Gröbner Bases for Coloured Operads
aa r X i v : . [ m a t h . C T ] A ug GR ¨OBNER BASES FOR COLOURED OPERADS
VLADISLAV KHARITONOV AND ANTON KHOROSHKIN
Abstract.
In this work we provide a definition of a coloured operad as a monoid in some monoidalcategory, and develop the machinery of Gr¨obner bases for coloured operads. Among the examplesfor which we show the existance of a quadratic Gr¨obner basis we consider the seminal Lie-Rinehartoperad whose algebras are pairs (functions, vector fields).
Contents
Introduction 2Acknowledgement 21. Notation and main definitions 31.1. Notation 31.2. Coloured sets 31.3. Classical definition of a coloured operad 31.4. Functorial definition of a coloured operad 41.5. Free Coloured Operads 61.6. Generating series 71.7. Forgetful functor 82. Combinatorial description 82.1. Coloured trees 82.2. Free nonsymmetric coloured operad 92.3. Free shuffle coloured operad 92.4. Free symmetric coloured operad 92.5. Gradings 102.6. Forgetful functor 103. Gr¨obner bases 113.1. Admissible orderings 113.2. QM-ordering 123.3. Divisibility 123.4. Reductions and S -polynomials 133.5. Gr¨obner bases 134. Examples 144.1. ICom operad 144.2.
AffHS operad 174.3.
MLie operad 194.4. H ( SC vor ) and LP operads 214.5. DCom operad 234.6.
Lie - Rinehart operad 254.7.
DerCom operad 29Appendix A. S -polynomial for Lie - Rinehart ntroduction Gr¨obner bases and the related concepts proved to be an extremely powerful tool for exploringdifferent properties of a wide range of algebraic objects. The list of objects for which this machineryhas been developed includes Lie algebras [17], commutative algebras [4], associative algebras [1, 2],symmetric operads [6] and nonsymmetric operads [8]. We are going to extend the Gr¨obner basesmachinery discovered in [6] to the case of coloured operads. The special case of coloured operadson colours called - -coloured operads was already worked out in [12], however, the general casehas additional complexity thanks to the action of symmetric group.The notion of a coloured operad generalizes the notion of a classical operad, allowing operationsto handle objects of different nature. Usually coloured operads are defined either through the typeof algebras they give rise to, or in purely combinatorial terms (as in the book by Yau [20]). However,neither of these approaches provides the notions required to define a Gr¨obner basis for an operad.The key ingredient for defining a Gr¨obner basis for a type of algebraic objects is an ordering ofthe monomial basis of the free object compatible with the algebraic structure. As in the case withclassical symmetric operads, the symmetric coloured operads do not admit any ordering compatiblewith operadic compositions, so we cannot hope to develop the desired notions directly.We start with the definition of a coloured operad introduced by van der Laan in [19] and then,in the spirit of [6], we introduce the notion of a shuffle coloured operad. The free shuffle colouredoperads has the canonical monomial basis which admits necessary orderings. There exists a forgetfulfunctor from the symmetric coloured operads to the shuffle coloured operads, which allows us totransfer the acquired information back to the symmetric operad.The approach discovered in [6] proved to be fruitful in the case of the classical operads, providingtools for concrete computations (see the book by Bremner and Dotsenko [3]) and enabling thealgorithmic realisation — a Haskell package
Operads [7].Section 4 is devoted to the description of a quadratic Gr¨obner basis in several natural operadson colours:§4.1 the operad ICom governing a pair– a commutative associative algebra and an ideal in it;§4.2 the operad
AffHS of affine homogeneous spaces discovered by Merkulov in [15];§4.4 The -th cohomology of the Swiss Cheese operad and its Koszul dual operad of Leibnizpairs.§4.6 The Lie-Rinehart operad and the operad DerCom governing pairs: a commutative algebraand a Lie algebra of its derivations.As a by-product we (re)prove that all aforementioned operads are Koszul and we compute the cor-responding generating series of dimensions of operations. Moreover, the structure of the symmetricgroup actions is also clear in all cases we consider.All statements regarding reducibility of certain S -polynomials in this work result from computa-tions performed on a computing with a Python script written for the purposes of this paper. Weprovide a sample of the script’s output for the operad Lie - Rinehart in the appendix and suggestdifferent extra arguments showing the reducibility of S -polynomials for other examples. Acknowledgement
We would like to thank V. Dotsenko for useful comments on the first draft of the text. Theresearch of A.Kh. was carried out within the HSE University Basic Research Program and funded(jointly) by the Russian Academic Excellence Project ’5-100’. The results of Section §3 have beenobtained under support of the RSF grant No.19-11-00275. . Notation and main definitions
We employ the definition of a coloured operad introduced by van der Laan in [19], rather thanthe more recent definitions presented in the book by Yau [20], for the former has the merit of beinga functorial one.1.1.
Notation. k — a field of characteristic 0. Vect — the category of finite-dimensional vector spaces over k . Fin — the category of finite sets with surjections as morphisms.
Ord — the category of finite ordered sets with order-preserving surjections as morphisms. n — the set { , . . . , n } . Σ n — symmetric group over a set of n elements.1.2. Coloured sets.
Fix a finite set I , called the colouring set. An I -coloured set (or I -set) S is afinite set S endowed with a map of sets χ : S → I called the colouring of S .Note that for a coloured set S , Σ | S | acts on S by precomposing the colouring of S with a givenpermutation σ . That is σ : ( S, χ ) ( S, χ ◦ σ ) .We denote by Fin I the category of I -sets with surjections of underlying sets as morphisms, andby Ord I – the category of ordered I -sets with order-preserving surjections of underlying sets asmorphisms.Denote by const c : S → I the constant colouring of S with the colour c , that is a colouring with const c ( s ) = c ∀ s ∈ S .Given a colouring χ of the set n and a colouring χ of the set m , define a colouring χ ◦ l χ ofthe set n + m − for any l ≤ n as follows: χ ◦ l χ ( k ) = χ ( k ) if k < l ; χ ( k − l + 1) if l ≤ k < l + m ; χ ( k − m ) if k ≥ l + m. Given a colouring set I = { c , . . . , c d } and a weight vector m = ( m , . . . , m d ) , we define thestandard colouring for m to be the colouring of the set with cardinality P m i , which assigns thefirst colour to the first m elements of the set, the second colour to the next m elements of the setand so on. We denote this colouring by st m .1.3. Classical definition of a coloured operad. An I -coloured collection P is a collection ofsets P ( n, χ, c ) indexed by all n > , all colourings χ of n , and all colours c ∈ I , endowed with aright Σ n -action on L χ P ( n, χ, c ) such that σ : P ( n, χ, c )
7→ P ( n, χσ, c ) for any σ ∈ Σ n .The colours χ (1) , . . . , χ ( n ) are called the input colours of P ( n, χ, c ) , and the colour c is calledthe output colour of P ( n, χ, c ) . Definition 1.1. A coloured operad is an I -coloured collection P endowed with a set of morphismscalled partial compositions: ◦ l : P ( n, χ , c ) ⊗ P ( m, χ , χ ( l )) −→ P ( n + m − , χ ◦ l χ , c ) for all l ≤ n, and a set of identity elements id c ∈ P (1 , const c , c ) , satisfying the following conditions: • Sequential composition axiom : ( λ ◦ t µ ) ◦ t − r ν = λ ◦ t ( µ ◦ r ν ) , for all t ≤ l, r ≤ m and λ ∈ P ( l, χ , c ) , µ ∈ P ( m, χ , χ ( t )) , ν ∈ P ( n, χ , χ ( r )) . Parallel composition axiom : ( λ ◦ r µ ) ◦ s − m = ( λ ◦ s ν ) ◦ r µ for all r < s ≤ l and λ ∈ P ( l, χ , c ) , µ ∈ P ( m, χ , χ ( r )) , ν ∈ P ( n, χ , χ ( s )) . • Identity axiom : id c ◦ ν = ν,µ ◦ s id χ ( s ) = µ for all s ≤ m and ν ∈ P ( n, χ , c ) , µ ∈ P ( m, χ , d ) . Remark . It is common to define a coloured operad of k colours by specifying the sets P ( m , . . . , m k , c ) of operations with m i arguments of i th colour and the output colour c , and partial compositions ofthese operations. Also one specifies the symmetries these operations have, so P ( m , . . . , m k ) is a Σ m × · · · × Σ m k -module.To refactor this definition into the definition of the above form, let n be equal to the sum of m i .Set P ( n, st m , c ) := P ( m , . . . , m k , c ) , and the Σ n -representation L χ P ( n, χ, c ) is isomorphic to theinduced representation from the Σ m × · · · × Σ m k -representation P ( m , . . . , m k , c ) . In particular,the I -coloured collection P is uniquely defined by the Σ I -collection ∪ m i ,c P ( m , . . . , m k , c ) .1.4. Functorial definition of a coloured operad.
Recall that ( Vect , ⊗ , k ) , ( Fin , ⊔ , ∅ ) , and ( Ord , ⊕ , ∅ ) are monoidal categories, where ⊕ denotes the ordered sum of sets. It is clear that Fin I and Ord I are also monoidal categories. Definition 1.3. (1) A nonsymmetric coloured collection P is a monoidal contravariant functor from thecategory Ord I to the category Vect .(2) A symmetric coloured collection P is a monoidal contravariant functor from the cate-gory Fin I to the category Vect . Remark . (1) The coloured sets ( n , χ ) and their morphisms form a skeleton of both categories Ord I and Fin I , so a collection P is completely determined by its values on all morphisms of the form ( m , χ ) ։ ( n , χ ) .(2) The coherence condition for a coloured collection P reads that: P (( m , χ ) ։ ( n , χ )) = O s ∈ n P (( f − ( s ) , χ | f − ( s ) ) ։ ( s, const χ ( s ) )) . Note that there is exactly one arrow from a coloured set ( n , χ ) to the coloured set ( , const c ) .The image of this arrow under P is the space P ( n, χ , c ) from the classical definition.Now we proceed to define the operadic compositions, which in this setting amount to the monoidalstructure on collections. Definition 1.5. (1) Let P and Q be two nonsymmetric coloured collections. Define theirnonsymmetric composition by the formula ( P◦Q )( n, χ , c ) := M ( m ,χ ) P ( m, χ , c ) ⊗ M f : n ։ m Q ( f − (1) , χ , χ (1)) ⊗ . . . ⊗ Q ( f − ( m ) , χ , χ ( m )) , where the inner sum is taken over all non-decreasing surjections f .
2) Let P and Q be two nonsymmetric coloured collections. Define their shuffle composition bythe formula ( P◦ sh Q )( n, χ , c ) := M ( m ,χ ) P ( m, χ , c ) ⊗ M f : n ։ m Q ( f − (1) , χ , χ (1)) ⊗ . . . ⊗ Q ( f − ( m ) , χ , χ ( m )) , where the inner sum is taken over all shuffling surjections f , that is surjections for which min f − ( i ) < min f − ( j ) whenever i < j (3) Let P and Q be two symmetric coloured collections. Define their symmetric compositionby the formula ( P◦Q )( n, χ , c ) := M ( m ,χ ) P ( m, χ , c ) ⊗ k Σ m M f : n ։ m Q ( f − (1) , χ , χ (1)) ⊗ . . . ⊗ Q ( f − ( m ) , χ , χ ( m )) , where the inner sum is taken over all surjections f .In all the above formulae one restricts χ to the respective set if necessary.(4) Define the functor I as follows: I ( n, χ , c ) = ( k if n = 1 and χ (1) = c ;0 otherwise. Remark . In the definition of symmetric composition the action of Σ m on the RHS if defined bypermuting the underlying coloured set for the left factor of the tensor product and by L f : n ։ m Q ( f − (1) , χ , χ (1)) ⊗ . . . ⊗ Q ( f − ( m ) , χ , χ ( m )) L σf : n ։ m Q ( f − ( σ − (1)) , σ − χ , χ ( σ − (1)) ⊗ . . . ⊗ Q ( f − ( σ − (1)) , σ − χ , χ ( σ − ( m )) on the right factor, thus ensuring that the colouring of outputs matches the colouring of inputs.It is straightforward to check that: Proposition 1.7.
Each of the compositions defined above, together with the functor I , endows theunderlying category with a structure of a strict monoidal category. Definition 1.8. (1) A nonsymmetric coloured operad is a monoid in the category of nonsym-metric coloured collections with the monoidal structure given by the nonsymmetric compo-sition. We denote the category of nonsymmetric coloured operads by
NSymOp I .(2) A shuffle coloured operad is a monoid in the category of nonsymmetric coloured collectionswith the monoidal structure given by the shuffle composition. We denote the category ofshuffle coloured operads by ShfOp I .(3) A symmetric coloured operad is a monoid in the category of symmetric coloured collectionswith the monoidal structure given by the symmetric composition. We denote the categoryof symmetric coloured operads by SymOp I . Remark . Given an operad P thus defined, one can retrieve the coloured operad structure on P in the sense of the definition 1.1. Firstly, by the merit of the unit morphism I → P one obtains theidentity elements id c . Then the partial composition ◦ l : P ( n, χ , c ) ⊗ P ( m, χ , χ ( l )) −→ P ( n + m − , χ ◦ l χ , c ) is the component of P ◦ P lying in P ( n, χ , c ) ⊗ hM P (1 , χ | , χ (1)) ⊗ . . . ⊗ P ( m, χ , χ ( l )) ⊗ . . . ⊗ P (1 , χ | n , χ ( n )) i . roposition 1.10. Two definitions of a coloured operad are equivalent.Proof.
To prove this proposition one may repeat the proof of Proposition 5.3.4 in [14] verbatim,keeping track of input-output colouring compatibility. (cid:3)
Remark . The combinatorial constructions presented further in this work stem from the thirdway of describing an operad, that is regarding an operad as an algebra over a monad of rooted trees.We do not provide this description here, limiting ourself to the aspects required for computation.This approach is explored more thoroughly in [14] for the uncoloured case, and the coloured case isthe same up to substituting coloured rooted trees for uncoloured rooted trees.1.5.
Free Coloured Operads.
The first step on the path to the Gr¨obner bases is the notion ofa free object. The free coloured operad and an operadic ideal in it are defined analogously to therespective notions for uncoloured operads. Here we provide definitions in the framework of theclassical definition 1.1, and in the next section we will introduce a combinatorial realisation of thesenotions.
Definition 1.12.
The free I -coloured operad F (Υ) generated by the Σ I -collection of operations Υ := ∪ Υ( m , . . . , m d ; c ) is the result of applying all possible operadic compositions to all pairs ofelements from the induced I -coloured collection of Σ n representations ⊕ χ Υ( n, χ, c ) .An operadic ideal in the free I -coloured operad F is the result of repetitively applying all possibleoperadic composition to all pairs of the form ( α, β ) and ( β, α ) , where β is already in the ideal and α is an arbitrary element of F . Remark . The notation F ( a , . . . , a k | b , . . . , b m ) of a presentation of an operad by a givenset of operations { a , . . . , a k } and relations b j ’s with known symmetries and the I -colouring ofinputs/outputs has the following meaning. First, with each generator a s ∈ F ( m , . . . , m k , c ) onehas to assign a linear basis of the induced representation k [Σ m + ... + m k ] a s and similarly one has tochoose the basis of the representations of symmetric group generated by defining relations b j ’s.In particular the quantity of generators and relations is much more comparing to the sym-metric case. For example, with a generator a ∈ F ( m , . . . , m k , c ) which is symmetric in eachcolour one has to assign (cid:0) ( m + ... + m k )! m ! ...m k ! (cid:1) different generators that correspond to different colourings χ : { , . . . , P m i } → I with | χ − ( j ) | = m j . Definition 1.14.
An operadic ideal generated by the set B = { b , . . . , b k } in a free coloured operad F is the result of repetitively applying all possible operadic composition to all pairs of the form ( α, β ) and ( β, α ) , where β is already in the ideal and α is an arbitrary element of F .Suppose we have a free coloured operad F with the set of generating operations G . Consider afree (uncoloured) operad F unc generated by the same set of operations G disregarding the matchingof the colours rule. Then the set of possible compositions of F unc includes the set of possiblecompositions of F , and the resulting operations are the same. This observation yields the following Proposition 1.15.
A free coloured operad F (as a monoid, disregarding the colour grading) is thefactor of the corresponding uncoloured operad F unc by the operadic ideal generated by all colour-matching relations. In particular, we have an inclusion of monoids F ֒ → F unc . In practice it is convenient to have an explicit combinatorial description for operads. To obtainsuch a description, we regard an operad as an algebra over the monad of rooted trees. We provideall combinatorial definitions needed for our purposes in the next section §2. For greater detail onthis we refer to [14]§5.6. .6. Generating series.
Suppose that the cardinality of the set of colours is equal to d , so we say I = { c , . . . , c d } . With each I -coloured symmetric collection P we assign a collection of d formalpower series:(1.1) F i P ( t , . . . , t d ) := X m ,...,m d ≥ dim P ( m + . . . + m d , χ, c i ) m ! . . . m d ! t m . . . t m d d , with | χ − ( c j ) | = m j for j = 1 , . . . , d The vector F I P ( t , . . . , t d ) := ( F P , . . . , F d P ) is called the generating series of the I -coloured sym-metric collection P . Proposition 1.16.
The composition of generating series of I -coloured collections equals the gener-ating series of the composition of collections: F I P◦Q = F I P ◦ F I Q Proof.
Let the generating series of P be in the variables t j and the generating series of Q in thevariables s k . Choose an arbitrary colour c i and consider the monomial dim P (Σ m i ,χ,c i ) m ! ...m d ! t m . . . t m d d in F i P . Substituting t j for F j Q ( s , . . . , s d ) we have:(1.2) dim P (Σ m i , χ, c i ) m ! . . . m d ! ( F Q ) m . . . ( F d Q ) m d = dim P (Σ m i , χ, c i ) m ! . . . m d ! ×× (cid:0) X p ,...,p d ≥ dim Q (Σ p i , χ, c ) p ! . . . p d ! s p . . . s p d d | {z } m times · . . . · X p d ,...,p dd ≥ dim Q (Σ p di , χ, c d ) p d ! . . . p dd ! s p d . . . s p dd d | {z } m d times (cid:1) Note that if χ = σ st w for a weight vector w and a permutation σ , then dim Q (Σ p di ; χ, c d ) = dim Q (Σ p di ; st w , c d ) , so the coefficient of s p j . . . s p jd d in the respective sum is equal to dim Q (Σ p di ; st w , c d ) .Let’s trace where some fixed monomial s r . . . s r d d appears in this expression. It comes from a choiceof a summand in each of the inner sums, that is from a partition of each s r j into Σ m i summands.Defining such partition for all s r j ’s is the same as defining a surjective map of the coloured set rrr for r = Σ r j (the colouring is read from the exponents in the partition) onto some coloured set M such that M has m i elements of i th colour. Choosing any particular M accounts for an orderingof the inner sums. There are m ! . . . m d ! such sets, and we denote by M st the one that has thecolouring st mmm .Denote by C f the coefficient given by the map f , namely: C f = Σ m i Y k =0 dim Q (Σ f − ( e k ); st w k , c k ) , where e k denotes the k th element of M and w k denotes the weight vector corresponding to f − ( e k ) . o the coefficient of s r . . . s r d d is equal to:(1.3) C ( r , . . . , r d ) = X m ,...,m d ≥ χ (cid:2) dim P (Σ m i ; χ, c i ) m ! . . . m d ! · X f : rrr → M C f (cid:3) == X m ,...,m d ≥ χ (cid:2) dim P (Σ m i ; χ, c i ) · X f : rrr → M st C f (cid:3) This coefficient accounts for all colourings with colouring vector ( r , . . . , r d ) , so for any such colouring χ we should multiply this coefficient by r ! ...r d ! , and the result is exactly the coefficient correspondingto ( P ◦ Q )(Σ r j , χ, c ) in F P◦Q . (cid:3) Note that one can also consider the generating series of characters of the product of symmetricgroups. Namely let F I P ( t ij ) be a collection of d := | I | formal power series on d families of variables { t i , t i , t i , . . . } with ≤ i ≤ d := | I | that are symmetric in each collection of variables: F I P ( t ij ) := X m ,...,m k char ( P ( m , . . . , m k , c )) . The composition of collections corresponds to the plethystic substitution of characters: F I P◦Q = F I P ◦ F I Q Recall, that the plethystic composition written in the basis of Newton power sums p k ( x , x , . . . ) := x k + x k + . . . can be written in the following way p d ◦ F ( . . . , p ( t i , t i , t i ) , . . . , p m ( t i , t i , . . . t i ) , . . . ) == F ( . . . , p d ( t i , t i , t i ) , . . . , p dm ( t i , t i , . . . t i ) , . . . ) . Forgetful functor.
The forgetful functor
Ord I → Fin I disregarding the ordering of sets givesrise to a forgetful functor F from the category of symmetric coloured collections to the category ofnonsymmetric coloured collections which forgets the Σ -module structure of the vector space. By thesame considerations as in Prop. 3 from [6], this functor commutes with the operadic compositionsin the following sense: for two symmetric coloured collections P and Q : F ( P ◦ Q ) = F ( P ) ◦ sh F ( Q ) . So this is in fact a functor from the category of symmetric coloured operads to the category ofshuffle coloured operads. We will explore this functor in greater detail in section 2.6.2.
Combinatorial description
In this section we give combinatorial descriptions for the free operad of each type.2.1.
Coloured trees.Definition 2.1. A coloured rooted tree is a non-empty directed tree such that: • Every vertex has at least one incoming edge (its inputs) and exactly one outgoing edge (itsoutput). • Edges are allowed tobe connected with only one vertex, such (half)edges are called external. • There is exactly one outgoing external edge, this edge is called the output of the tree. Thefree endpoint of the output is called the root of the tree. • The free endpoints of the incoming external edges are called the leaves of the tree. Wesuppose the tree to be decorated, meaning that the leaves of the tree are bijectively markedwith the elements of the set n (called labels) for some n . All edges of the tree are coloured with the set I .A coloured rooted tree with one vertex is called a corolla. A coloured rooted tree with no verticesis called a degenerate tree.We picture the trees to be growing from the root upward, so following the direction of edges onegoes down the tree.A planar representation of a directed tree is equivalent to an ordering of inputs for each vertexof the tree. We compare two inputs of a vertex by comparing the minimal label reachable goingthrough each input up the tree, the input with the lesser reachable label is lesser.Now our goal is, given a coloured collection P , construct a realisation of the free operad F ( P ) generated by P . In all three cases the realisation will be given in terms of coloured rooted trees andthe grafting operation on them. From now on by a tree we will mean a coloured rooted tree.2.2. Free nonsymmetric coloured operad.
Let P be a nonsymmetric coloured collection. Fixa basis B of P . Now we assign a planar tree to each element of B .First, to each identity element id c we assign a degenerate tree of the corresponding colour. Thento an element p of B belonging to P ( n, χ, c ) we assign a corolla with n leaves with labels increasingfrom left to right, and we colour the leaves’ edges according to χ . We mark the vertex of the corollaby p .We define the partial composition T ◦ l T of two trees by grafting T on the input of T labelledwith l , provided that this input and the output of T have the same colour. Otherwise we set thecomposition to be zero.The basis of the free operad F ( P ) consists of all trees obtained by grafting procedure startingfrom the set of corollas. By definition, this basis is closed under partial composition. We will referto the elements of this basis as the tree monomials.2.3. Free shuffle coloured operad.
Let P be a nonsymmetric coloured collection. We constructthe set of degenerate trees and corollas similarly to the previous case. We define the partial com-position T ◦ l,σ T of two trees by grafting T on the input of T labelled with l , provided that thisinput and the output of T have the same colour. Otherwise we set the composition to be zero.We label the inputs of the resulting tree the same way as with nonsymmetrical composition, andafter that we act by σ on labels coming from T and the labels coming from T to the right of thegrafting site.Note that the trees resulting from this procedure satisfy the shuffle condition:For each inner vertex of the tree, the smallest descendants in each subtree growing from thisvertex form an increasing sequence.Such trees are called shuffle trees.The basis of the free operad F ( P ) consists of all trees obtained by this grafting procedure startingfrom the set of corollas.2.4. Free symmetric coloured operad.
Let P be a symmetric coloured collection. We constructthe set of degenerate trees and corollas similarly to the previous cases, but now we render our treesas not equipped with planarization.As in the previous cases, we define the partial composition T ◦ l,σ T of two trees by grafting T on the input of T labelled with l , provided that this input and the output of T have the samecolour, and otherwise set the composition to be zero.We label the inputs of the resulting tree the same way as with nonsymmetrical composition, andafter that we act by σ on all labels of our tree.Again, the basis of the free operad F ( P ) consists of all trees obtained by this grafting procedurestarting from the set of corollas. .5. Gradings.
A tree in the basis of the free operad F has three separate gradings:(1) Arity degree – the number of leaves of the tree. The space of elements of arity degree n is F ( n ) .(2) Operation degree – the number of inner vertices of the tree.(3) Colour degree – a vector c with c i equal to the number of inputs coloured with i minus thenumber of outputs coloured with i (which is or ).Note that all three gradings are additive under operadic compositions. Definition 2.2.
An element of the free operad is said to be homogeneous if it is a sum of basiselements with the same arity degree.2.6.
Forgetful functor.
We now return to the forgetful functor F : SymOp I → NSymOp I definedin section 1.7. As all our computations will involve transferring from a symmetric coloured operadto the corresponding shuffle coloured operad, we would like to provide a more concrete descriptionof this functor.In our setting, operads are usually defined through generators and relations. First we need todetermine how F acts on the set of generators of an operad.Recall that each space of operations P ( n, χ, c ) of a symmetric operad is an Σ n -module, so eachgenerator g comes with the orbit of g under the action of the permutation group. The forgetfulfunctor erases this action, so we need to introduce a new generator for each operation in the orbit of g . After this we need to choose a planarization of the resulting generators so that they are legitimateelements in the shuffle operad. Example 2.3.
Suppose we are given two generators α and r of arity 2, such Σ acts trivially on α and non-trivially on r . Then we will need to introduce a new generator l : l ( x, y ) = r ( y, x ) : α , r F −→ α , r , l Now we need to do the same with relations, which also come with their Σ -orbits. For each relationand each permutation, act by the permutation on the relation, and then choose a planarization ofthe result such that all trees are shuffle trees. Example 2.4.
Suppose we have the following quadratic relation on the generators from the previousexample: rr
21 3 − r α The identity permutation will give us just the relation itself. The transposition (12) will give us: l r − lα
21 3 he transposition (23) yields: rr
31 2 − r α The transposition (13) yields: rl
21 3 − lα
31 2
The cycle c = (123) will give us: rl
31 2 − lα
21 3
And the cycle c = (132) will give us: l l − lα
31 2 Gr ¨obner bases
In this section we define all entities needed for the definition of a Gr¨obner basis.3.1.
Admissible orderings.
From now on by operad we mean shuffle coloured operad unlessspecified otherwise.
Definition 3.1.
Let F be a free operad. An ordering of the tree monomials of F is said to beadmissible if the following holds:(1) If n < m then α < β for all α ∈ F ( n ) , β ∈ F ( m ) . (2) For α, α ′ ∈ F ( m ) , β, β ′ ∈ F ( n ) , if α ≤ α ′ , β ≤ β ′ then α ◦ i,ω β ≤ α ′ ◦ i,ω β ′ for all possibleoperadic compositions.Our goal is to construct an admissible ordering of the monomials in the free operad. We claimthat the construction of path-lexicographic ordering from [6] can be transferred to the colouredsetting. Recall that the path-lexicographic ordering for (non-coloured) shuffle operad is constructedas follows: • For a tree monomial α ∈ F ( n ) construct a vector a = ( a , . . . , a n ) , where a i is the wordcomposed of vertex labels on the path from the root of the tree to the i th leaf, and apermutation s ∈ S n which is read from leaves left to right (recall that shuffle tree is planar). • To compare two monomials, first compare their arities (the lengths of the sequence ( a , . . . , a n ) ). • If arities are equal, compare the vectors a component-wise using degree-lexicographic order-ing on words. • If vectors a are equal, compare the permutations using reverse lexicographic order. xample 3.2. For the tree monomial rl
21 3 one has ( a | s ) = (( rl, r, rl ) | (132)) .Remark . Given the vector a and the colouring data of the generating operations one can restorethe colourings of all edges of the tree, so this construction accounts for the colouring data as well. Proposition 3.4.
The path-lexicographic ordering is admissible.Proof.
Recall the definition of F unc from Proposition 1.15. In [6] it is shown that the path-lexicographic ordering is admissible for F unc . But the requirements for being admissible in terms ofcoloured composition are less strict than for being admissible in terms of uncoloured composition,as the former is the subset of the latter. The trees of F are also a subset of trees of F unc , and therestriction of an ordering on any subset is again an ordering. (cid:3) QM-ordering.
The path-lexicographic ordering and its variations turn out to be inconvenientfor calculating the Gr¨obner bases of some operads (e.g. the operad of Poisson algebras). In [5] a newfamily of orderings was introduced, and we will employ an ordering of this type in our examples.We will call this type of orderings QM-orderings, which stands for Quantum Monomial.The path-lexicographic ordering is based on the comparison of words in a free noncommutativealgebra generated by the set of generators of the operad. The idea of a QM-ordering is to replacemonomials in the free noncommutative algebra by monomials in the algebra of quantum polynomials.For our purposes it will suffice to construct QM-orderings for the operad with two generators x, y , so the algebra of quantum monomials is k h x, y i / ( xq − qx, yq − qy, yx − xyq ) where q is a formalparameter that commutes with x and y . To compare two monomials in this algebra, first writethem in the standard form x a y b q c <> x a y b q c . Then use the following rule x a y b q c < x a y b q c ⇔ a > a , ( a = a ) & ( b < b )( a = a ) & ( b = b ) & ( c < c ) Having a comparison for words in the algebra, that is compatible with multiplication, we expandthis ordering to an ordering on the free operad by associating a vector of words corresponding topaths from the root to leaves, same as we did for path-lexicographic ordering. We refer to [5] for theproof that this is indeed an admissible ordering. In all computations were are dealing with a QM -ordering the choice of the extension of the partial QM -ordering does not affect the story becausethe monomials that are not comparable with respect to a given QM -ordering do not interact witheach other under the Buchberger algorithm.3.3. Divisibility.
Consider a tree monomial α with the underlying tree T . For a subtree T ′ of T ,containing all edges adjacent to the vertices of the subtree, we define a tree monomial α ′ as follows:the underlying coloured tree of α ′ is T ′ and the labelling of the leaves is determined by the smallestdescendant ordering, that is, the leaf with the smallest leaf label among its descendants gets thelabel , the leaf with the same property among the yet unlabelled leaves gets the label and so on. Definition 3.5.
A tree monomial α is divisible by a tree monomial β if there is a subtree T ′ of theunderlying tree of α , such that β is the corresponding tree monomial for T ′ .As β corresponds to a proper subtree of α , we can obtain α by applying operadic compositions to β . This sequence of compositions can be applied to any tree monomial with the same number andcolouring of the inputs and the output as β . This yields an operator on tree monomials which wedenote by m α,β . Note that since the ordering of the tree monomials is compatible with the operadiccompositions, if γ < β then m α,β ( γ ) < α . .4. Reductions and S -polynomials. In this section we recall the notions introduced in [6], asthey also suit the case of coloured operads.
Definition 3.6.
For an element f of the free operad its leading term lt( f ) is the largest (in termsof the chosen admissible ordering) tree monomial in the expansion of f . The coefficient of lt( f ) iscalled the leading coefficient and denoted by c f . Definition 3.7.
For two homogeneous element f and g such that lt( f ) is divisible by lt( g ) we definereduction of f modulo g by the formula: rd g ( f ) = f − c f c g m lt( f ) , lt( g ) ( g ) By construction we have lt(rd g ( f )) < lt( f ) . Definition 3.8.
A tree monomial γ is called a common multiple of the tree monomials α and β , ifit is divisible by both α and β . Tree monomials α and β are said to have a small common multiple,if they have a common multiple that is a union of two overlapping trees with one of these treesbeing isomorphic to α and another isomorphic to β as a shuffle tree. In particular, the the numberof vertices of the underlying tree is less than the total number of vertices for α and β .Assume we have two homogeneous elements f and g whose leading terms have a small commonmultiple γ . In this setup we give the following definition: Definition 3.9.
The S -polynomial of f and g corresponding to γ is defined by the formula: s γ ( f, g ) = m γ, lt( f ) ( f ) − c f c g m γ, lt( g ) ( g ) . Gr¨obner bases.Definition 3.10.
Let M be an operadic ideal in a free I -coloured shuffle operad F with a chosenadmissible ordering of monomials in F I and let G be a set generating M . G is called a Gr¨obnerbasis of M if for any element f in M the leading term of f is divisible by the leading term of someelement of G .This setting allows us to implement the classical Buchberger algorithm (for the description of thealgorithm in operadic context we refer to Section §3.7 of [6].Proposition 1.15 says that the I -coloured shuffle operad F I / M is isomorphic to the quotient ofthe free shuffle (uncoloured) operad F by the ideal f M that is a union of M and all compositionsthat contradicts the colouring. Theorem 3.11.
Let G be a Gr¨obner basis of an operadic ideal M in a free I -coloured shuffleoperad F I ( a , . . . , a k ) and let B be the set { a i ◦ l a j } of all partial composition of generators with theinconsistent colouring of the l ’th input of a i and the output of a j considered as quadratic monomialsin the free uncoloured shuffle operad F ( a , . . . , a n ) . Then the union G ⊔ B constitutes the Gr¨obnerbasis of the ideal f M ⊂ F .Proof.
Note that the set of colour mixing compositions constitute an ideal e B ⊂ F generated by B .Therefore, each small common multiple γ of the colour mixing relations a i ◦ l a j and any element α ∈ G belongs to e B and, in particular, the corresponding S -polynomial associated with γ is reducedto zero using the relations from B . (cid:3) P. van der Laan explained in [19] that I -coloured (co)operads admit Bar and coBar constructionsand quadratic coloured operads admit the Koszul duality functor. One says that an I -colouredoperad P is Koszul whenever the coBar construction of its Koszul dual cooperad P ! is quasi-isomorphic to P . In particular, the latter coBar construction Ω( P ! ) coincides with the minimalresolution of P in the category of I -coloured operads. heorem 3.12. Suppose that an I -coloured operad P generated by the given set { a , . . . , a n } ad-mits a quadratic Gr¨obner basis with respect to an admissible ordering ≺ of monomials in thefree I -coloured shuffle operad F ( a , . . . , a n ) . Then the I -coloured operad P is Koszul and itscoloured Koszul dual operad P ! generated by the dual set of generators { a ∨ , . . . , a ∨ n } admits aquadratic Gr¨obner basis of relations with respect to the reverse admissible ordering of monomials a ∨ ≺ op b ∨ def ⇔ a ≻ b of the same arity/homogeneity in F ( a ∨ , . . . , a ∨ n ) . Note that the uncoloured shuffle operad associated with P ! differs from the shuffle operad that isKoszul dual to the uncoloured operad associated with P . Therefore, Theorem 3.12 does not followfrom the analogous statement known for ordinary shuffle operads. However, the strategy of theproof is the same: Proof. If G is a linear basis of quadratic relations in the I -coloured operad P and b G is the set ofleading monomials of G with respect to the partial ordering ≺ then the dual space Ann ( G ) admitsa linear basis ¯ G whose leading monomials with respect to the reverse ordering ≻ op consists of thecomplement of G in the set of quadratic monomials. The associated graded I -coloured shuffle operad gr P has monomial quadratic relations b G and therefore is Koszul, its shuffle I -coloured Koszul dualoperad (gr P ) ! has also monomial relations that are indexed by the aforementioned complement of G .What follows that the operad P is Koszul and the dimensions of graded components of the colouredKoszul dual operad P ! coincides with the dimensions of the corresponding graded components of (gr P ) ! . Consequently, ¯ G constitutes a Gr¨obner bases of P ! . (cid:3) Examples
ICom operad.
The
ICom operad is a symmetric coloured operad on two colours generated bythree operations: i , α = α , r (4.1)subject to the following quadratic relations: αα
31 2 = αα
21 3 = α α (4.2) rr
31 2 = rr
21 3 = r α (4.3) i = ir (4.4) r i = r i (4.5)Relation (4.2) means that α is a commutative associative multiplication, Relation (4.3) says that r defines an action of this commutative algebra and Relation (4.4) says that i is a map of modules ofthis algebra. A typical algebra over the operad ICom is a pair ( A, I ) of a commutative algebra A and an ideal I ֒ → A . α corresponds to the multiplication in A , r to the multiplication of an elementof the ideal by an element of A , and i corresponds to the inclusion of I into A .The corresponding coloured shuffle operad has four generating operations, we denote them by i, α, r and l : i , α , r , l (4.6) Theorem 4.1.
The Σ and Σ orbits of the defining quadratic relations (4.2) - (4.5) constitute aquadratic Gr¨obner basis of the ideal of relations of the -coloured operad ICom if one considers thepath lexicographic ordering of the monomials associated with the following ordering of generators: α < i < l < r .The generating series of dimensions of
ICom are equal to −−−→ F ICom ( t , t ) := ( F ICom ( t , t ); F ICom ( t , t )) = (cid:0) e t + t − e t + t − e t (cid:1) First, let us act by the symmetric group on each of the relations to obtain the relations in theshuffle operad, and then find the leading term in each acquired relation. For each relation in thesymmetric coloured operad
ICom we list the leading terms of relations in the shuffle operad producedby it.Relation (4.2) yields: αα
31 2 , αα
21 3 (4.7)Relation (4.3) yields: rr
31 2 , rr
21 3 , rl
31 2 , rl
21 3 (4.8) nd: lα
31 2 , lα
21 3 (4.9)Relation (4.4) yields: ir , il (4.10)And the relation (4.5) yields: li (4.11)Define gr ICom as the factor of the free operad generated by i, α, r, l by the operadic ideal spannedby all the leading terms listed above. We claim that
Proposition 4.2.
Starting with arity , all trees in gr ICom have the following general form: thetree grows only to the right; from the root up, first come N l ≥ vertices of type l ; then either treeterminates or comes exactly one vertex of type r ; then come N α ≥ vertices of type α . Additionally,any of the α -type vertices and the r -type vertex may have vertices of type i grafted upon them (thus i -type vertices are always leaves): l l · · · l N l r N l + 1 αi · · · N l + 2 αi N l + N α + 2 N l + N α + 1 (4.12) Proof.
The presence of the terms (4.7) restricts the subtrees composed of α ’s to those growingrightwards (as in the case of the Com operad).From (4.10) we deduce that i can only be a leaf. The terms (4.8) and ( ?? ) ensure that l -subtreescan only grow rightwards, and that we can’t graft α upon l . Also we can’t graft l upon r , and, by(4.11), i upon l . Gathering this data together we prove the claim. (cid:3) Let’s make some observations about a tree of the form (4.12). First, if it has the output of thesecond colour, it must have at least one leaf of the second colour. Second, given two numbers n ≥ nd m > and a colouring χ of the set m + n of type ( n, m ) , there is exactly one tree of this type,with the output of the second colour, whose inputs are coloured with χ . Namely, if the first elementof the second colour in χ is not the last element of the set, it corresponds to the only vertex of type r (and the rest of χ is acquired by grafting or not grafting i ’s on α ’s); and if it is the last elementin the set, the corresponding tree consist solely of l ’s.Otherwise, if the output of the tree is of the first colour, it means that the tree is composed from α ’s and ι ’s, and there is no restriction for m to be greater than zero (so, any χ is feasible).Thus we conclude that for n + m ≥ :(4.13) dim ICom ( n + m, χ, ... ) ≤ dim gr ICom ( n + m, χ, ... ) = 1;dim ICom ( n + m, χ, | ) ≤ dim gr ICom ( n + m, χ, | ) = ( , if m = 0 , , if m ≥ Note that one can easely construct an
ICom -algebra consisting of a commutative algebra A andits ideal I such that each operation of the type (4.12) is different from zero, so our bounds are infact tight what follows that the defining relations of the shuffle operad ICom constitute a Gr¨obnerbasis. The generating series of
ICom coincides with the generating series of gr ICom and are easilycomputed thanks to (4.13) what finishes the proof of Theorem 4.1.
Corollary 4.3.
The operad
ICom is Koszul and its Koszul dual operad admits a quadratic Gr¨obnerbasis.
AffHS operad.
In [15] Merkulov introduced a notion of affine homogeneous space, facilitatingthe study of deformation theory:
Definition 4.4.
An affine homogeneous space is a collection of data ( g , h , h , i , ϕ ) consisting of: • a Lie algebra g with Lie bracket [ , ] ; • a vector space h with a g -module structure h , i : g ⊗ h → g ; • a linear map ϕ : g → h , satisfying the equation ϕ ([ a, b ]) = h a, ϕ ( b ) i − ( − | a || b | h b, ϕ ( a ) i for any a, b ∈ g .The operad AffHS governing affine homogeneous spaces has three generators: i , β = − β , m (4.14)subject to the following set of relations: ββ
31 2 − ββ
21 3 − β β (4.15) mβ
31 2 + m m − m m (4.16) β + m i − m i (4.17)The corresponding shuffle operad has four generators: i , β , m , n (4.18)We will employ a modification of QM-ordering (see (3.2)), in which m and n play the role of x , and b and i play the role of y . So our ordering will based of the ordering on monomials in thefollowing algebra: A = k h m, n, β, i, q i , mq − qm, βq − qβ, βm − mβq,nq − qm, βn − nβq,iq − qi, im − miq,in − niq The monomials in A have the following normal form: first comes an ( m, n ) -word of total degree d x , then a ( β, i ) -word of total degree d y and then q d q . Before comparison we present all involvedmonomials in the normal form.To compare two monomials, we first compare their arities. If equal, we compare their d x ’s, themonomial with greater d x is smaller. If equal, we compare the ( m, n ) -words lexicographically. Ifequal, compare the d y ’s, the monomial with greater d y is greater. If equal, compare the ( β, i ) -wordslexicographically. If equal, compare the d q ’s, the monomial with greater d q is greater. Theorem 4.5.
The operad
AffHS admits a quadratic Gr¨obner basis with respect to the aforemen-tioned QM -ordering. It is not difficult to show that all S -polynomials for the set of relations (4.14)-(4.18) can bereduced to zero. However, we want to explain another proof of this result below. Proof.
The QM -ordering we defined leads to the following choice of the leading terms.The relation (4.15) yields: ββ
31 2 (4.19)The relation (4.16) yields all the trees in the S orbit of the first tree in the relation: mβ
31 2 , n β , mβ
21 3 (4.20) nd the relation (4.17) yields: iβ (4.21)Therefore, the element of the coloured Koszul dual operad AffHS ! are spanned by common mul-tiples of the aforementioned leading monomials. What follows that(4.22) dim AffHS ! ( m, n, | ) ≤ dim gr AffHS ! ( m, n, | ) = 1 , if n = 0;dim AffHS ! ( m, n, ... ) ≤ dim gr AffHS ! ( m, n, ... ) = 1 if n ≤ . and dim gr AffHS ! ( m, n, ... ) equals zero in all other cases. Using a particular algebra over the operad AffHS ! one can show that the left hand side in Inequalities (4.22) is bounded from below by and,therefore, AffHS ! admits a quadratic Gr¨obner basis. Theorem 3.12 implies that the same happensfor AffHS . (cid:3) Corollary 4.6.
The suboperad of
AffHS generated by m and i is free and there is an isomorphismof coloured symmetric collections: AffHS ≃ ( Lie ; F ( m, i )) ⇒ F AffHS ( t , t ) = (cid:18) − ln(1 − t ); t + t − t (cid:19) Proof.
Follows from the description of normal forms. For example, each shuffle monomial in thefree operad F ( m, i ) is not divisible by any leading term of the given Gr¨obner basis. (cid:3) MLie operad.
The operad
MLie has two generators: β = − β , d (4.23)subject to the following two relations: ββ
31 2 − ββ
21 3 − β β (4.24) dβ
31 2 − d d + d d (4.25)Algebra over MLie is a pair of a Lie algebra L and an L -module. The colouring of inputs/outputsmatches the colouring of the inputs of the generators of the operad LP considered in the succeedingExample 4.4, since the operad MLie is a suboperad of LP .The corresponding shuffle operad has three generators: , d , e = d (4.26) Theorem 4.7.
The operad
MLie admits a quadratic Gr¨obner basis with respect to the QM -orderingwith d and e playing the role of x , and β playing the role of y .Proof. The QM -ordering leads to the following list of leading terms:The relation (4.24) yields: ββ
31 2
And the relation (4.25) yields: dβ
31 2 , dβ
21 3 , e β It is well known that the uncoloured S -polynomial corresponding to the small common multiple oftwo Jacobi identities can be reduced to zero. The remaining S -polynomial is assigned to the smallcommon multiple of the Jacobi identity (4.24) and (4.25): We write the small common multiple ofthe Jacobi relation (4.24) and module structure (4.25) as well as reductions of the corresponding S -polynomial using the language of composition of operations with numbers indexing outputs: d ( B ( B (1 , , ,
4) := dB B
31 2
The corresponding S -polynomial µ is equal to µ := d (1 , ◦ [ B ( B (1 , , − B ( B (1 , , − B (1 , B (2 , − [ d ( B (1 , , − d (1 , d (3 , e ( d (1 , , ◦ B (1 , We underline all monomials that admits further reduction (rewritings) of the S -polynomial µ : µ = − d ( B ( B (1 , , , − d ( B (1 , B (2 , ,
4) + d ( B (1 , , d (3 , − e ( d ( B (1 , , ,
3) = (4.25) = − d ( B (1 , , d (2 , e ( d ( B (1 , , , − d (1 , d ( B (2 , , e ( d (1 , , B (2 , d (1 , d (2 , d (3 , − e ( d (1 , d (3 , , − e ( d (1 , d (2 , ,
3) + e ( e ( d (1 , , ,
3) = (4.25) = − d ( B (1 , , d (2 , e ( d (1 , , B (2 , − e ( d (1 , d (2 , ,
3) + e ( e ( d (1 , , , − e ( e ( d (1 , , ,
2) + d (1 , e ( d (2 , , All remaining S -polynomials corresponds to the action of symmetric group on the colourings of thelatter one. (cid:3) orollary 4.8. There is an isomorphism of coloured collections
MLie ≃ ( F ( d ) , Lie ) , where by F ( d ) we denote the free -coloured operad generated by a single element d . In particular, −−−→ F MLie ( t , t ) = (cid:18) t − t + t ; − log(1 − t ) (cid:19) Proof.
The set of leading monomials explains the structure of normal words in
MLie . What followsthat
MLie consists of two disjoint parts. The first part is the operad
Lie generated by β , and thesecond part is the free shuffle coloured operad generated by d and e . The latter is the shuffle operadassigned to the free symmetric coloured operad generated by a single element d . (cid:3) H ( SC vor ) and LP operads. In [11] Hoefel and Livernet provide a description of the operad SC (Swiss Cheese) and its zeroth homology H ( SC vor ) . For the latter the authors proved its Koszulnessand provide the Koszul dual operad – the operad of Leibniz pairs LP . We present a quadraticGr¨obner basis for the latter operad and hence present another proof of the koszulness of H ( SC vor ) and LP . Moreover, we computed the generating series of LP and H ( SC vor ) . Definition 4.9.
A Leibniz pair is a pair of a Lie algebra L and an associative algebra A togetherwith a morphism of Lie algebras L → Der( A ) .The operad LP has three generators: β = − β , a , d (4.27)subject to the following relation:The Jacobi relation: ββ
31 2 − ββ
21 3 − β β (4.28)The associativity relation for a : aa
31 2 − a a (4.29)The derivation relation: d a − a d − ad
31 2 (4.30)The Lie algebra morphism relation: β
31 2 − d d + d d (4.31)The corresponding shuffle operad has five generators: β , a , b = a , d , e = d (4.32) Theorem 4.10.
The defining relations of the operad LP forms a quadratic Gr¨obner bases of relationswith respect to the QM (partial) ordering with a and b being variables of the type x , β , m , and n are y -type variables, and in addition a > b , m > n lexicographically.Proof. The S -polynomials for relations (4.31) and (4.28) can be reduced to zero in the same way asin the previous example. We are left to check the reducibility of the S -polynomials for (4.30) and(4.29). We provide the reduction of one of these polynomials associated with the small commonmultiple d (1 , a (2 , a (3 , , as they lie in a single Σ n -orbit. d (1 , a ( a (2 , , − b ( d (1 , a (3 , , − a ( d (1 , , a (3 , (4.29) = d (1 , a ( a (2 , , − b ( d (1 , a (3 , , − a ( a ( d (1 , , , (4.30) = − a ( a ( d (1 , , ,
4) + b ( d (1 , , a (2 , a ( d (1 , a (2 , , − b ( b ( d (1 , , , − b ( a ( d (1 , , , (4.30) = b ( d (1 , , a (2 , − b ( b ( d (1 , , , − b ( a ( d (1 , , ,
2) + a ( b ( d (1 , , , (4.29) = b ( d (1 , , a (2 , − b ( b ( d (1 , , , (4.29) = 0 (cid:3) The QM -ordering leads to the following choice of the leading terms.The relation (4.28) yields: ββ
31 2
The relation (4.29) yields: a a , ab
31 2 , b b , a b , b a , ab
21 3
The relation (4.30) yields: a , ea
21 3 , eb
31 2 , d b , eb
21 3 , ea
31 2
And the relation (4.31) yields: dβ
31 2 , ee
31 2 , dβ
21 3
The choice of the leading terms in the derivation relation (4.30) ensures that any element of theoperad can be rewritten in the following normal form: a two-level tree with the bottom level consistsof vertices a and b , and the top level consists of the vertices d , e , and β .From this observation and the generating relations of LP we conclude that LP = As ◦ MLie , assymmetric coloured collections, where As is the associative operad generated by a , the operad MLie was described in the previous example.Now we can compute the generating series for LP : LP ≃ As ◦ MLie ⇒ −−→ F LP ( t , t ) = −→ F As ◦ −−−→ F MLie == (cid:18) t − t ; t (cid:19) ◦ (cid:18) t − t + t ; − log (1 − t ) (cid:19) = (cid:18) − t t + t ; − log(1 − t ) (cid:19) . DCom operad.
Pairs of the form ( A, D ) , consisting of a commutative algebra A and a space D of its derivation are governed by the following two-coloured operad DCom generated by twogenerators of arity 2, α and d : α = α , d (4.33)with the relations for α being an associative commutative product and d being the derivation of α (the Leibniz rule): aa
31 2 − a a (4.34) dα
31 2 = α d + αd
21 3 (4.35)This example is rather contrived, as such pairs fit more naturally in the uncoloured framework. Weconsider this two-coloured version with a view to use the computations for it in Example 4.6. heorem 4.11. The -coloured operad DCom admits a quadratic Gr¨obner basis with respect to the QM -ordering described in §3.2 with y = d, x = α .There is an isomorphism of coloured symmetric collections DCom = Com ◦ F ( d ) , where F ( d ) isthe free operad generated by d and Com consists of operations of the first colour and the generatingseries −−−−→ F DCom ( t , t ) is equal to (cid:18) e t − t , t (cid:19) .Proof. The corresponding shuffle operad has three generators: α , d , e = d (4.36)The following list of leading terms do appear with respect to the aforementioned QM -ordering: αα
21 3 , α α , e α , dα
21 3 , dα
31 2 (4.37)Thus, there are the S -polynomials of the first colour that deal with the commutative associativeproduct and are known to be reducible to and there is an S -polynomial associated with the smallcommon multiple e (1 , α ( α (2 , , of Relations (4.34) and (4.35): e (1 , α ( α (2 , , − α ( e (1 , α (2 , , − α ( e (1 , , α (2 , (4.34) = e (1 , α ( α (2 , , − α ( e (1 , α (2 , , − α ( α ( e (1 , , , (4.35) = − α ( α ( e (1 , , ,
4) + α ( e (1 , α (2 , ,
4) + α ( e (1 , , α (2 , − α ( α ( e (1 , , , − α ( α ( e (1 , , , (4.35) = α ( e (1 , , α (2 , − α ( α ( e (1 , , , − α ( α ( e (1 , , ,
3) + α ( α ( e (1 , , , (4.34) = α ( e (1 , , α (2 , − α ( α ( e (1 , , , (4.34) = 0 All other S polynomials differ from this one by the action of symmetric group what affects thereplacement d by e .It is immediate to see that the elements of DCom have the following normal form: a leftwardgrowing tree of α ’s with arbitrary compositions of d and e plugged into it. It means that assymmetric coloured collections DCom = Com ◦ F ( d ) , where F ( d ) is the free operad generated by d and Com consists of operations of the first colour.Now we can compute the generating series for
DCom : −−−−→ F DCom ( t , t ) = ( e t − t ) ◦ ( t − t ; t ) = (cid:18) e t − t − t (cid:19) (cid:3) .6. Lie - Rinehart operad.
Following [16], [13] we say that a Lie-Rinehart algebra is a pair ( S, L ) of a commutative algebra S and a Lie algebra L , such that L acts on S by derivations, L is an S module, and the following relations hold: ( sα )( t ) = s · ( α ( t )) , [ α, sβ ] = s [ α, β ] + α ( s ) β ; for s, t ∈ S , α, β ∈ L . With each algebraic variety or smooth manifold X one can assign aLie-Rinehart algebra consisting of the commutative algebra of functions on X and the Lie alge-bra of vector fields on X .We define the operad Lie - Rinehart as the coloured symmetric operad on two colours generated byfour operations: α = α , β = − β , d , m (4.38)subject to the following list of relations.The associativity relation for α : aa
31 2 − a a (4.39)The Jacobi relation for β : ββ
31 2 − ββ
21 3 − β β (4.40)The Leibniz rule: dα
31 2 = α d + αd
21 3 (4.41)The relation describing the morphism of Lie algebras L → Der( S ) : d β = dd
21 3 − dd
31 2 (4.42)The relation stating that L is an S -module: α
31 2 = m m (4.43)And two relation specific for the Lie-Rinehart algebras: d m = αd
21 3 (4.44) βm
21 3 = m β + md
31 2 (4.45)The corresponding shuffle operad has six generators: α , β , d , e = d , m , n = m (4.46) Theorem 4.12. • The operad
Lie - Rinehart admits a quadratic Gr¨obner basis; • The -coloured symmetric collection Lie - Rinehart is isomorphic to the composition N m ◦ ( DCom ; Lie ) .Here DCom is the subset of DCom spanned by operations with the output of the first(straight) colour and N m is a nilpotent quadratic operad generated by a single element m subject to the relation m m = 0 . Proof.
In this example we employ a further modification of a QM-ordering. We divide the generatorsinto three groups: light ( β , n , m ), heavy ( d , e ), and superheavy ( α ). The light generators play therole of y , the heavy ones play the role of x , and the superheavy play the role of x relatively tothe heavy ones. Namely, we base our ordering on the ordering of the monomials in the algebra A defined as: A = k h α, d, e, m, n, β, q i , αq − qα, dα − αdq, eα − αeq,mα − αmq, nα − αnq, βα − αβq,dq − qd, md − dmq, nd − dnq, βd − dβq,eq − qe, me − emq, ne − enq, βe − eβq,mq − qm, nq − qn, βq − qβ A word in A has the following normal form: w = w α w d,e w m,n,β q d q , where w α is an α -word of degree d α , w d,e is a ( d, e ) -word of degree d ( d,e ) , and w m,n,β is an ( m, n, β ) -word of degree d ( m,n,β ) . To compare two such words, first compare d α , the word with smaller d α is reater. If equal, the word with smaller d ( d,e ) is greater. If equal, the word with greater d ( m,n,β ) isgreater. If equal, the word with greater d q is greater.This ordering leads to the following choice of the leading terms: The relation (4.39) yields: αα
21 3 , α α (4.47)The relation (4.40) yields: ββ
31 2 (4.48)The relation (4.41) yields: e α , dα
21 3 , dα
31 2 (4.49)The relation (4.43) yields: m m , nm
21 3 , m n , nn
31 2 , nn
21 3 , nm
31 2 (4.50)The relation (4.42) yields: d β , eβ
21 3 , eβ
31 2 (4.51)The relation (4.44) yields: em
31 2 , en
31 2 , d n , em
21 3 , d m , en
21 3 (4.52)And the relation (4.45) yields: m , βm
21 3 , βn
31 2 , β n , βn
21 3 , βm
31 2 (4.53)Note that for every relation excluding the Jacobi identity for β and associativity relations for α all the leading terms constitute the entire Σ -orbit acting on different colourings of inputs. Thisobservation shortens the number of S -polynomials whose reductions one has to verify. We work outall reductions (one for each Σ -orbit) in the Appendix A.Contemplating on the choice of the leading terms, one can conclude, that the elements of theoperad Lie - Rinehart have the following normal form: • On the first level they have one vertex of type n or m (or none of those) • Two blocks can be grafted on n or m : block consisting of α ’s ( Com -block) and block con-sisting of β ’s ( Lie -block). • Additionally, any free input of the
Com -block may be decorated with an arbitrary ( d, e ) -tree. m or n Com -block ( d, e ) -tree Lie -block ( d, e ) -tree ( d, e ) -tree(4.54)So the elements of the corresponding symmetric operad have the form: m DCom -block
Lie -block(4.55)Note that the action of the permutation σ on the set of inputs of a normal word will give againa normal word whenever σ will not interact with the Lie block. On the other hand, the action ofsymmetric group on the operad Lie is also well known. Thus, we conclude that the description ofthe normal words implies the isomorphism of the coloured symmetric collections Lie - Rinehart andthe composition N m ◦ ( DCom , Lie ) . Here we denote by N m the -coloured operad generated bya single operation m in arity and all non-trivial compositions are equal to zero. The colouredsymmetric collection assigned with N m consists of m in arity , two identity elements of two coloursin arity , and in all other arities.This allows us to compute the generating series: −−−−−−−→ F Lie - Rinehart ( t , t ) = ( t ; t + t t ) ◦ (cid:18) e t − t − − log(1 − t ) (cid:19) = (cid:18) e t − t − − log(1 − t ) · e t − t (cid:19) (cid:3) Corollary 4.13.
The map of coloured operads
DCom → Lie - Rinehart is an embedding. .7. DerCom operad.
The
DerCom operad is an operad governing pairs of a commutative algebra S and a Lie algebra L , such that L acts on S by derivations and L has a structure of S -module.In combinatorial terms the operad DerCom is an operad on colours { c, l } , is generated by thefollowing list of binary operations: • α ( - , - ) ∈ DerCom (2 , , c ) – a commutative associative product; • [ - , - ] ∈ DerCom (0 , , l ) – a Lie bracket of the derivations, yielding the Jacobi identity; • d ( - , - ) ∈ DerCom (1 , , c ) – the action of the derivation on the elements of a commutativealgebra; • m ( - , - ) ∈ DerCom (1 , , l ) – the action of a commutative algebra on the Lie algebra of deriva-tions.It is immediate to see that DerCom is essentially the operad
Lie - Rinehart without two relations(4.44) and (4.45).
Theorem 4.14.
The same choice of ordering and the leading terms also leads to a quadratic Gr¨obnerbasis for
DerCom .Proof.
The set of S -polynomials for DerCom is the subset of S -polynomials computed for Lie - Rinehart and the corresponding reductions do not involve relations (4.44) and (4.45) as one can see from thecomputations presented in Appendix A. (cid:3)
Appendix A. S -polynomial for Lie - Rinehart
In the appendices we provide a sample of computations for the operads LP and Lie - Rinehart . Thefull computation is too voluminous to include here, but by the merit of Σ n -symmetry of the set ofthe leading terms, it suffice to provide one example for every pair of relations with a non-trivial S -polynomial.The corresponding shuffle coloured operad has the following list of relations where we underlinethe leading monomial in each relation:(Com1) α ( α (1 , , − α ( α (1 , ,
2) = 0 (Com2) α ( α (1 , , − α (1 , α (2 , (Lie) β ( β (1 , , − β ( β (1 , , − β (1 , β (2 , (Leib1) e (1 , α (2 , − α ( e (1 , , − α ( e (1 , ,
2) = 0 (Leib2) d ( α (1 , , − α ( d (1 , , − α (1 , e (2 , (Leib3) d ( α (1 , , − α (1 , d (2 , − α ( d (1 , ,
2) = 0 (Mor1) d (1 , β (2 , d ( d (1 , , − d ( d (1 , ,
2) = 0 (Mor2) e ( β (1 , ,
2) + d ( e (1 , , − e (1 , d (2 , (Mor3) − d (1 , β (2 , d ( d (1 , , − d ( d (1 , ,
3) = 0 (Mor4) − e ( β (1 , ,
3) + e (1 , e (2 , − d ( e (1 , ,
2) = 0 (Mor5) − e ( β (1 , ,
2) + e (1 , d (2 , − d ( e (1 , ,
3) = 0 (Mor6) e ( β (1 , ,
3) + d ( e (1 , , − e (1 , e (2 , (SMod1) m ( α (1 , , − m (1 , m (2 , SMod2) m ( α (1 , , − n ( m (1 , ,
2) = 0 (SMod3) m ( α (1 , , − m (1 , n (2 , (SMod4) n (1 , α (2 , − n ( n (1 , ,
3) = 0 (SMod5) n (1 , α (2 , − n ( n (1 , ,
2) = 0 (SMod6) m ( α (1 , , − n ( m (1 , ,
3) = 0 (LR-A1) e ( m (1 , , − α (1 , e (2 , (LR-A2) e ( n (1 , , − α ( e (1 , ,
2) = 0 (LR-A3) d (1 , n (2 , − α ( d (1 , ,
3) = 0 (LR-A4) e ( m (1 , , − α (1 , d (2 , (LR-A5) d (1 , m (2 , − α ( d (1 , ,
2) = 0 (LR-A6) e ( n (1 , , − α ( e (1 , ,
3) = 0 (LR-B1) β (1 , m (2 , − m ( e (1 , , − n ( β (1 , ,
2) = 0 (LR-B2) − β ( m (1 , , − m ( d (1 , , − m (1 , β (2 , (LR-B3) − β ( n (1 , , − n (1 , d (2 , n ( β (1 , ,
2) = 0 (LR-B4) β (1 , n (2 , − m ( e (1 , , − n ( β (1 , ,
3) = 0 (LR-B5) − β ( n (1 , , − n (1 , e (2 , n ( β (1 , ,
3) = 0 (LR-B6) − β ( m (1 , , − m ( d (1 , ,
2) + m (1 , β (2 , Below is the list of representatives of Σ -orbits of the set of all reductions. eduction of the S -polynomial for relations LR-B1 and LR-B2 associated with the small commonmultiple β ( m (1 , , m (2 , : − m ( e ( m (1 , , , − n ( β ( m (1 , , , − m ( d (1 , m (2 , , − m (1 , β ( m (2 , , LR − B = − m ( e ( m (1 , , , − m ( d (1 , m (2 , , − m (1 , β ( m (2 , , n ( m ( d (1 , , , n ( m (1 , β (3 , , LR − B = − m ( e ( m (1 , , , − m ( d (1 , m (2 , ,
4) + n ( m ( d (1 , , ,
2) + n ( m (1 , β (3 , , m (1 , m ( d (2 , , − m (1 , m (2 , β (3 , LR − A = − m ( d (1 , m (2 , ,
4) + n ( m ( d (1 , , ,
2) + n ( m (1 , β (3 , ,
2) + m (1 , m ( d (2 , , − m (1 , m (2 , β (3 , − m ( α (1 , d (2 , , LR − A = n ( m ( d (1 , , ,
2) + n ( m (1 , β (3 , ,
2) + m (1 , m ( d (2 , , − m (1 , m (2 , β (3 , − m ( α (1 , d (2 , , − m ( α ( d (1 , , , SMod = n ( m ( d (1 , , ,
2) + n ( m (1 , β (3 , , − m ( α ( d (1 , , , − m ( α (1 , , β (3 , SMod = 0 Reduction of the S -polynomial for relations LR-B1 and LR-B2 associated with the small commonmultiple d (1 , β (2 , m (3 , : − d (1 , m ( e (2 , , − d (1 , n ( β (2 , , − d ( d (1 , , m (3 , d ( d (1 , m (3 , , LR − A = − d (1 , m ( e (2 , , − d ( d (1 , , m (3 , d ( d (1 , m (3 , , − α ( d (1 , β (2 , , LR − A = − α ( d (1 , β (2 , , − α ( d (1 , , e (2 , − α ( d ( d (1 , , ,
3) + d ( α ( d (1 , , , Mor = − α ( d (1 , , e (2 , d ( α ( d (1 , , , − α ( d ( d (1 , , , Leib = 0 Reducing S-polynomial for relations LR-B1 and SMod1 associated with small common multiple β (1 , m (2 , m (3 , : − m ( e (1 , , m (3 , − n ( β (1 , m (3 , ,
2) + β (1 , m ( α (2 , , LR − B = − m ( e (1 , , m (3 , − n ( m ( e (1 , , , − n ( n ( β (1 , , ,
2) + m ( e (1 , α (2 , , n ( β (1 , , α (2 , SMod = − n ( m ( e (1 , , , − n ( n ( β (1 , , ,
2) + m ( e (1 , α (2 , ,
4) + n ( β (1 , , α (2 , − m ( α ( e (1 , , , SMod = − n ( n ( β (1 , , ,
2) + m ( e (1 , α (2 , ,
4) + n ( β (1 , , α (2 , − m ( α ( e (1 , , , − m ( α ( e (1 , , , SMod = m ( e (1 , α (2 , , − m ( α ( e (1 , , , − m ( α ( e (1 , , , Leib = 0 educing S-polynomial for relations LR-B1 and Lie associated with the small common multiple β ( β (1 , , m (3 , : − m ( e ( β (1 , , , − n ( β ( β (1 , , ,
3) + β ( β (1 , m (3 , ,
2) + β (1 , β (2 , m (3 , LR − B = − m ( e ( β (1 , , , − n ( β ( β (1 , , ,
3) + β ( m ( e (1 , , ,
2) + β ( n ( β (1 , , , β (1 , m ( e (2 , , β (1 , n ( β (2 , , LR − B = − m ( e ( β (1 , , , − n ( β ( β (1 , , ,
3) + β ( m ( e (1 , , ,
2) + β ( n ( β (1 , , , β (1 , n ( β (2 , , m ( e (1 , e (2 , ,
4) + n ( β (1 , , e (2 , LR − B = − m ( e ( β (1 , , , − n ( β ( β (1 , , ,
3) + β ( n ( β (1 , , ,
2) + β (1 , n ( β (2 , , m ( e (1 , e (2 , ,
4) + n ( β (1 , , e (2 , − m ( d ( e (1 , , , − m ( e (1 , , β (2 , LR − B = − m ( e ( β (1 , , , − n ( β ( β (1 , , ,
3) + β ( n ( β (1 , , ,
2) + m ( e (1 , e (2 , , n ( β (1 , , e (2 , − m ( d ( e (1 , , ,
4) + n ( β (1 , β (2 , , LR − B = − m ( e ( β (1 , , , − n ( β ( β (1 , , ,
3) + m ( e (1 , e (2 , , − m ( d ( e (1 , , , n ( β (1 , β (2 , ,
3) + n ( β ( β (1 , , , Mor = − n ( β ( β (1 , , ,
3) + n ( β (1 , β (2 , ,
3) + n ( β ( β (1 , , , Lie = 0
Reducing S-polynomial for relations LR-A1 and SMod1 associated associated with the smallcommon multiple e ( m (1 , m (2 , , : − α (1 , e ( m (2 , , e ( m ( α (1 , , , LR − A = − α (1 , α (2 , e (3 , α ( α (1 , , e (3 , Com = 0 Reducing S-polynomial for relations LR-A1 and Leib1 associated with the small common multiple e ( m (1 , , α (3 , : − α (1 , e (2 , α (3 , α ( e ( m (1 , , ,
4) + α ( e ( m (1 , , , LR − A = − α (1 , e (2 , α (3 , α ( α (1 , e (2 , ,
4) + α ( α (1 , e (2 , , Leib = α ( α (1 , e (2 , ,
4) + α ( α (1 , e (2 , , − α (1 , α ( e (2 , , − α (1 , α ( e (2 , , Com = 0 Reducing S-polynomial for relations Mor1 and Lie associated with the small common multiple d (1 , β ( β (2 , , : d ( d (1 , β (2 , , − d ( d (1 , , β (2 , d (1 , β ( β (2 , , d (1 , β (2 , β (3 , Mor = − d ( d ( d (1 , , ,
4) + d ( d ( d (1 , , ,
4) + d ( d ( d (1 , , , − d ( d ( d (1 , , , − d ( d (1 , β (2 , ,
3) + d ( d (1 , , β (2 , − d ( d (1 , , β (3 , d ( d (1 , β (3 , , Mor = 0 educing S-polynomial for relations Mor1 and Leib2 associated with the small common multiple d ( α (1 , , β (2 , : d ( d ( α (1 , , , − d ( d ( α (1 , , ,
2) + α ( d (1 , β (2 , ,
4) + α (1 , e ( β (2 , , Mor = d ( d ( α (1 , , , − d ( d ( α (1 , , ,
2) + α (1 , e ( β (2 , , − α ( d ( d (1 , , , α ( d ( d (1 , , , Mor = d ( d ( α (1 , , , − d ( d ( α (1 , , , − α ( d ( d (1 , , ,
4) + α ( d ( d (1 , , , α (1 , e (2 , e (3 , − α (1 , d ( e (2 , , Leib = − α ( d ( d (1 , , ,
4) + α ( d ( d (1 , , ,
4) + α (1 , e (2 , e (3 , − α (1 , d ( e (2 , , d ( α ( d (1 , , ,
3) + d ( α (1 , e (2 , , − d ( α ( d (1 , , , − d ( α (1 , e (3 , , Leib = − α (1 , d ( e (2 , , d ( α (1 , e (2 , , − α ( d (1 , , e (2 , Leib = 0 Reducing S-polynomial for relations SMod1 and SMod2 associated with the small common mul-tiple n ( m (1 , m (3 , , : n ( m ( α (1 , , , − m ( α (1 , , m (3 , SMod = n ( m ( α (1 , , , − m ( α ( α (1 , , , SMod = − m ( α ( α (1 , , ,
4) + m ( α ( α (1 , , , Com = 0 Reducing S-polynomial for relations Com1 and Com2 associated with the small common multiple α ( α (1 , , α (2 , : α ( α (1 , α (2 , , − α ( α ( α (1 , , , Com = α ( α (1 , α (2 , , − α ( α ( α (1 , , , Com = − α ( α ( α (1 , , ,
4) + α ( α ( α (1 , , , Com = 0 Reducing S-polynomial for relations Com1 and Leib1 associated with the small common multiple e (1 , α ( α (2 , , : e (1 , α ( α (2 , , − α ( e (1 , α (2 , , − α ( e (1 , , α (2 , Com = e (1 , α ( α (2 , , − α ( e (1 , α (2 , , − α ( α ( e (1 , , , Leib = − α ( α ( e (1 , , ,
4) + α ( e (1 , α (2 , ,
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E-mail address : [email protected] National Research University Higher School of Economics, 20 Myasnitskaya street, Moscow101000, Russia & Institute for Theoretical and Experimental Physics, 25 Bolshaya Cheremushkin-skaya, Moscow 117259, Russia;
E-mail address : [email protected]@hse.ru