 that have the same nature as the onein hoPois kd . First, one considers the filtration of the Def( Pois kd − → Graphs k ↓ d ) such that the associatedgraded differential is the operadic commutator with the generator µ . The freeness of the algebra ∗ Graphs k ↓ d ( n ) implies that the first term of the corresponding spectral sequence can be identified withthe graph subcomplex c GC ↓ ,kd ⊂ c GC ↓ d spanned by directed connected graphs Γ with no oriented cycles,such that in addition the remainder modulo k of • the number of outgoing edges in each external vertex of a graph Γ and • the full amount of hairs (also called sometimes outgoing external legs )is equal to 1. Consequently the loop order of Γ ∈ c GC ↓ ,kd is divisible by k .A graph Γ ∈ c GC ↓ ,kd with nk + 1 hairs is considered as a homomorphism α Γ : hoPois kd ( nk + 1) → Graphs ↓ ,kd ( nk + 1) that maps the generator ν nk +1 to the odd graph ˜Γ whose inner part is isomorphic tothe inner part of Γ and hairs are connected with external vertices such that the external vertices becomeunivalent. Of course, one has to sum up over these possibilities to make this map S nk +1 -invariant. Allother generators of the Koszul resolution of Pois kd are mapped to zero.Let us notice, that the differential in c GC ↓ ,kd is more complicated then the simple vertex splittingdifferential and can be defined pictorially in the following way: δ Γ = X ν Γ • ν ± X j > kj + 1)! Γ ± X j > kj + 1)! Γ (4.2)Here the first sum runs over vertices of Γ, and the symbol • ν shall mean that the graph on the right isinserted at vertex ν . In the second term the black vertex has valence kj + 1, and it is necessary to sumup over all possibilities to connect the edge pictorially ending in Γ to vertices of Γ. In the last term onesums over all external legs of Γ and connects one to the new vertex.Second, one defines a quasiisomorphism GC ↓ ,kd → c GC ↓ ,kd given by the following pictorial presenta-16ion: Γ ∞ X j =0 kj + 1)! Γ . . . | {z } kj +1 × (4.3)where the picture on the right means that one should sum over all ways of connecting kj + 1 outgoingedges to the graph Γ such that the resulting graph belongs to c GC ↓ ,kd . Corollary 4.4.
The Lie algebra of homotopy derivations of the operad
Pois kd is isomorphic to the Liesubalgebra GC kd ⊂ GC d spanned by graphs with the loop order divisible by k .Proof. The equivalence of oriented graph complex and the nonoriented one of consecutive degrees wasdiscovered in [29]: GC ↓ d ≃ GC d − ⊕ loops (4.5)and known to preserve the loop order grading. The ”loops” elements has loop order 1 and we have anisomorphism of subalgebras of graphs of loop order divisible by k : GC ↓ ,kd ≃ GC kd − A Combinatorial computations with graphs
A.1 Computing homology of
Graphs k ↓ In this section we prove Theorem 3.19 which states that the operads of graphs
Graphs kd +1 is equivalentto the operad Pois kd . Proof.
It is easy to see that the combinatorial map ψ : hoPois kd → Graphs k ↓ d +1 is a map of operads. Let usdefine a natural surjective inverse map of complexes of S -modules ǫ : Graphs k ↓ d +1 → hoPois kd which sendsall graphs that contain a vertex with more than one incoming edge to zero and what remains is uniquelypresented as an operadic tree. For example:1 2 3 5 4 6 7 ǫ → ν ( x , ν ( x , x , x ) , x , x , x )The map ǫ is not a map of operads, however, this is a one-sided inverse to the map of S -modules hoPois kd → Graphs k ↓ d +1 and in order to finish the proof of Theorem 3.19 it is enough to show that thekernel of ǫ is an acyclic complex. Let us define a combinatorial filtration on the kernel of ǫ , such that itis easy to show that the corresponding associated graded differential is acyclic. Definition A.1.
For each oriented graph Γ ∈ Graphs k ↓ d +1 containing a vertex with more than one incomingedge we can assign a unique directed path (called the bad path ) yielding the following properties: • the bad path starts in a bad vertex (= a vertex containing more than one incoming edge); • the bad path ends in an external vertex and this is the lowest possible external vertex with a possibleincoming path from a bad vertex; • the bad path contains a unique bad vertex (the source of a path). In the example below we draw the bad path in red and fill the bad vertices in grey:1 2 3 4 5 6 717he vertex splitting differential may either increase the length of the bad path or increase the number ofan external vertex in the target of a path. Consider a filtration on co ker( ǫ X ) by the number of the badexternal vertex and consider a grading by the length of the bad path on the associated graded complex.The associated graded differential preserves the index of the bad external vertex and increases the lengthof the bad path. Moreover, the associated graded differential does not interact with edges that startsnot at the bad path. Consequently, the associated graded complex is a direct sum of complexes ⊕ G C q G .Where the summand C q G is spanned by the set of graphs that coincide after contracting the bad path.Let us show that each of these subcomplexes C q G is acyclic. Consider a particular graph Γ ∈ C q G . Let v be an vertex in the bad path of Γ then v has the following restriction on the set of incoming and outgoingedges outside of a bad path: • if v is not the source vertex of a bad path then v has even positive number of outgoing edges(outside of a bad path) and no incoming edges; • if v is the starting vertex of a bad path then v has even number of outgoing (outside of a bad path)edges and at least two incoming edges; • if v is a target external vertex in a bad path then v has zero number of outgoing edges.Let us order the set S (Γ) of edges starting at vertices in the bad path and ending outside of a badpath. Let A := Q [ e , . . . , e m ] be the free commutative (polynomial) algebra generated by the variablesindexed by S (Γ). Let A ( k ) := ⊕ n > A kn ⊂ A be the k -th Segre power consisting of polynomials of degreesdivisible by k . The latter is known to be a quadratic Koszul algebra. Consider the Bar resolution: B ( A ( k ) ) := M m > A ( k ) ⊗ A ( k )+ ⊗ . . . ⊗ A ( k )+ | {z } m , d ( a ⊗ a ⊗ . . . ⊗ a m ) = m − X i =0 ( − i a ⊗ . . . ⊗ a i a i +1 ⊗ . . . which is the resolution of a trivial left A ( k ) -module. We notice that the S m -invariants of the multilinearpart of the bar resolution B ( A ( k ) ) is dual to the subcomplexes C q G spanned by graphs which has thesame shape as Γ after contracting the bad path. The multilinear part of B ( A ( k ) ) is acyclic and thanksto Mashke’s theorem the subspace of S m -invariants form an acyclic subcomplex and consequently, C q G isalso acyclic.The case of an absence of outgoing edges outside of a bad path has to be considered separately, butin this case the length of a bad path may be either 0 or 1 and it is clear that it has no homology. A.2 Computing homology of
ICG k ↓ This section is devoted to give a proof of the key Lemma 3.23. The proof is based on a collection ofconsecutive spectral sequence arguments such that the associated graded differential for each particularspectral sequence is a vertex splitting (edge contraction) that does not brake certain symmetries definedcombinatorially in terms of graphs. Maschke’s theorem is the key argument that helps in this type ofcomputations. One also should have in mind that any given pair of integers n > l > ICG k ↓ ( n ) with a given loop order l is finite. Consequently all spectral sequences we aredealing with converge because all complexes are splitted to the direct sums of finite dimensional ones.Let us start from certain combinatorial definition that leads to the generalization of Lemma 3.23: Definition A.2.
For each subset S ⊂ [1 n ] consider the subspace ICG k ↓ S ( n ) ⊂ ICG k ↓ ( n ) spanned byinternally connected graphs Γ yielding the two following properties:( ı ) for all s ∈ S the graph Γ has a unique edge ending in the external vertex s and the source vertexof this edge has more than one outgoing edges: s . . . ( ıı ) we do not allow an internal vertex of valency k + 1 such that with it has no incoming edges and k (of k + 1 ) outgoing edges are connected to the k different external vertices s , . . . , s k ∈ S : s s s k . . . . (A.3)18 emma A.4. All non-trivial cohomology classes in
ICG k ↓ S ( n ) can be represented by linear combinationsof internally trees whose all inner vertices either have one output and two inputs or have k + 1 outgoingedges and no incoming edges.In particular, for each given loop degree the homology is concentrated in a unique homological degree. Note that Lemma 3.23 is a particular case of Lemma A.4 when S = ∅ . Proof.
The proof is the simultaneous (increasing) induction on the number n of externally connectedvertices, (decreasing) induction on the cardinality of the subset S and (increasing) induction on the loopgrading of a graph.For the base of induction ( | S | = n ) we notice that there are no oriented graphs (with no orientedloops) in ICG k ↓ [1 n ] ( n ). Indeed, each internal vertex of a graph Γ ∈ ICG k ↓ has at least one outgoing edge andif, in addition, each external vertex of Γ is connected by a unique edge and the source of this edge hasat least one edge ending in an internal vertex then one can construct a directed path of arbitrary lengththat avoids external vertices. However, the number of vertices is finite and hence this path contains aloop what is not allowed in ICG k ↓ . (Induction step). Suppose m ∈ [1 n ] \ S and thanks to the induction on n we may suppose that weare dealing with the ideal of graphs connected with m . For each graph Γ ∈ ICG k ↓ S ( n ) we can define themaximal subtree T m := T m (Γ) ⊂ Γ whose vertices are defined by the following property: w ∈ T m def ⇔ There exists a unique (nondirected) path (with no selfintersections) that starts in w andends in m . Let T ↓ m be the subtree of T m spanned by those vertices v of T m yielding the conditions:( a
1) the unique path starting at v and ending in m is a directed path;( a
2) if the vertex v is internal (differs from m ) then it has a unique outgoing edge.Let Γ ,. . . ,Γ k be the set of internally connected components of the complementary graph Γ \ T ↓ m (internalvertices of T ↓ m are considered as external vertices of the complementary graph Γ \ T ↓ m ). Moreover, ∀ i k there exists exactly one vertex v i ∈ T ↓ m such that there exists not less than one edge endingin v i that belongs to Γ i . The uniqueness of v i is goverened by definition of T m ⊃ T ↓ m . In Picture (A.5)we consider an example of a graph Γ. The complementary graph Γ \ T ↓ m has three connected componentdrawn in different collors (green, yellow and blue). We draw the tree T ↓ m together with the incomingedges that correspond to connected components that is designed by corresponding colloring:Γ Γ \ T ↓ m (Γ) T ↓ m (Γ) p i j m l m m p i j m l (A.5)Consider the filtration F p of ICG k ↓ S ( n ) by the number p of internally connected components of thegraph Γ \ T ↓ m (Γ). While ordering the corresponding graded components we ends up with the followingisomorphism of symmetric collections: gr F ICG k ↓ S ≃ L ∞ ◦ (cid:16) F / F > ( ICG k ↓ S ) (cid:17) Thanks to the Koszulness of the operad
Lie it remains to explain that the quotient complex
ICG k ↓ S ( n ) := F / F > ( ICG k ↓ S ) spanned by graphs Γ ∈ ICG k ↓ S ( n ) with unique internally connected component Γ ofΓ \ T ↓ m (Γ) has appropriate cohomology.Note that the graph Γ \ T ↓ m (Γ) may have a unique internally connected component if and only if T ↓ m (Γ) either is equal to m or consists of one edge v m . Therefore, the complex ICG k ↓ S ( n ) admitsa decomposition ICG k ↓ S ( n ) = ICG k ↓ S ( n ) ⊕ ICG k ↓ S ( n ) where the additional rightmost lower index corre-sponds to the number of edges in T ↓ m . Consider the homotopy h : ICG k ↓ S ( n ) ։ ICG k ↓ S ( n ) to the first19ifferential in the corresponding spectral sequence given by contraction of the edge v m if allowed.The kernel of the surjective homotopy h is spanned by graphs having more than one outgoing edge fromthe unique internal vertex v ∈ T ↓ m . In particular, the complex ICG k ↓ S ⊔{ m } ( n ) is a direct summand of thekernel of h and can be reached by the decreasing induction on the cardinality of S .Thus, it remains to show that the complement of ICG k ↓ S ⊔{ m } ( n ) in the kernel of h (which we denoteby K ) is acyclic. Note that the complement K consists of graphs that has the forbidden vertex (A.3)connected with certain external vertices t , . . . , t k − ∈ S and m . Consider the filtration by the lexico-graphical order of the subset T of first k − m (via exactly one internalvertex v ). The associated graded complex K T admits additional twostep filtration K T = K k +1 T ⊕ K >k +1 T ,where K k +1 T is spanned by graphs with the vertex v of valency k + 1. Note that in the latter case thevertex v has only outgoing edges connected with T , m and the remaining part of the graph. Consider thehomotopy h ′ to the associated graded differential given by contraction of the unique edge connecting v and the remaining part of the graph: h ′ : K k +1 T → K >k +1 T v tk − . . . t m v tk − . . . t m The homotopy h ′ defines a bijection between graphs spanning K k +1 T and K >k +1 T except one particularcase when the remaining part of the graph consists of one external vertex t and if S = T or S = T ⊔ { t } the cohomology of K T is spanned by the image of the generator t t ,...,t k − ,t,m with respect to the map ψ defined in (3.22). As promised we finished with the increasing induction on the cardinality of the subset S of external vertices. References [1] V. I. Arnold, ”The cohomology ring of the group of dyed braids”, Mat. Zametki 5 (1969), 227–231[2] ”Word operads and admissible orderings” preprint arXiv:1907.03992[3]
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