Endofunctors and Poincaré-Birkhoff-Witt theorems
aa r X i v : . [ m a t h . C T ] D ec ENDOFUNCTORS AND POINCAR´E–BIRKHOFF–WITTTHEOREMS
VLADIMIR DOTSENKO AND PEDRO TAMAROFF
Abstract.
We determine what appears to be the bare-bones categori-cal framework for Poincar´e–Birkhoff–Witt type theorems about univer-sal enveloping algebras of various algebraic structures. Our languageis that of endofunctors; we establish that a natural transformation ofmonads enjoys a Poincar´e–Birkhoff–Witt property only if that transfor-mation makes its codomain a free right module over its domain. Weconclude with a number of applications to show how this unified ap-proach proves various old and new Poincar´e–Birkhoff–Witt type theo-rems. In particular, we prove a PBW type result for universal envelopingdendriform algebras of pre-Lie algebras, answering a question of Loday. Introduction
It is well known that the commutator [ a, b ] = ab − ba in every associativealgebra satisfies the Jacobi identity. Thus, every associative algebra may beregarded as a Lie algebra, leading to a functor from the category of asso-ciative algebras to the category of Lie algebras assigning to an associativealgebra the Lie algebra with the same underlying vector space and the Liebracket as above. This functor admits a left adjoint U ( − ), the universalenveloping associative algebra of a Lie algebra. The classical Poincar´e–Birkhoff–Witt (PBW) theorem identifies the underlying vector space of theuniversal enveloping algebra of any Lie algebra with its symmetric algebra;the precise properties of such an identification depend on the proof onechooses.More generally, a functor from the category of algebras of type S to thecategory of algebras of type T is called a functor of change of structure if itonly changes the structure operations, leaving the underlying object of analgebra intact. Informally, one says that such a functor has the PBW prop-erty if, for any T -algebra A , the underlying object of its universal enveloping S -algebra U S ( A ) admits a description that does not depend on the algebrastructure, but only on the underlying object of A .This intuitive view of the PBW property is inspired by the notion ofa PBW pair of algebraic structures due to Mikhalev and Shestakov [47]. Mathematics Subject Classification.
Key words and phrases. endofunctor, monad, universal enveloping algebra, Poincar´e–Birkhoff–Witt theorem.
There, the algebraic setup is that of varieties of algebras. The authorsof [47] define, for any T -algebra A , a canonical filtration on the universalenveloping algebra U S ( A ) which is compatible with the S -algebra structure,and establish that there is a canonical surjection π : U S (Ab A ) ։ gr U S ( A ) , where Ab A is the Abelian S -algebra on the underlying vector space of A .They say that the given algebraic structures form a PBW pair if that canon-ical surjection is an isomorphism. Furthermore, they prove a result statingthat this property is equivalent to U S ( A ) having a basis of certain monomi-als built out of the basis elements of A , where the definition of monomialsdoes not depend on a particular algebra A . This latter property is definedin a slightly more vague way than the former one; trying to formalise it,we discovered a pleasant categorical context where PBW theorems belong.The approach we propose is to use the language of endofunctors, so that afully rigorous way to say “the definition of monomials does not depend ona particular algebra” is to say that the underlying vector space of U S ( A ) isisomorphic to X ( A ), where X is an endofunctor on the category of vectorspaces, with isomorphisms U S ( A ) ∼ = X ( A ) being natural with respect to alge-bra maps. Our main result (Theorem 3.1) states that if algebraic structuresare encoded by monads, and a functor of change of structure arises from anatural transformation of monads φ : M → N , then the PBW property holdsif and only if the right module action of M on N via φ is free; moreover thespace of generators of N as a right M -module is naturally isomorphic to theendofunctor X above.In the context of the classical PBW theorem for Lie algebras, the conditionof freeness of a module does emerge in a completely different way: whenworking with Lie algebras over rings, one would normally require the Liealgebra to be free as a module over the corresponding ring in order for thePBW theorem to hold, see [17]. We feel as though we have to emphasizethat our “freeness of a module” condition is of entirely different nature: itis freeness of action of one monad on another, which only makes sense whenone goes one level up in terms of categorical abstraction and considers allalgebras of the given type as modules over the same monad. This conditionis not expressible if one looks at an individual algebra, and this is preciselywhat makes our main result completely new in comparison with existingliterature on PBW type theorems. It is also worth mentioning that one classof operads for which the free right module condition is almost tautologicallytrue is given by those obtained by means of distributive laws [46]; however,for many interesting examples it is definitely not the case. (The exampleof post-Poisson algebras in the last section of this paper should be veryinstructional for understanding that.)It is worth remarking that there is a number of other phenomena whichare occasionally referred to as PBW type theorems. One of them dealswith various remarkable families of associative algebras depending on one NDOFUNCTORS AND POINCAR´E–BIRKHOFF–WITT THEOREMS 3 or more parameters, and is completely out of our scope; we refer the readerto the survey [56] for further information. The other one deals with univer-sal enveloping algebras defined as forgetful functors as above, but considerssituations where the universal enveloping algebra admits what one wouldagree to consider a “nice” description. One important feature of such a“nice” description is what one can call a “baby PBW theorem” stating thatthe natural map from an algebra to its universal enveloping algebra is anembedding. By contrast with our result which in particular shows that thePBW property holds for all algebras if and only if it holds for free algebras,checking the baby PBW property requires digging into intricate propertiesof individual algebras: there exist examples of algebraic structures for whichthe baby PBW property holds for all free algebras but nevertheless fails forsome non-free algebras. A celebrated example where the baby PBW prop-erty holds but the full strength PBW property is not available is given byuniversal enveloping diassociative algebras of Leibniz algebras [40]; furtherexamples can be found in [13, 30] and [9, 10].We argue that our result, being a necessary and sufficient statement,should be regarded as the bare-bones framework for studying the PBWproperty; as such, it provides one with a unified approach to numerousPBW type results proved —sometimes by very technical methods— in theliterature, see e.g. [16, 17, 31, 35, 36, 48, 49, 53, 54, 57]. Most of those PBWtype theorems tend to utilise something extrinsic; e.g., in the case of Liealgebras, one may consider only Lie algebras associated to Lie groups andidentify the universal enveloping algebra with the algebra of distributionson the group supported at the unit element (see [55], this is probably theclosest in spirit to the original proof of Poincar´e [50]), or use the additionalcoalgebra structure on the universal enveloping algebra (like in the proof ofCartier [12], generalised by Loday in [41] who defined a general notion ofa “good triple of operads”). Proofs that do not use such deus ex machina devices normally rely on an explicit presentation of universal enveloping al-gebras by generators and relations (following the most famous applicationof Bergman’s Diamond Lemma [6], in the spirit of proofs of Birkhoff [7] andWitt [58]); while very efficient, those proofs break functoriality in a ratherdrastic way, which is highly undesirable for objects defined by a universalproperty. Finally, what is often labelled as a categorical approach to thePBW theorem refers to proving the PBW theorem for Lie algebras in an ar-bitrary k -linear tensor category (over a field k of characteristic zero) recordedin [18]; this approach is indeed beautifully functorial but does not at all clar-ify what property of the pair of algebraic structures ( Lie , Ass ) makes it work.Our approach, in addition to being fully intrinsic and functorial, unravelsthe mystery behind that very natural question.Towards the end of this paper, we present a few applications of our frame-work. In particular, we prove a new PBW theorem for universal envelopingdendriform algebras of pre-Lie algebras (Theorem 4.6), thus answering a
VLADIMIR DOTSENKO AND PEDRO TAMAROFF question of Loday that remained open for a decade. The proof of that re-sult demonstrates that our monadic approach to PBW type theorems opensa door for utilising a range of operadic techniques which previously weremainly used for purposes of homotopical algebra [11, 44]. Another appli-cation of our main result was recently obtained in [34] where a PBW typetheorem for associative universal enveloping algebras of operadic algebras isproved; a hint for importance of operadic right modules for such a statementto hold can be found in [25, Sec. 10.2].To conclude this introduction, it is perhaps worth noting that our defi-nition of the PBW property exhibits an interesting “before/after” dualismwith that of [47]: that definition formalises the intuitive notion that “op-erations on A do not matter before computing U S ( A )”, so that operationson U S ( A ) have some canonical “leading terms”, and then corrections thatdo depend on operations of A , while our approach suggests that “operationson A do not matter after computing U S ( A )”, so that the underlying vectorspace of U S ( A ) is described in a canonical way. In Proposition 4.3, we showthat our formalisation, unlike that of [47], shows that the extent to whicha PBW isomorphism may be functorial depends on the characteristic of theground field, rather than merely saying “certain strategies of proof are notavailable in positive characteristic”. Acknowledgements.
We thank Dmitry Kaledin and Ivan Shestakov forextremely useful and encouraging discussions of this work. These discussionshappened when the first author was visiting CINVESTAV (Mexico City);he is grateful to Jacob Mostovoy for the invitation to visit and to presentthis work. We also thank Vsevolod Gubarev, Pavel Kolesnikov and BrunoVallette for useful comments, and Anton Khoroshkin for informing us ofthe preprint [34] that builds upon our work. Special thanks due to MartinHyland whose questions greatly helped to make the proof of the main resultmore comprehensible.2.
Recollections: monads, algebras, modules
In this section, we recall some basic definitions and results from categorytheory used in this paper, referring the reader to [37, 38, 45] for furtherdetails.2.1.
Monads.
Let C be a category. Recall that all endofunctors of C forma strict monoidal category ( End ( C ) , ◦ , ). More precisely, in that categorymorphisms are natural transformations, the monoidal structure ◦ is the com-position of endofunctors, ( F ◦ G )( c ) = F ( G ( c )), and the unit of the monoidalstructure is the identity functor, ( c ) = c . A monad on C is a monoid( M , µ M , η M ) in End ( C ); here we denote by µ M : M ◦ M → M the monoidproduct, and by η M : → M the monoid unit. NDOFUNCTORS AND POINCAR´E–BIRKHOFF–WITT THEOREMS 5
Algebras. An algebra for the monad M is an object c of C , and astructure map γ c : M ( c ) → c for which the two diagrams M ( M ( c )) M ( γ c ) / / µ M ( c ) (cid:15) (cid:15) M ( c ) γ c (cid:15) (cid:15) M ( c ) γ c / / c ( c ) η M ( c ) / / c ( ( PPPPPPPPPPPPPPPP M ( c ) γ c (cid:15) (cid:15) c commute for all c . The category of algebras over a monad M is denoted by C M .2.3. Modules.
The notion of a module over a monad follows the generaldefinition of a module over a monoid in monoidal category. We shall primar-ily focus on right modules; left modules are defined similarly. A right moduleover a monad M is an endofunctor R together with a natural transformation ρ R : R ◦ M → R for which the two diagrams R ( M ( M ( c ))) R ( µ M ( c )) / / ρ R ( M ( c )) (cid:15) (cid:15) R ( M ( c )) ρ R ( c ) (cid:15) (cid:15) R ( M ( c )) ρ R ( c ) / / R ( c ) R ( ( c )) R ( η M ( c )) / / R ( c ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ R ( M ( c )) ρ R ( c ) (cid:15) (cid:15) R ( c )commute for all c . The category of right modules over a monad M is denotedby Mod M . The forgetful functor from the category Mod M to End ( C ) hasa left adjoint, called the free right M -module functor; the free right M -module generated by an endofunctor X is X ◦ M with the structure map X ◦ M ◦ M → X ◦ M given by 1 X ◦ µ M .2.4. Coequalizers in categories of algebras.
Recall that a reflexive pair in a category C is a diagram c f ) ) g c , d o o where f d = gd = 1 c . Throughout this paper, we shall assume the fol-lowing property of the category C : for every monad M , the category C M has coequalizers of all reflexive pairs. There are various criteria for that tohappen, see, for instance, [1] and [5, Sec. 9.3] (both relying on the seminalwork of Linton on coequalizers in categories of algebras [39]). In particular,this property holds for any complete and cocomplete well-powered regularcategory where all regular epimorphisms split. This holds, for instance, for VLADIMIR DOTSENKO AND PEDRO TAMAROFF the category
Set and the categories
Vect k (the category of vector spacesover k , for any field k ) and Vect Σ k (the category of symmetric sequencesover k , for a field k of zero characteristic), as well as their “super” ( Z - or Z / Categorical PBW theorem
The adjunction between change of structure and direct image.
Suppose that M and N are two monads on C , and that φ : M → N is a naturaltransformation of monads. For such data, one can define the functor ofchange of algebra structure φ ∗ : C N → C M for which the algebra map M ( c ) → c on an N -algebra c is computed as thecomposite M ( c ) φ (1 c ) −−−→ N ( c ) γ c −→ c. By [39, Prop. 1], under our assumptions on C the functor φ ∗ has a leftadjoint functor, the direct image functor φ ! , and for every M -algebra c , the N -algebra φ ! ( c ) can be computed as the coequalizer of the reflexive pair ofmorphisms N ( M ( c )) N ( φ (1 c )) / / N ( γ c ) N ( N ( c )) µ N (1 c ) / / N ( c ) , which is reflexive with the arrow d : N ( c ) → N ( M ( c )) given by N ( c ) ∼ = −→ N ( ( c )) N ( η M (1 c )) −−−−−−−→ N ( M ( c )) . Let us give a toy example of this general construction which would befamiliar to a reader without a systematic categorical background. Let C = Vect k be a category of vector spaces over a field k , and let A be an as-sociative algebra over k . Consider the endofunctor M A of Vect k given by M A ( V ) = A ⊗ V . It is easy to see that the associative algebra structureon A leads to a monad structure on M A , and algebras over the monad M A are left A -modules. Moreover, if ψ : A → B is a morphism of associative al-gebras, we have a natural transformation of monads φ : M A → M B , and thefunctors φ ∗ and φ ! are the usual restriction and induction functors betweenthe categories of left modules.In general, the direct image functor is well understood and frequently usedin the case of analytic endofunctors [32], i. e. in the case of operads [44]; inthat case this formula for the adjoint functor fits into the general frameworkof relative composite products of operadic bimodules [26, 52]. Relative prod-ucts of arbitrary endofunctors do not, in general, satisfy all the propertiesof relative composite products; however, in some situations all the necessary NDOFUNCTORS AND POINCAR´E–BIRKHOFF–WITT THEOREMS 7 coequalizers exist (and are absolute); as a consequence, for our purposesthere is no need to restrict oneself to analytic endofunctors.3.2.
The main result.
As we remarked above, our goal is to give a cate-gorical formalisation of an intuitive view of the PBW property according towhich “the underlying object of the universal enveloping algebra of c doesnot depend on the algebra structure of c ”. Suppose that φ : M → N is a nat-ural transformation of monads on C . We shall say that the datum ( M , N , φ ) has the PBW property if there exists an endofunctor X such that the under-lying object of the universal enveloping N -algebra φ ! ( c ) of any M -algebra c is isomorphic to X ( c ) naturally with respect to morphisms in C M . Using thisdefinition, one arrives at a very simple and elegant formulation of the PBWtheorem. Note that using the natural transformation φ , we can regard N asa right M -module via the maps N ◦ M −−−→ N ◦ φ N ◦ N −−→ µ N N . Theorem 3.1.
Let φ : M → N be a natural transformation of monads. Thedatum ( M , N , φ ) has the PBW property if and only if the right M -moduleaction on N via φ is free.Proof. Let us first suppose that the datum ( M , N , φ ) has the PBW property,and let X be the corresponding endofunctor. Let us take an object d of C and consider the free M -algebra c = M ( d ); we shall now show that the directimage φ ! ( c ) is the free N -algebra N ( d ). To that end, we note that there isan obvious commutative diagram C N ❅❅❅❅❅❅❅❅ φ ∗ / / C M ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ C where the arrows to C are obvious forgetful functors from the categoriesof algebras. All the three functors in this commutative diagram are rightadjoint functors, so the corresponding diagram of the left adjoint functorsalso commutes, meaning that free M -algebras are sent under φ ! to free N -algebras: we have φ ! ( M ( d )) ∼ = N ( d ) naturally in d . Combining this resultwith the PBW property, we see that we have a natural isomorphism N ( d ) ∼ = φ ! ( M ( d )) ∼ = X ( M ( d )) = ( X ◦ M )( d ) , which shows that N ∼ = X ◦ M on the level of endofunctors. Finally, we notethat the pair of arrows N ( M ( M ( d ))) N ( φ (1 M ( d ) )) / / N ( γ M ( d ) ) N ( N ( M ( d ))) µ N (1 M ( d ) ) / / N ( M ( d )) . VLADIMIR DOTSENKO AND PEDRO TAMAROFF that defines φ ! ( M ( d )) as a coequaliser arises from evaluating the diagram N ◦ M ◦ M N ◦ φ ◦ M / / N ◦ γ M N ◦ N ◦ M µ N ◦ M / / N ◦ M . of right M -modules and their maps on the object c . This shows that the iso-morphism of endofunctors we obtained agrees with the right module action,and hence N is a free right M -module.The other way round, suppose that N is a free right M -module, so that N ∼ = X ◦ M for some endofunctor X . To prove that the datum ( M , N , φ ) hasthe PBW property, we shall utilize a very well known useful observation: inany split fork diagram c f ( ( g c t o o e + + d s l l where es = 1 d , f t = 1 c , and gt = se , d is the coequalizer of the pair f, g .The N -algebra φ ! ( c ) is the coequalizer of the reflexive pair N ( M ( c )) N ( φ (1 c )) / / N ( γ c ) N ( N ( c )) µ N (1 c ) / / N ( c ) , Note that the composition of the arrows N ( M ( c )) N ( φ (1 c )) −−−−−−→ N ( N ( c )) µ N (1 c ) −−−−→ N ( c ) is the definition of the right module action of M on N , so under theisomorphism of right modules N ∼ = X ◦ M , the above pair of arrows becomes X ( M ( M ( c ))) X ( µ M (1 c )) - - X ◦ M ( γ c ) X ( M ( c )) , Let us prove that φ ! ( c ) ∼ = X ( c ) by demonstrating that this pair of arrowscan be completed to a split fork with X ( c ) as the handle of the fork. Tothat end, we define the arrow e : X ( M ( c )) → X ( c ) to be 1 X ( γ c ), the arrow s : X ( c ) → X ( M ( c )) to be the composite X ( c ) ∼ = −→ X ( ( c )) X ( η M (1 c )) −−−−−−−→ X ( M ( c )) , and the arrow t : X ( M ( c )) → X ( M ( M ( c ))) to be the composite X ( M ( c )) ∼ = −→ X ( ( M ( c ))) X ( η M (1 M ( c ) )) −−−−−−−−−→ X ( M ( M ( c ))) , so the property es = 1 X ( M ( c )) follows from the unit axiom for the algebra c ,the property f t = 1 X ( M ( c )) follows from the unit axiom for the monad M ,and also se = gt by a direct inspection. This verification was natural in c with respect to morphisms in C M , so we have φ ! ( c ) ∼ = X ( c ) naturally in c ,and the datum ( M , N , φ ) has the PBW property. (cid:3) NDOFUNCTORS AND POINCAR´E–BIRKHOFF–WITT THEOREMS 9
Continuing with the toy example of endofunctors M A of Vect k , freeness of M B as as a right M A -module corresponds (at least for augmented algebras)to freeness of B as a right A -module. If we have a right A -module isomor-phism B ∼ = X ⊗ A , the underlying space of the induced module B ⊗ A M isisomorphic to X ⊗ M , and does not depend on the module structure on M .For instance, this is frequently used in representation theory to obtain anexplicit description for the underlying spaces of induced representations ofgroups and of Lie algebras; in the latter case freeness follows from the clas-sical PBW theorem. Our result offers another PBW-flavoured viewpoint forsuch an explicit description.4. Case of analytic endofunctors
Most interesting instances where our results have so far found applicationsdeal with the case where the endofunctors M and N are analytic [32], sothat the monads are in fact operads [44]. In this section, we shall mainlydiscuss the case C = Vect k , where k is a field of characteristic zero. Ingeneral, for analytic endofunctors to make sense and satisfy various familiarproperties, it is enough to require that the category C is symmetric monoidalcocomplete (including the hypothesis that the monoidal structure distributesover colimits). To state and prove a homological criterion for freeness likethe one in Section 4.1, one has to make some extra assumptions, e.g. assumethat the category of symmetric sequences C Σ is a concrete Abelian categorywhere epimorphisms split.4.1. Homological criterion of freeness.
We begin with setting up ourmain technical tool, a homological criterion of freeness of right modules. Itis well known that operadic right modules are generally easier to work withthan left modules, since the composite product of analytic endofunctors islinear in the first argument. In particular, one has the wealth of homologicalalgebra constructions that are applicable to the Abelian category of rightmodules, see [25] for details. Moreover, for connected weight graded operadsover a field of characteristic zero, one can define the notion of a minimal freeresolution of a weight graded module and prove its existence and uniquenessup to isomorphism, like it is done for modules over rings in the seminal paperof Eilenberg [24]. This leads to a homological criterion for freeness of a right M -module R .Recall that for an operad M , its left module L , and its right module R ,there is a two-sided bar construction B • ( R , M , L ). In somewhat concreteterms, it is spanned by rooted trees where for each tree the root vertexis decorated by an element of M , the internal vertices whose all childrenare leaves are decorated by elements of N , and other internal vertices aredecorated by elements of P ; the differential contracts edges of the tree anduses the operadic composition and the module action maps. For an operadwith unit, this bar construction is acyclic; moreover, for a connected weightgraded operad M the two-sided bar construction B • ( M , M , M ) is acylic. This leads to a free resolution of any right-module R as R ◦ M B • ( M , M , M ) ∼ = B • ( R , M , M ) . This resolution can be used to prove the following result.
Proposition 4.1.
Let M be a connected weight graded operad acting on Vect k , and let R be a weight graded right M -module. The right module R is free if and only if the positive degree homology of the bar construction B • ( R , M , ) vanishes; in the latter case, R is generated by H ( B • ( R , M , )) .Proof. This immediately follows from the existence and uniqueness of theminimal free right M -module resolution of R . (cid:3) This result is usually applied in one of the following ways. First, onecan define a filtration on R that is compatible with the right M -action, andprove freeness of the associated graded module, which then by a spectralsequence argument proves freeness of R . Second, one may apply the forgetfulfunctor from symmetric operads to shuffle operads, and prove freeness in theshuffle category; since the forgetful functor is monoidal and does not changethe underlying vector spaces, this guarantees vanishing of homology in thesymmetric category; this approach was introduced by the first author in [20].4.2. Aspects of the classical PBW theorem.
Let us first discuss howthe classical Poincar´e–Birkhoff–Witt theorem for Lie algebras fits in ourframework. For that, we consider the morphism of operads φ : Lie → Ass which is defined on generators by the formula [ a , a ] a · a − a · a . Case of a field of zero characteristic.
As a first step, let us outline a proofof (a version of) the classical PBW theorem (Poincar´e [50], Birkhoff [7],Witt [58]) over a field k of characteristic zero. Theorem 4.2.
Let L be a Lie algebra over a field k of characteristic zero.There is a vector space isomorphism U ( L ) ∼ = S ( L ) which is natural with respect to Lie algebra morphisms. Here S ( L ) , as usual,denotes the space of symmetric tensors in L .Proof. According to Theorem 3.1, it is sufficient to establish freeness of theassociative operad as a right
Lie -module. For that, one argues as follows.There is a filtration on the operad
Ass by powers of the two-sided idealgenerated by the Lie bracket a · a − a · a . The associated graded op-erad gr Ass is easily seen to be generated by two operations that togethersatisfy the defining relations of the operad
Poisson encoding Poisson al-gebras and, possibly, some other relations. It is well known that for theoperad
Poisson , we have
Poisson ∼ = Com ◦ Lie on the level of endofunctors,so it is a free right
Lie -module with generators
Com . By a straightfor-ward computation with generating functions of dimensions, this implies thatdim
Poisson ( n ) = n ! = dim Ass ( n ), and consequently there can be no other NDOFUNCTORS AND POINCAR´E–BIRKHOFF–WITT THEOREMS 11 relations. By a spectral sequence argument, it is enough to prove the homol-ogy vanishing required by Proposition 4.1 for the associated graded operad,so the
Lie -freeness of
Poisson implies the
Lie -freeness of
Ass , with the samegenerators
Com . Noting that
Com ( L ) = S ( L ) completes the proof. (cid:3) Non-functoriality of PBW in positive characteristic.
A useful feature of theexample of the morphism
Lie → Ass is that it highlights a slight differencebetween our approach and the one of [47]. It turns out that by talking aboutPBW pairs, one does not detect an important distinction between the caseof a field of characteristic zero and a field of positive characteristic; moreprecisely, the following result holds. (As the proof of Theorem 4.2 shows, inthe characteristic zero case, such issues do not arise, and the two approachesare essentially equivalent.)
Proposition 4.3.
Let the ground field k be of characteristic p > . Thenthe pair of operads ( Ass , Lie ) is a PBW-pair in the sense of [47] , so that S ( L ) = U (Ab L ) ∼ = gr U ( L ) for any Lie algebra L , but there is no way tochoose vector space isomorphisms S ( L ) ∼ = U ( L ) to be natural in L .Proof. The previous argument shows that gr
Ass ∼ = Poisson over any field k . This easily implies that the canonical surjection π is an isomorphism,establishing the PBW pair property. However, if we had vector space iso-morphisms S ( L ) ∼ = U ( L ) which are functorial in L , then by Theorem 3.1 wewould have Ass ∼ = Com ◦ Lie as analytic endofunctors, and as a consequencethe trivial submodule of
Ass ( n ) ∼ = k S n would split as a direct summand,which is false in positive characteristic. (cid:3) To have a better intuition about the second part of the proof, one maynote that the proof of equivalence of two definitions in [47] goes by sayingthat if we have a PBW pair of algebraic structures, then, first, the universalenveloping algebra of an Abelian algebra has a basis of monomials whichdoes not depend on a particular algebra, and then derive the same for anyalgebra using the PBW property. The latter step requires making arbitrarychoices of liftings that cannot be promoted to an endofunctor.4.3.
Enlarging the category of algebra objects.
Let us record a verysimple corollary of Theorem 3.1 for the case of operads.
Proposition 4.4.
Let φ : M → N be a morphism of augmented operads thatare analytic endofunctors of a category C . Assume that the datum ( M , N , φ ) has the PBW property, and let D be a category of which the category C is afull subcategory. Then the datum ( M , N , φ ) has the PBW property if M and N are regarded as analytic endofunctors of D .Proof. The only remark to make is that for a free module X ◦ M the spaceof generators X can be recovered as the quotient by the right action ofthe augmentation ideal, hence the space of generators is also an analyticendofunctor of C . An analytic endofunctor of C gives rise to an analyticendofunctor of D , and freeness for the enlarged category follows. (cid:3) As a first application of this result, the PBW property for the morphismof operads
Lie → Ass over a field k of characteristic 0 implies that thesame holds for associative algebras and Lie algebras in various symmet-ric monoidal categories that extend the category Vect k ; for example, thisimplies that the PBW theorem for Lie superalgebras (proved in [53] andre-discovered in [17]) and the PBW theorem for twisted Lie algebras [57] donot need to be proved separately, as already indicated by Bernstein’s proofof the PBW theorem [18] mentioned in the introduction.A slightly less obvious application for the same morphism of operads Lie → Ass is to the so called Leibniz algebras, the celebrated “noncommu-tative version of Lie algebras” [8]. Recall that a Leibniz algebra is a vectorspace with a bilinear operation [ − , − ] without any symmetries satisfying theidentity [ a , [ a , a ]] = [[ a , a ] , a ] − [[ a , a ] , a ]. For a Leibniz algebra L ,the space L spanned by all squares [ x, x ] is easily seen to be an ideal, andthe quotient L/L has a natural structure of a Lie algebra. Moreover, it isknown that the quotient map L → L/L is a Lie algebra in the symmetricmonoidal “category of linear maps” of Loday and Pirashvili [42]. In this cat-egory, by the classical PBW theorem, the underlying object of the universalenveloping algebra of L → L/L is isomorphic to S ( L → L/L ) ∼ = (cid:0) S ( L/L ) ⊗ L → S ( L/L ) (cid:1) . This gives a conceptual categorical explanation of appearance of the vectorspace S ( L/L ) ⊗ L in the context of universal enveloping algebras of Leibnizalgebras [43, Th. 2.9].4.4. The PBW non-theorem for Leibniz algebras.
A well known in-stance where the direct image functor φ ! can be computed explicitly butdepends on the algebra structure is the case of the morphism Leib → Diass from the aforementioned operad of Leibniz algebras to the symmetric operadof diassociative algebras. Here diassociative algebras refer to the algebraicstructure introduced by Loday [40] for the purpose of studying periodicityphenomena in algebraic K-theory; a diassociative algebra is a vector spacewith two bilinear operations ⊢ and ⊣ satisfying the identities( a ⊣ a ) ⊣ a = a ⊣ ( a ⊣ a ) , ( a ⊣ a ) ⊣ a = a ⊣ ( a ⊢ a ) , ( a ⊢ a ) ⊣ a = a ⊢ ( a ⊣ a ) , ( a ⊣ a ) ⊢ a = a ⊢ ( a ⊢ a ) , ( a ⊢ a ) ⊢ a = a ⊢ ( a ⊢ a ) . The morphism φ : Leib → Diass is defined by the formula φ ([ a , a ]) = a ⊣ a − a ⊢ a . In fact, this pair of operads and the morphism betweenthem come from the morphism Lie → Ass via a certain endofunctor of thecategory of operads, the tensor product with the operad usually denoted by
Perm , see [15]. It is known [27] that the universal enveloping diassociativealgebra of a Leibniz algebra L is, as a vector space, isomorphic to the tensorproduct S ( L/L ) ⊗ L mentioned above, and hence very much depends on theLeibniz algebra structure of L . (As we saw in Section 4.3, it happens because NDOFUNCTORS AND POINCAR´E–BIRKHOFF–WITT THEOREMS 13
Leibniz algebras give rise to Lie algebras in a larger category where
L/L isincluded as a part of the object.) It is natural to ask what exactly breaks inthis case if one attempts to mimic our proof of the classical PBW theorem.The associated graded operad of Diass with respect to the filtration definedby the Leibniz operation is easily seen to be generated by an operation a , a a · a satisfying the identities of the operad Perm and an operation a , a [ a , a ] satisfying the Leibniz identity; these operations are relatedby several identities including[ a , a · a ] = [ a , a ] · a +[ a , a ] · a and [ a · a , a ] = [ a , a ] · a − a · [ a , a ] . Expanding the operadic monomial [ a · a , a · a ] in two different ways, oneobtains the identity a · [ a , a ] · a + a · [ a , a ] · a = 0 , showing that the right Leib -module is not free, and that the obstruction tofreeness does indeed arise from the symmetric part of the Leibniz bracket(that vanishes on the Lie level). This identity can be lifted to a slightly lessappealing identity in the operad
Diass , which we do not include here.4.5.
Universal enveloping pre-Lie algebras of Lie algebras.
In a littleknown paper [54], a PBW type theorem is proved for universal envelopingpre-Lie algebras of Lie algebras. Let us explain how this result fits into ourformalism. We denote the operad encoding pre-Lie algebras by
PreLie . It iswell known that there exists a morphism of operads φ : Lie → PreLie definedby φ ([ a , a ]) = a · a − a · a . Proposition 4.5.
The datum ( Lie , PreLie , φ ) has the PBW property.Proof. We shall once again utilise the filtration argument, considering thefiltration of the operad
PreLie by powers of the two-sided ideal generated bythe Lie bracket. In [19], the associated graded operad was studied. Exam-ining the proof of the main result of [19], we see that the associated gradedoperad is free as a right
Lie -module, since that proof exhibits an explicit basisof tree monomials in the associated graded operad, and the shape of thosemonomials allows to apply an argument identical to that of [20, Th. 4(2)].A standard spectral sequence argument completes the proof. (cid:3)
It is interesting that
PreLie is also free as a left
Lie -module, which wasused in [14] to establish that for a free pre-Lie algebra L , the result of changeof algebra structure φ ∗ ( L ) is free as a Lie algebra.4.6. A new PBW theorem: solution to an open problem of Loday.
We conclude this paper with new PBW type result answering a questionthat Jean-Louis Loday asked the first author around 2009. Namely, theoperad
Dend of dendriform algebras admits a morphism from the operad
PreLie , which we shall recall below. It has been an open problem to prove aPBW-type for dendriform universal enveloping algebras of pre-Lie algebras,which we do in this section. Since this paper was produced, an alternative proof (however without functoriality) was obtained by Gubarev [28]. Inthe same paper [28], some PBW-type results involving post-Lie algebrasare proved; their functorial versions are obtained, using rewriting theory forshuffle operads, in a separate note by the first author [21].Recall that the dendriform operad
Dend is the operad with two binarygenerators denoted by ≺ and ≻ that satisfy the identities( a ≺ a ) ≺ a = a ≺ ( a ≺ a + a ≻ a ) , ( a ≻ a ) ≺ a = a ≻ ( a ≺ a ) , ( a ≺ a + a ≻ a ) ≻ a = a ≻ ( a ≻ a ) . In this section, we shall consider a different presentation of the operad
Dend via the operations a ◦ a = a ≺ a − a ≻ a and a · a = a ≺ a + a ≻ a . By a direct computation, all relations between these operations are conse-quences of the identities( a ◦ a ) ◦ a − a ◦ ( a ◦ a ) = ( a ◦ a ) ◦ a − a ◦ ( a ◦ a ) , ( a · a ) · a = a · ( a · a ) + a · ( a · a ) − ( a ◦ a ) ◦ a , ( a · a ) ◦ a = ( a ◦ a ) · a − a · ( a ◦ a ) + a · ( a ◦ a ) , ( a ◦ a ) · a + ( a ◦ a ) · a = a ◦ ( a · a ) + a ◦ ( a · a ) . In particular, this implies the well known statement that the operation a ◦ a = a ≺ a − a ≻ a satisfies the pre-Lie identity, so that there isa morphism φ : PreLie → Dend sending the generator of
PreLie to a ≺ a − a ≻ a . We can now state the promised new PBW theorem. Theorem 4.6.
The datum ( PreLie , Dend , φ ) has the PBW property.Proof. The proof of this theorem utilises the operad
PrePoisson controllingpre-Poisson algebras [2] which we shall recall below. Since by the operad
PreLie we always mean the operad controlling the right pre-Lie algebras, weshall work with right pre-Poisson algebras (opposite of those in [2]).For the first step of the proof, we consider the filtration F • Dend of theoperad
Dend by powers of the two-sided ideal generated by the operation a ◦ a . In the associated graded operad, the relations determined abovebecome( a ◦ a ) ◦ a − a ◦ ( a ◦ a ) = ( a ◦ a ) ◦ a − a ◦ ( a ◦ a ) , ( a · a ) · a = a · ( a · a ) + a · ( a · a ) , ( a · a ) ◦ a = ( a ◦ a ) · a − a · ( a ◦ a ) + a · ( a ◦ a ) , ( a ◦ a ) · a + ( a ◦ a ) · a = a ◦ ( a · a ) + a ◦ ( a · a ) . These are precisely the defining relations of the operad controlling rightpre-Poisson algebras. Thus, the associated graded operad gr F Dend admitsa surjective map from the operad
PrePoisson ; this result is in agreement
NDOFUNCTORS AND POINCAR´E–BIRKHOFF–WITT THEOREMS 15 with [2, Sec. 4] where it is shown that for a filtered dendriform algebrawhose associated graded algebra is a Zinbiel algebra, that associated gradedacquires a canonical pre-Poisson structure.We shall now look at the shuffle operad
PrePoisson f associated to theoperad PrePoisson via the usual forgetful functor [11, 23]. It is generated byfour elements · , ◦ , ¯ · , ¯ ◦ which are the two operations and their opposites. Weconsider the ordering which is the superposition of the quantum monomialordering [22, Sec. 2] for which every degree two monomial with · or ¯ · atthe root and ◦ or ¯ ◦ at the non-root vertex is smaller than every degree twomonomial with ◦ or ¯ ◦ at the root and · or ¯ · at the non-root vertex, andthe path-lexicographic ordering induced by the ordering · < ¯ · < ¯ ◦ < ◦ . Aslightly tedious computation shows for this choice of ordering this operadhas a quadratic Gr¨obner basis; moreover, we have dim PrePoisson f (4) = 336.The surjection mentioned above leads to a surjection of vector spaces PrePoisson f (4) ։ Dend f (4) , and if we note that dim Dend f (4) = 4! ·
14 = 336, we conclude that thissurjection must be an isomorphism. In particular, when we pass from theoperad
Dend to its associated graded, no new cubic relations arise in theassociated graded case (our operads are generated by binary operations, socubic elements live in arity 4). Repeating mutatis mutandis the argument of[51, Th. 7.1], we see that the operad gr F Dend is quadratic, and
PrePoisson ∼ =gr F Dend .By direct inspection of our Gr¨obner basis of the operad
PrePoisson f , [20,Th. 4(2)] applies, showing that this operad is free as a right PreLie f -module.By Proposition 4.1 and a spectral sequence argument, the same is true forthe operad Dend . (cid:3) Corollary 4.7.
The operad of pre-Poisson algebras is Koszul.Proof.
This follows immediately from the fact that the associated shuffleoperad has a quadratic Gr¨obner basis. (cid:3)
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Institut de Recherche Math´ematique Avanc´ee, UMR 7501, Universit´e deStrasbourg et CNRS, 7 rue Ren´e-Descartes, 67000 Strasbourg, France
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