Derived Poincaré-Birkhoff-Witt theorems (with an appendix by Vladimir Dotsenko)
aa r X i v : . [ m a t h . K T ] J un D E R I V E D P O I N C A R É – B
I R K H O FF – W
I T TT H E O R E M S space with an appendix by Vladimir Dotsenko A NTON K HOROSHKIN AND P EDRO T AMAROFF
Abstract
We propose a new general formalism that allows us to study Poincaré–Birkhoff–Witt type phenomena for universal enveloping algebras in the differential gradedcontext. Using it, we prove a homotopy invariant version of the classical Poincaré–Birkhoff–Witt theorem for universal envelopes of Lie algebras. In particular, ourresults imply that all the previously known constructions of universal envelopesof L ∞ -algebras (due to Baranovsky, Lada and Markl, and Moreno-Fernández) rep-resent the same object of the homotopy category of differential graded associ-ative algebras. We also extend Quillen’s classical quasi-isomorphism C −! BU from differential graded Lie algebras to L ∞ -algebras; this confirms a conjecture ofMoreno-Fernández. Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. The derived PBW property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Lie ∞ −! Ass ∞ — 2.2. Quillen the-orem for L ∞ -algebras — 2.3. The universal envelope as a functor on the homotopycategory — 2.4. Cohomology groups — 2.5. Derived PBW theorems for associativeenvelopes
3. Further directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A. Models of operads via homological perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 18 D ERIVED
PBW
THEOREMS
Introduction
It is a classical result going back to Poincaré, Birkhoff and Witt that, over a field ofcharacteristic zero, the universal enveloping algebra U ( g ) of a Lie algebra g is iso-morphic, as a vector space, to the symmetric algebra S ( g ) on g . One can, in fact,extend without changes the definition of the functor U to the category of dg Lie al-gebras, and a consequence of the PBW theorem is that for any dg Lie algebra we havea natural isomorphism U H ( g ) −! HU ( g ). This result implies that the underlying ho-mology group of the universal envelope of a dg Lie algebra does not depend on theLie algebra structure of g but only on the homology group H ( g ), and that this functordescends to the homotopy category of dg Lie algebras. This category can be modeledthrough minimal L ∞ -algebras and their morphisms up to L ∞ -quasi-isomorphismand, similarly, the homotopy category of dga algebras can be modeled through min-imal A ∞ -algebras and their morphisms up to A ∞ -quasi-isomorphism, so it is reas-onable to consider the problem of finding a functor at the level of L ∞ -algebras rep-resenting U on the homotopy category.This program has been carried out by V. Baranosvky [2] and later by J. Moreno-Fernandez [24], and their results imply that to every minimal L ∞ -algebra one canassign a “universal enveloping” minimal A ∞ -algebra Υ ( g ) that enjoys many proper-ties similar to those of the classical universal enveloping algebra functor.its underlying vector space is the symmetric algebra S ( g ), independently of the L ∞ -structure of g ,there exists a Quillen type A ∞ -quasi-isomorphism Ω C ( g ) −! Υ ( g ) (so the al-gebra Υ ( g ) does have the correct homotopy type),the canonical inclusion g −! Υ ( g ) is a strict L ∞ -morphism: the antisymmet-rized higher multiplication maps on Υ ( g ) restrict to the higher brackets of g .One of the goals of this paper is to explain, using the methods of V. Dotsenko andsecond author, introduced in [11] and suitably extended to the dg setting, that theresults of [2] and [24] both follow from a derived version of the classical Poincaré-Birkhoff-Witt theorem for Lie algebras. To accomplish this, we study, through the the-ory of operads, the Lada–Markl functor that assigns to an A ∞ -algebra the L ∞ -algebraobtained by antisymmetrising all the product operations. As in the non-dg case, thisfunctor has a left adjoint, and we prove it satisfies a derived Poincaré–Birkhoff–Witttheorem . Our main result implies the following; here U ( g ) denotes the left adjointabove, while U H ( g ) is the universal envelope of the Lie algebra H ( g ): . K HOROSHKIN AND
P. T
AMAROFF Theorem.
For every minimal L ∞ -algebra g :(1) there is an isomorphism between the symmetric algebra S ( g ) and the homologyHU ( g ) of the A ∞ -universal envelope U ( g ) . This isomorphism is natural with re-spect to strict L ∞ -morphisms.(2) S ( g ) can be endowed with a minimal A ∞ -algebra structure A ∞ -quasi-isomorphicto U ( g ) so that the inclusion g −! S ( g ) is a strict map of L ∞ -algebras. Î From this theorem, we deduce the three proposed models for universal envelopesof L ∞ -algebras are, up to homotopy, one and the same. This means, in particular,that the two existing tentative models for universal envelopes are in fact models. Thisfollows from the following: Corollary.
Let g be a minimal L ∞ -algebra. Every minimal A ∞ -algebra structure onS ( g ) for which the restriction of the antisymmetrized structure operations coincide withthe operations of g is A ∞ -quasi-isomorphic to U ( g ) . Thus, the models of Baranovskyand Moreno-Fernandez are A ∞ -isomorphic, and A ∞ -quasi-isomorphic to U ( g ) . Î For a final application, consider an L ∞ -algebra g . Then C ( g ) is a commutative dgccoalgebra, where C is the Quillen construction on g . Since U ( g ) is an A ∞ -algebra, itis natural to compare C ( g ) to BU ( g ), which is a (non-commutative) dgc coalgebra.This result was conjectured in [24]. Corollary.
There is an acyclic cofibration C ( g ) −! BU ( g ) of dgc coalgebras naturalwith respect to strict L ∞ -morphisms. Moreover, for any minimal L ∞ -algebra g , thereare acyclic cofibrations C ( g ) −! B S ( g ) . In particular, BU ( g ) always admits a com-mutative model. Î Structure.
The paper is organised as follows. In Section (1) we prove our maintheorem relating almost-free and derived PBW morphisms between dg operads. Wethen use it in Section (2) to show that the map from the homotopy Lie operad to thehomotopy associative operad is derived PBW and deduce from this several resultson universal envelopes of L ∞ -algebras, which recover results from Baranovsky andMoreno-Fernandez, and extend the Quillen quasi-isomorphism C −! BU to L ∞ -algebras. We also apply our main theorem to show associative universal envelopessatisfy the derived PBW property as soon as they are PBW. The appendix, written byV. Dotsenko, contains a general result on models of operads obtained by homologicalperturbation which we use in the particular case of the homotopy associative and thehomotopy Poisson operad. D ERIVED
PBW
THEOREMS
Notation and conventions.
We work over a field of characteristic zero, which wewrite k . We assume the reader is familiar with the theory of algebraic operads, aspresented, for example, in [23], with the elements of model theory, as presented, forexample, in [19], and with the basic tools of homological algebra. Whenever a newdefinition is provided, it will appear in boldfaced italics . Acknowledgements.
This paper saw great progress while the second author wasvisiting the Higher School of Economics in Moscow. We thank the Higher School ofEconomics for their hospitality and wonderful working conditions. We thank Vladi-mir Dotsenko for his constant support and valuable advice during the preparation ofthese notes, for his insight into PBW theorems for operads that motivated this sequelto the joint work in [11], and for his careful reading of the manuscript.We thank Alexander Efimov for useful discussions that encouraged us to write Sec-tion 2.3 and Ricardo Campos, Daniel Robert-Nicoud and Luis Scoccola for their use-ful comments and suggestions. We also thank Ben Knudsen for answering somequestions about his work [16] on enveloping E n -algebras of spectral Lie algebras, andGuillermo Tochi for pointing us to this paper in the first place.The research of A. Kh. was carried out within the HSE University Basic ResearchProgram and supported in part by the Russian Academic Excellence Project ’5-100’and in part by the Simons Foundation. . K HOROSHKIN AND
P. T
AMAROFF For convenience, we remind the reader of the language of [11]. Let us fix a symmet-ric monoidal category C that can be either of Vect k , the category of graded vectorspaces, Ch k , the category of complexes over k , Σ Mod , the category of Σ -modules un-der the Cauchy tensor product or Ch Σ , the category of dg Σ -modules under the sameproduct. We say a map of operads f : P −! Q over C satisfies the Poincaré-Birkhoff-Witt property if there is an endofunctor T : C −! C so that universal enveloping al-gebra functor f ! is naturally isomorphic to T with respect to P -algebra maps. In otherwords, we demand that f ! depend only on the underlying object of that algebra in C ,but not on the operations of that particular P -algebra. With this at hand, the mainresult of [11] is the following (we refer the reader to [17] for details on operads andtheir modules): Theorem.
The map f : P −! Q satisfies the Poincaré-Birkhoff-Witt property if andonly if Q is a free right P -module. Moreover, in this case, if T is a basis for Q as a rightP -module, then f ! is isomorphic to T , naturally with respect to P -algebra maps. Î The new formalism we propose for dg operads is as follows. Fix a dg operad P . Wesay a right dg P -module X is almost-free if it admits a bounded below exhaustivefiltration so that its associated graded module is chain equivalent to a free right P -module on a basis of cycles. A morphism of dg operads f : P −! Q is almost-free if Q is an almost-free right P -module. Note that for every dg module X we have acorresponding right H P -module
H X , where H is the homology functor. It followsthat we have both a left adjoint f ! to the restriction functor f ∗ : Q Alg −! P Alg and aleft adjoint (
H f ) ! to the restriction functor ( H f ) ∗ : HQ Alg −! H P
Alg . In this way, weobtain two functors H ◦ f ! : P Alg −! HQ Alg , (
H f ) ! ◦ H : P Alg −! HQ Alg and a natural transformation F : ( H f ) ! ◦ H −! H ◦ f ! . The map f is derived Poincaré–Birkhoff–Witt if F is a natural isomorphism. Our main result is the following: Theorem 1.1.
Every morphism that is almost-free is derived PBW. Moreover, if T isa basis of cycles for such a morphism, the homology H f ! of its universal envelope isnaturally isomorphic to T H as a functor of algebras on its domain to algebras over thehomology of its codomain. D ERIVED
PBW
THEOREMS
Thus, in the same way that a classical PBW theorem gives an amenable descriptionof universal enveloping algebra dependent only on the underlying object of the input,a derived PBW theorem gives us an amenable description of the homology of theuniversal enveloping algebra dependent only on the homology of the input.
Proof.
We split our proof into three steps. Our main tool is a classical spectral se-quence argument.
Step 1.
Suppose that Q is in fact P -free on a basis of cycles, so that Q = T ◦ P with d T =
0. Since T has trivial differential, the Künneth theorem for the circleproduct gives a natural isomorphism HQ −! T ◦ H P , and shows that HQ is a freeright H P -module. Moreover, for every left P -module X , we have natural isomorph-isms H ( f ! ( X )) = H ( Q ◦ P X ) −! H ( T ◦ X ) −! T ◦ H X . We also have natural isomorph-isms T ◦ H X −! T ◦ H P ◦ H P X −! HQ ◦ H P
H X = ( H f ) ! ( H X ), which gives what wewanted: the natural map F X : ( H f ) ! ( H X ) −! H ( f ! ( X )) is a natural isomorphism, sothat f is derived PBW in this case. Step 2.
Let us consider now the situation where we have an almost-free filtration F on Q , and consider the induced filtration on f ! ( X ). Linearity on the left of the com-posite product gives us that gr F ( f ! ( X )) is of the form T ◦ X where T has trivial differen-tial, so that the domain of the E -page of the maps of spectral sequences convergingto F X : ( H f ) ! ( H ( X )) −! H ( f ! ( X )) looks like T ◦ H P . The arguments above now showthat the induced morphism at the E -page is an isomorphism. Step 3.
To conclude, let us suppose that we have a filtration on Q such that gr ( Q )is chain equivalent to a free right P -module Q ′ . Arguing as before, we have a mapof spectral sequences converging to F X : ( H f ! )( H X ) −! H ( f ! ( X )) with E equal to( gr Q ) ◦ P X , and a map ( gr Q ) ◦ P T −! Q ′ ◦ P X = T ◦ X . Since the circle product isleft linear, this map is still a chain equivalence, and thus induces an isomorphism onhomology.The last claim is already part of the content of the main result in [11]. This concludesthe proof of the theorem. Î We point out that the work of V. Hinich [12, Section 4.6.3] proves that universal en-velopes preserve acyclic cofibrations between algebras: a derived PBW theorem ex-tends this to arbitrary weak equivalences.
Corollary 1.1.
If f : P −! Q is derived PBW, then f ! preserves weak equivalences. Î . K HOROSHKIN AND
P. T
AMAROFF It is worth pointing out that the PBW theorem of V. Dotsenko and the second authorin [11] already implies the following classical result, since there we show that the mapof operads
H f : Lie −! Ass is free and thus PBW in the classical sense.
Corollary.
Let g be a dg Lie algebra. Then the natural map U H ( g ) −! HU ( g ) is anisomorphism of algebras, so that a map of dg Lie algebras is a quasi-isomorphism ifand only if the map on universal envelopes is one. Î Relation to a theorem of Adams.
It is interesting to point out that in [1] (see also [4,Chapter 9]), the author shows that if f : Λ −! Γ is an inclusion of Hopf algebrasand Λ is central in Γ , then B Γ admits a filtration whose graded coalgebra is chainequivalent to ( B Λ ⊗ B Ω , d ⊗ B f is almost-free according to our definition.In fact, this result was the main inspiration for our definition of almost-free morph-isms. We remark that in the context of differential graded homological algebra, theobjects which posses useful homological properties and replace, in a way, the freeobjects of the classical theory, are sometimes called “semi-free modules”. We chosenot to use this terminology to avoid any kind of confusion. It is useful to note that onecan just assume that Γ is Λ -free along with the fact that Λ is central to deduce this.Hence, Adams’ result states precisely that, under this last extra hypothesis, B pre-serves almost-free maps. Having appropriate hypotheses and a similar result for Ω ,one would obtain results relating classical PBW maps of operads, as defined in [11],to derived PBW maps between cofibrant replacements. We intend to pursue theseideas in the future. In this section we prove the derived version of the classical PBW theorem, whichwe then use to deduce results of V. Baranosvky and J. M. Moreno-Fernández, bothwho constructed universal envelopes for L ∞ -algebras, and answer in the positive aconjecture of Moreno-Fernández regarding the universal envelope construction con-sidered by Lada–Markl in [21]. Observe that the motto of [11] that “the universal en-velope of g is independent of the Lie algebra structure of g ” is now replaced by “thehomology of the universal envelope of g is independent of the homotopy Lie algebrastructure of g ”. Since all three constructions have their particular intricacies, let usbegin by recalling the essential definitions of the respective universal envelopes for L ∞ -algebras. D ERIVED
PBW
THEOREMS
The universal envelope as a left adjoint.
Let us recall the work of Lada–Markl [21]that generalizes the well-known fact the antisymmetrization of the product of an as-sociative algebra yields a Lie algebra. In [21], the authors show that if A is an A ∞ -algebra with higher products ( m , m , m , . . .) and if we set, for each n ∈ N , l n = X σ ∈ S n ( − σ m n · σ (1)these maps define on A an L ∞ -algebra structure. This implies there is a map of op-erads f : Lie ∞ −! Ass ∞ defined by ((1)). As in the classical case, it makes sense todefine the universal envelope of an L ∞ -algebra through the left adjoint f ! of the mapthat assigns an A ∞ -algebra A to the corresponding L ∞ -algebra f ∗ ( A ), which we write A ◦ , following the prescription above: this is the quotient of the free A ∞ -algebra on g by the relations imposed by the equation ((1)). We will write U ( g ) for f ! ( g ) and call it the universal enveloping algebra of g ; since this universal algebra is given unequi-vocally by the same formalism that defines the classical universal envelope of dg Liealgebras, we refrain from giving it any other name. Note that by construction there isa unit map g −! U ( g ) ◦ which is a strict morphism of L ∞ -algebras. The construction of Baranosvky.
Let us recall that if g is a dg Lie algebra, thereis a natural quasi-isomorphism of dga algebras q : Ω C ( g ) −! U ( g ) which assigns agenerator of the Chevalley–Eilenberg complex C ( g ) to its antisymmetrization in U ( g ).Baranosvky [2] shows that there is a contraction of complexes Ω C ( g ′ ) −! S ( g ′ ) where g ′ is the abelian algebra associated to g . To define U ( g ) for the more general class of L ∞ -algebras, Baranovsky resorts to a perturbative method, as follows. For such analgebra g , the key ingredients of his construction are:the contraction of complexes Ω C ( g ′ ) −! S ( g ′ ),the resulting contraction of dgc coalgebras B Ω C ( g ′ ) −! B S ( g ′ ).the fact the differential of B Ω C ( g ) is a perturbation of the differential of B Ω C ( g ′ ).This two facts imply, together with the homological perturbation lemma, that thereis on B S ( g ′ ) a dgc coalgebra structure that is quasi-isomorphic to B Ω C ( g ). In otherwords, there is on S ( g ′ ) an A ∞ -algebra structure that is A ∞ -quasi-isomorphic to Ω C ( g ).Moreover, Baranosvky shows the PBW inclusion g −! S ( g ) ◦ is a strict map of L ∞ -algebras. We call this the Baranosvky universal enveloping algebra of g . Note that Ω C ( g ) has the same homotopy type as U ( g ), so it makes sense to focus on this objectto elucidate a universal enveloping algebra, since Ω C ( g ) exists for any L ∞ -algebra. . K HOROSHKIN AND
P. T
AMAROFF The construction of Moreno-Fernández.
The approach of Moreno-Fernández [24]is slightly different from that of Baranovsky, but is also perturbative in nature. Let ustake a dg Lie algebra g , and assume we have a contraction of complexes from g onto H ( g ). The homotopy transfer theorem then guarantees there is an L ∞ -structure on H ( g ) which makes H ( g ) an L ∞ -algebra L ∞ -quasi-isomorphic to g . The author thenshows there there is an explicit contraction of complexes U ( g ) −! S ( H ( g )) which, bythe homotopy transfer theorem, gives us an A ∞ -algebra structure on S ( H ( g )) whichis A ∞ -quasi-isomorphic to U ( g ). Moreover, one can arrange it so that the inclusion H ( g ) −! S ( H ( g )) ◦ is a strict map of L ∞ -algebras.We have not explained yet how to define universal envelopes of L ∞ -algebras, how-ever; we do it only for minimal algebras. In this case, we can take the dg Lie algebra L C ( g ) that comes equipped with an L ∞ -quasi-isomorphism L C ( g ) −! g —this isthe so-called rectification theorem, obtained by the bar-cobar construction— so wecan proceed with the prescription of Moreno-Fernández to define on S ( g ) an A ∞ -structure so that the PBW inclusion g −! S ( g ) ◦ is a strict map of L ∞ -algebras. We callthis the Moreno-Fernández universal enveloping algebra of g . Lie ∞ −! Ass ∞ Theorem 2.1.
The morphism f : Lie ∞ −! Ass ∞ is almost-free, so it is derived PBW.Proof. We begin with recalling from [23, Prop. 9.1.5] that the Loday–Livernet present-ation of the associative operad is given by a commutative non-associative product x x and an anti-symmetric Lie bracket [ x , x ] that is a derivation of the product andsatisfies the identity( x x ) x − x ( x x ) = [ x , [ x , x ]].Let us consider the weight grading on the space of generators which assigns weightzero to the Lie bracket and weight one to the product. The associated graded relationswith respect to this filtration are the relations of the Poisson operad, and the under-lying Σ -modules of Pois and
Ass are isomorphic. We are therefore in the situationwhere result of Appendix applies: there exists a quasi-free resolution of
Ass whosedifferential is obtained from d Pois ∞ by a perturbation that lowers the weight grading.Moreover, the space of generators of this resolution can be identified with the Koszuldual cooperad of Pois whose underlying Σ -module is isomorphic to that of the Koszul D ERIVED
PBW
THEOREMS dual cooperad of
Ass ; therefore, this resolution has to be minimal and isomorphic to
Ass ∞ . The dg suboperad Lie ∞ is in weight filtration zero, so the weight filtration is afiltration of right Lie ∞ -modules whose associated graded operad is Pois ∞ .To complete the proof, we will show that there is a chain homotopy equivalenceof right Lie ∞ -modules π : Pois ∞ −! Com ◦ Lie ∞ . For this, we use the language ofdistributive laws between operads [23, Sec. 8.6.3]. The distributive law λ that givesrise to the isomorphism Pois = Com ∨ λ Lie can be enhanced to a distributive law λ ′ between the operads Com and
Lie ∞ , for which all higher brackets are derivations withrespect to the commutative product; there is a surjective quasi-isomorphism Com ∨ λ ′ Lie ∞ ! Com ∨ λ Lie = Pois . Hence, we get a surjective quasi-isomorphism
Pois ∞ −! Com ◦ Lie ∞ = Com ∨ λ ′ Lie ∞ .Since we are working over a field of characteristic zero, we can produce a map i : Com −! Com ∞ such that pi =
1, where p is the projection onto homology. Thisthen gives us a map j : Com ◦ Lie ∞ −! Pois ∞ by composing with the composition of Pois ∞ , and this map is a section of π . Since Pois ∞ is free as an operad, and since we’reworking over a field of characteristic zero, we can produce an equivariant contractinghomotopy h for j π . This completes the proof that Lie ∞ −! Ass ∞ is almost-free and,by Theorem 1.1, it is derived PBW. Î The algebras over the operad
Com ∨ λ ′ Lie ∞ used in the proof are sometimes called homotopy Poisson algebras or P ∞ -algebras in the literature, even though this operadis not cofibrant. These have been considered by A. S. Cattaneo and G. Felder, andindependently by T. Voronov, and are related to the theory of Lie and Courant al-gebroids, and Poisson manifolds, see [7, 13, 15] for example. L ∞ -algebras We recall that if g is an L ∞ -algebra, the bar construction on g is the commutative dgccoalgebra C ( g ) with underlying coalgebra S c ( s g ), the free commutative coalgebra onthe suspension of g , and with differential d : C ( g ) −! C ( g ) induced from the higherbrackets of g ; the higher Jacobi identities for these higher brackets are equivalent tothe single equation d =
0. We write x ∧· · ·∧ x t a generic element from C ( g ), omittingthe suspensions signs for ease of notation. Observe this element is simply the anti-symmetrization of the corresponding elementary tensor s x ⊗ · · · ⊗ s x t in T c ( s g ). . K HOROSHKIN AND
P. T
AMAROFF Similarly, if A is an A ∞ -algebra, the bar construction on A is the dgc coalgebra B A with underlying coalgebra T c ( s A ), the free coalgebra on the suspension of A , andwith differential d : B A −! B A induced from the higher products of A ; the Stasheffidentitis for these higher products are equivalnt to the single equation d . We writea [ x | · · · | x t ] a generic element of B A . Observe that we can apply both constructions,in particular, to dg Lie algebras and dga algebras, of course. Finally, if C is a dgccoalgebra, the cobar constuction on C is the dga algebra Ω C with underlying algebra T ( s − C ), the free algebra on the desuspension of C , with differential induced fromthe comultiplication and differential of C . Concretely, it is the unique derivation of Ω C that extends the map s − C −! Ω C such that d ( s − c ) = s − d c − s − ⊗ s − ∆ c .Let us recall from [14] that for a dg-Lie algebra ( g , d ) there is a quasi-isomorphism q : C ( g ) −! BU ( g ) where the left hand side is the Quillen construction on g (thatcoincides with the Chevalley-Eilenberg complex of g ) and the right hand side is theassociative bar construction on the universal envelope of g . We reminder the readerthat q is determined uniquely by a map τ : C ( g ) −! U ( g ), which we call the twistingcochain associated to q . That this be a twisting cochain is equivalent to the Maurer–Cartan equation d τ + τ ⋆ τ =
0. Here, the star product of the convolution dga algebra A = hom( C ( g ), U ( g )) is defined by ⋆ = µ ( − ⊗ − ) ∆ .We also remind the reader that τ simply sends a generator s g ∈ s g to the class ofits desuspension in U ( g ). The following theorem extends this picture to the case of L ∞ -algebras —we will observe below τ also defines a twisting morphism in this moregeneral setting. In this case, the Maurer–Cartan equation is replaced by a higher ana-log: since U ( g ) is now an A ∞ -algebra, the convolution algebra above is in fact againan A ∞ -algebra, so that for each t ∈ N and each f , . . . , f t ∈ A , we have higher multi-plications defined by m t ( f , · · · , f t ) = µ t ( f ⊗· · ·⊗ f t ) ∆ ( t ) . The Maurer–Cartan equationincorporates these higher products and now reads: d τ + X t Ê µ t ( τ , . . . , τ ) = C ( g ) −! BU ( g ) is one ofdgc coalgebras, and that this assignment defines a bijection between dgc coalgebramaps C ( g ) −! BU ( g ) and twisting cochains C ( g ) −! U ( g ). For details, we refer thereader to the book [23]. D ERIVED
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THEOREMS
Before stating the result, we recall that the category of dgc coalgebras
Cog is a modelcategory where the cofibrations are the degree-wise monomorphisms, the weak equi-valences are the quasi-isomorphisms and cofibrations satisfy the right lifting prop-erty with respect to acyclic fibrations; all objects are cofibrant, and the fibrant dgccoalgebras are the quasi-free ones, see [25]. A map of coalgebras is a weak equival-ence if and only if its image under the cobar construction is a quasi-isomorphism ofdga algebras. This class is strictly contained in the class of quasi-isomorphisms.
Theorem 2.2.
For any L ∞ -algebra g the map q : C ( g ) −! BU ( g ) corresponding tothe twisting cochain C ( g ) −! U ( g ) that assigns a generator s g ∈ C ( g ) the class of itsdesuspension is an acyclic cofibration of coalgebras. If g is minimal, then there areacyclic cofibrations of the form C ( g ) −! B S ( g ) .Proof. Let us begin by observing that the Maurer–Cartan equation for τ is simply arestatement that the higher products in U ( g ) antisymmetrize to the higher bracketsof g . Indeed, for a generator x = x ∧ · · · ∧ x t in the domain, we have that τ d x is equalto the higher bracket [ x , · · · , x t ], up to signs, while the only non-zero term involvinghigher products of τ involves ∆ ( t ) ( x ). This is just the signed sum over σ ∈ S t of σ x , andthen m t ( τ , . . . , τ )( x ) is precisely the antisymmetrized higher product of x ⊗ · · · ⊗ x t .To see that q is a weak equivalence one can show that Ω C ( g ) −! S ( g ) is one. Todo this, one can argue as in [2, Theorem 3], or note that there is a morphism of A ∞ -algebras BU ( g ) −! B S ( g ), where the right hand side is Baranovsky’s universal en-velope, that is a quasi-isomorphism. Indeed, the map H ( U ( g )) −! S ( g ) is an auto-morphism of the enveloping associative algebra of ( g , l ). This implies that Ω q is aquasi-isomorphism, since ε U ( g ) : Ω BU ( g ) −! U ( g ) is one, and Ω C ( g ) −! U ( g ) is aquasi-isomorphism that factors as ε U ( g ) Ω q . Î We write A ¡ for the homology of B A and, for an L ∞ -algebra g , we write g ¡ for thehomology of C ( g ). An immediate corollary of the previous theorem is the following;all claims follow from the general theory of twisting cochains; see [2, Theorem 2]. Corollary 2.1.
Let g be a minimal L ∞ -algebra. Then(1) the twisted complex C ( g ) ⊗ τ S ( g ) is quasi-isomorphic to k [0] ,(2) we have an isomorphism g ¡ −! S ( g ) ! of minimal A ∞ -coalgebras,(3) the categories of g -modules and U ( g ) -modules are equivalent and,(4) the functors ? ⊗ τ S ( g ) : D ( C ( g )) ⇆ D ( S ( g )) : ? ⊗ τ C ( g ) are mutuallyinverse derived equivalences. Î . K HOROSHKIN AND
P. T
AMAROFF Suppose that f : C ( g ) −! C ( g ′ ) is an L ∞ -morphism. The coalgebra BU ( g ) is fibrantand the Quillen map of g ′ is an acyclic cofibration, so we can produce a lift U ( f ) : BU ( g ) −! BU ( g ′ ). The following lemma implies this assignment has good homotop-ical properties; in particular, it is well defined up to homotopy. Lemma 2.1.
Let ϕ , ϕ ′ : BU ( g ) −! BU ( g ′ ) be such that ϕ q = ϕ ′ q = q ′ f . Then ϕ ≃ ϕ ′ asmaps of dgc coalgebras. In particular, for a second map g : C ( g ′ ) −! C ( g ′′ ) , we havethat U ( g ) U ( f ) ≃ U ( g f ) for any choice of lifts of f , g and g f .Proof. The reader can consult [19, Sections 1.1-1.2] for details on the elements ofmodel categories used in this proof. We remind the reader
Cog is the model categoryof coalgebras where the cofibrations are the injections and the weak equivalences arecreated by the cobar functor.The map q is a weak equivalence between cofibrant objects and BU ( g ′ ) is fibrant. Itfollows from Lemma 1.1.12 that we have an induced isomorphism q ∗ : Cog ( BU ( g ), BU ( g ′ ))/ ≃ r −! Cog ( C ( g ), BU ( g ′ ))/ ≃ r Since all objects in
Cog are cofibrant and B ( − ) has image in fibrant objects, we de-duce from Proposition 1.2.5 (v) that we can replace the right homotopy relation ≃ r unambiguously by the homotopy relation ≃ , and finally Theorem 1.2.10 (ii) impliesthat this isomorphism identifies naturally with an isomorphism q ∗ : [ BU ( g ), BU ( g ′ )] −! [ C ( g ), BU ( g ′ )]which implies that if q ∗ ( ϕ ) = q ∗ ( ϕ ′ ) then these two maps must be homotopic. Î With this at hand we have the following result.
Theorem 2.3.
The universal envelope preserves weak equivalences of L ∞ -algebras, soit descends to a functor on the homotopy category. Moreover, if for a minimal L ∞ -algebra g we identify S ( g ) with the universal envelope of the Lie algebra ( g , l ) , then(1) there exists a minimal A ∞ -algebra structure on S ( g ) for which the inclusion g −! S ( g ) ◦ is a strict map of L ∞ -algebras and, moreover,(2) any such A ∞ -structure is A ∞ -isomorphic to this one, so that the universal envel-opes of Baranovsky and Moreno-Fernández are A ∞ -isomorphic and A ∞ -quasi-isomorphic to U ( g ) . D ERIVED
PBW
THEOREMS
Proof.
The claim about weak equivalences is immediate since the Quillen map is aweak equivalence. The first claim follows from the homotopy transfer theorem, andthe second one follows from the universal property of the enveloping algebra and ourmain result. Indeed, suppose that S ( g ) is endowed with an A ∞ -algebra structure asin the statement of the theorem. The strict map of L ∞ -algebras g −! S ( g ) gives us an A ∞ -quasi-isomorphism BU ( g ) −! B S ( g ), for the map U ( g ) −! S ( g ) induces an iso-morphism of algebras: we have shown that H ( U ( g )) is naturally identified with theuniversal envelope of the Lie algebra ( g , b ), and so does S ( g ) in Baranosvky’s con-struction. Î Let us recall that one can rectify every L ∞ -algebra g to a bona-fide dg Lie algebra.In this way, one can show that the homotopy category of dg Lie algebras ho ( dgLie )and the category of L ∞ -algebras up to quasi-isomorphism are equivalent. An analog-ous result holds for dga algebras. Hence, we can view the universal envelope functor Lie ∞ Alg −! Ass ∞ Alg as a choice for a representative of the functor that assigns to a dgLie algebra its universal envelope. A restatement of our results is the following.
Corollary 2.2.
The constructions of Baranovsky, Lada–Markl and Moreno-Fernándezgive representatives for the universal envelope functor ho ( U ) : ho ( dgLie ) −! ho ( dgAlg ) .Moreover, if F : Lie ∞ Alg −! Ass ∞ Alg satisfies the conditions:(1) the A ∞ -algebra F ( g ) is minimal whenever the L ∞ algebra g is minimal,(2) the underlying vector space to F ( g ) is the symmetric algebra S ( g ) and,(3) the higher products of F ( g ) induce the higher brackets on g ⊆ S ( g ) ,then F descends to the homotopy category and ho ( F ) = ho ( U ) . Î Recall that if P is an operad and A is a P -algebra, operadic cohomology of A is, bydefinition, the cohomology of the complex of P -derivations Der( B , A ) where B is acofibrant resolution of A in the model category of P -algebras. This complex is quasi-isomorphic to the dg Lie algebra Der( B ) of derivations of B to itself. More generally,the cohomology of A with values in an operadic A -module M is, by definition, thecohomology of the complex of P -derivations Der( B , M ) where M is given a B -modulestructure through the map B −! A . We refer the reader to [17, 23] for details. . K HOROSHKIN AND
P. T
AMAROFF Let us fix a minimal L ∞ -algebra g , a minimal A ∞ -algebra S ( g ) that models U ( g ), andan acyclic twisting cochain C ( g ) −! S ( g ). Observe that then Ω C ( g ) is a quasi-freemodel and hence a cofibrant replacement for S ( g ), so that the operadic cohomology H ∗ Ass ( S ( g )) can be computed through the complex of derivations of the dg-algebra Ω C ( g ) with values in S ( g ). This receives a map from the complex of derivationsDer( Ω C ( g ), g ) through the strict map of L ∞ -algebras g −! S ( g ). We then obtain thefollowing result, which is expected. Theorem 2.4.
The maps above induce an isomorphism H ∗ Ass ( S ( g )) −! H ∗ Lie ( g , S ( g )) and an injection H ∗ Lie ( g ) −! H ∗ Lie ( g , S ( g )) in operadic cohomology groups. Î To each symmetric operad P one can assign an associative universal envelopingfunctor U P : P Alg −! Alg from the category of P -algebras to the category of asso-ciative algebras, see [18] and [20, Section 1] for details. The associative algebra U P ( V )associated with a P -algebra V satisfies the universal property that the category of left U P ( V )-modules is equivalent to the category of left modules over a P -algebra V . Thefirst author explains in detail in [20, Section 1.2] that one can also interpret this as-sociative universal envelope as a left adjoint to a restriction functor, so that one maystudy it using the formalism of [11].Indeed, the universal enveloping functor U P is obtained from the left adjoint f ! cor-responding to the restriction functor for the map of colored operads f : ( P , k ) ! ( P , ∂ P ).Here ( P , k ) is the two-colored operad that governs pairs of the form ( V , M ) where V isa P -algebra and M is a vector space, while ( P , ∂ P ) is the two-colored operad that gov-erns pairs of the form ( V , M ) where V is a P -algebra and M is a left V -module. Recallthat ∂ P is the derivative of the symmetric sequence P , and it is obtained from P byadding an extra color to the output and one of the inputs of each operation in P . Withthis at hand, one can check that f ! ( V , k ) = ( V , U P ( V )).The fact that U P ( V ) is an associative algebra (and not merely a vector space) comesfrom an additional structure on ∂ P : although this collection is no longer an operad inan obvious way, it is a twisted associative algebra —that is, an associative algebra forthe Cauchy product in symmetric sequences— where the product is given by gratingthe root on the unique new colored input. In this way U P ( V ) inherits an associativealgebra structure, and one can check that the datum of a left U P ( V )-modules is thesame as that of a left V -module. D ERIVED
PBW
THEOREMS
The first author has shown in [20] that the functor U P ( − ) satisfies the PBW prop-erty whenever P admits a Gröbner basis whose leading monomials are given by leftcombs. Recall that a shuffle monomial is called a left comb if and only if all its innervertices belong to the leftmost branch of a shuffle tree, that is, the path connectingthe first input and the output. Another criterion was proposed in [20] in case P is a Koszul operad : the functor U P satisfies the PBW property if and only if the twistedassociative algebra ∂ P ! is quadratic, Koszul and generated by ∂ X as atwisted associative algebra.
Here X is the generating symmetric sequence of the symmetric operad P . We claimthat the same conditions are sufficient in the derived setting; we still assume that P is Koszul. Theorem 2.5.
If the associative universal envelope U P satisfies the PBW property thenthe corresponding derived associative universal envelope U P ∞ satisfies the derived PBWproperty. In particular, if an operad P admits a quadratic Gröbner basis whose leadingmonomials are given by left combs then the universal enveloping functor U P ∞ satisfiesderived PBW.Proof. Let P ∞ : = Ω P ¡ be the minimal cofibrant model of P generated by the Koszul-dual cooperad P ¡ and denote by ( ∂ P ¡ ) ¡ Tw - As the twisted associative algebra that isKoszul dual to the twisted associative algebra ∂ P ¡ . One of the main observationsin [20] is the commutation of the coloring procedure P ∂ P and cobar construc-tions. So that there is a natural isomorphism ∂ ( Ω P ¡ ) −! Ω ( ∂ P ¡ ). Each element of a(colored) cobar construction of a (colored) (co)operad is represented by a (colored)operadic tree T . The colored cobar construction of the colored operad ∂ P admits thePBW-filtration given by the number of edges connecting the branch colored in a newcolor and the remaining part of an operadic tree T . The associated graded complexis quasi-isomorphic to the composition Ω Tw - As ( ∂ P ¡ ) ◦ Ω P ¡ ≃ Ω Tw - As ( ∂ P ¡ ) ◦ P ∞ .As shown in [20] the Koszulness of the twisted associative algebra ∂ P ! is a necessarycondition for U P to satisfy the ordinary PBW criterion. Hence the PBW property for U P implies the existence of a chain equivalence s : ( ∂ P ¡ ) ¡ Tw - As ! Ω Tw - As ( ∂ P ¡ ), whichshows the map ∂ P ∞ is almost-free, and finishes our proof of the theorem. Î . K HOROSHKIN AND
P. T
AMAROFF An interesting consequence of the above “PBW-rigidity” phenomenon for associ-ative universal envelopes is the following result, which shows that to compute thederived associative universal envelope of a P -algebra, one may only resolve only onevariable in the functor U P ( − ). Corollary 2.3.
Let P be Koszul. Given a P ∞ -algebra A and a dg P -model B of A, thecorresponding associative universal envelopes are quasi-isomorphic, namely, there isalways a quasi-isomorphism U P ( B ) −! U P ∞ ( A ) of dga algebras. Î Let us fix a map f : P −! Q of dg operads over a field k . Although we focused on P -algebras —left P -modules concentrated in arity 0—, the universal envelope defines amap f ! : P Mod −! Ch k , which is given explicitly by f ! ( X ) = Q ◦ P X . According to [17],the category Mod P of right P -modules admits a cofibrantly generated model struc-ture where fibrations and weak-equivalences are defined point-wise. In particular,we can consider a cofibrant replacement Q ∗ of Q , for example, the two sided bar con-struction B ( Q , P , P ), and define L f ! ( X ) = Q ∗ ◦ P X , which gives us the object Tor P ( Q , X ).Following the procedure of [4, Chapter 7], we can produce a Eilenberg–Moore typespectral sequence which gives a fine tool to study derived PBW phenomena; our ar-guments essentially consider the situation when there is an immediate collapse ofthis sequence due to Q begin almost-free. It would be interesting to consider situ-ations where certain restrictions on Q , other than almost-freeness, allow us to obtainderived PBW theorems. Although the canonical map α : S ( g ) −! U ( g ) is not a map of algebras (since thesource is commutative, but the target is not), we can consider the adjoint action of g on both spaces. It is well known that the map above is then one of g -modules andthus induces a map α g : S ( g ) g −! U ( g ) g , where U ( g ) g is just the center of U ( g ), acommutative algebra. This map is, however, not an isomorphism of algebras either. Aremarkable result of M. Duflo shows that one can construct from this an isomorphism D ERIVED
PBW
THEOREMS of algebras α g ◦ J : S ( g ) g −! U ( g ) g ,known as the Duflo isomorphism , through a suitable (and quite involved) modifica-tion of this map through an automorphism J of S ( g ). In fact, M. Pevzner and C. Toros-sian proved that the Duflo isomorphism is part of an isomorphism of Lie cohomologygroups H ∗ ( g , S ( g )) −! H ∗ ( g , U ( g ))induced from a quasi-isomorphism C ∗ ( g , S ( g )) −! C ∗ ( g , U ( g )) between the corres-ponding Chevalley–Eilenberg complexes, following insight of M. Kontsevich. We referthe reader to [6] for details and useful references. In [5] the authors define the ∞ -centre of a minimal A ∞ -algebra, which can be used, for example, to describe theimage of the “wrong way” map H ∗+ d ( L X ) −! H ∗ ( Ω X )onto the Pontryagin algebra of a simply connected smooth oriented d -manifold X :the image of this map is precisely the ∞ -centre of H ∗ ( Ω X ). In particular we can con-sider, for any minimal L ∞ -algebra, the ∞ -centre Z ∞ S ( g ), which is a commutativealgebra. It would be interesting to understand this higher centre and explore the pos-siblity of extending the results of Duflo and Pevzner–Torossian to this setting. A Models of operads via homological perturbation This section records an instance of a general homological perturbation argumentwhich allows one to obtain, in a range of cases, a resolution of a filtered object fromthe one of its associated graded object. A similar argument for a perturbative con-struction of a resolution of a shuffle operad with a Gröbner basis is featured in [9,Th. 4.1]. We keep the assumption on the characteristic of the ground field.Let us consider a (non-dg) symmetric operad Q = T ( X )/( R ) generated by a finitedimensional Σ -module X concentrated in arities greater than one, subject to a finitedimensional space of relations R . Suppose that the Σ -module X is equipped with a by Vladimir Dotsenko . K HOROSHKIN AND
P. T
AMAROFF non-negative weight grading, X = M n ≥ X ( n ) .This weight grading gives rise to a weight grading of the free operad T ( X ), and hencean increasing filtration F • T ( X ) such that F k T ( X ) is spanned by all elements of weightat most k . This filtration gives rise to a filtration on each Σ -submodule of T ( X ). Inparticular, we may consider the operad P = T ( X )/( gr F R ). In general, there is an iso-morphism of Σ -modules between Q and gr F Q = gr F T ( X )/ gr F ( R ), and a surjection of Σ -modules from P onto Q . Theorem.
Suppose that the underlying Σ -modules of the operads P and Q are iso-morphic. Consider the minimal quasi-free resolution P ∞ −! P in the category ofweight graded operads. There exists a quasi-free resolution Q ∞ −! Q whose under-lying free operad is the same and the differential is obtained from the differential of P ∞ by a perturbation that lowers the weight grading.Proof. Assume that P ∞ is of the form ( T ( W ), d ), where W is an Σ -module that isbi-graded, by weight and by homological degree. We shall prove that there exists aquasi-free resolution Q ∞ = ( T ( W ), d + d ′ ) of Q so that d ′ is strictly weight decreasing.Since the resolution P ∞ = ( T ( W ), d ) is minimal, it follows in particular that W = X , W = k s ⊗ gr F R . The operad T ( W ) can be mapped to both the operad P andthe operad Q : one may project it onto its part of homological degree 0, the latter isisomorphic to the free operad T ( X ) which admits obvious projection maps to P andto Q . Let us choose splittings for those projections; this amounts to exhibiting twoidempotent endomorphisms ¯ π and π of T ( W ) such that both of which annihilateall elements of positive homological degree, and such that the former annihilates theideal ( gr F R ) ⊂ T ( X ) = T ( W ) and the latter annihilates the ( R ) ⊂ T ( X ) = T ( W ) .Since P is finitely generated and has no generators of arity 1, components of thefree operad T ( W ) are finite-dimensional, and there exists a weight graded homotopy h : T ( W ) −! T ( W ) such that h = d , h ] = − ¯ π .We are going to define a derivation D : T ( W ) −! T ( W ) of degree − H : ker( D ) ! T ( W ) of degree +
1. Note that a derivation is fullydetermined by the images of generators, and that D | W = T ( W ) has no ele-ments of negative homological degree. For each element x of T ( W ) of a certain ho-mological degree, we call the “leading term” of x the homogeneous part of x of max-imal possible weight grading; we denote it by b x . D ERIVED
PBW
THEOREMS
We shall prove by induction on k that one can define the values of D on generatorsof homological degree k + H on elements of ker( D ) of homologicaldegree k so that the following five conditions hold:(1) for all elements x ∈ T ( W ), the leading term of the difference D ( x ) − d ( x ) is ofweight lower than that of x ,(2) we have D = k + H ( x ) − h ( b x ) is of weight lower than that of x ,(4) we have D H = − π on elements of ker( D ) of homological degree k .As a basis of induction, we shall choose a basis of W = k s ⊗ gr F R , and set D ( s ⊗ r ′ ) = r where r is some element of R for which b r = r ′ . We note that D ( sr ′ ) − d ( sr ′ ) = r − r ′ hassmaller weight than r ′ , so Condition (1) is satisfied. Condition (2) is satisfied for de-gree reasons, as there are no elements of negative homological degree. Condition (1)together with the fact that ¯ π = π on elements of weight zero implies that the leadingterm of Dh ( b x ) is d h ( b x ) = (1 − ¯ π )( b x ) = (1 − π )( b x ) = b x − π ( b x ),and so we may define H on elements of homological degree zero by induction onweight as follows. On elements x of weight zero, we put H ( x ) = h ( x ), and on elements x of positive weight, we put H ( x ) = h ( b x ) + H ( x − π ( x ) − Dh ( b x )).Both Condition (3) and Condition (4) are proved by induction on weight. For formerone, the inductive argument is almost trivial; we shall show how to prove the latter.On elements of weight zero, we have H ( x ) = h ( x ) and ¯ π = π , so Condition (4) is true: D H ( x ) = d h ( x ) = [ d , h ]( x ) = (1 − ¯ π )( x ) = (1 − π )( x ).For elements of positive weight, we have, by induction, D H ( x ) = Dh ( b x ) + D H ( x − π ( x ) − Dh ( b x )) = Dh ( b x ) + (1 − π )((1 − π ) x − Dh ( b x )) = (1 − π ) ( x ) + π ( Dh ( b x )) = (1 − π )( x ), . K HOROSHKIN AND
P. T
AMAROFF since π vanishes on the image of D = ( R ) and 1 − π is a projector. To carry the inductivestep, we proceed in a similar way. To define the image under D of a generator ofhomological degree k + >
1, we put D ( x ) = d ( x ) − H D d ( x ). Condition (1) now easilyfollows by induction. For Condition (2), we note that D ( x ) = D ( d ( x ) − H D d ( x )) = D d ( x ) − D H ( D d ( x )) = D d ( x ) − (1 − π ) D d ( x ) = π ( D d ( x )) = D d ( x ) ∈ ker( D ), and π vanishes on the image of D . From that, we see thatwhenever x ∈ ker( D ), we have x − Dh ( b x ) ∈ ker( D ). Using Condition (1) and the factthat ¯ π vanishes on elements of positive homological degree, we see that the leadingterm of Dh ( b x ) is d h ( b x ) = (1 − ¯ π )( b x ) = b x , so the leading term of x − Dh ( b x ) is of weightlower than that of x . Consequently, we may define H on elements of ker( D ) of homo-logical degree k > x of weight zero,we put H ( x ) = h ( x ), and on elements x of positive weight, we put H ( x ) = h ( b x ) + H ( x − Dh ( b x )).Once again, a simple inductive argument shows that Conditions (3) and (4) are satis-fied, which completes the construction of D and H .We conclude that D makes T ( W ) a dg operad, that the homology of that operad isisomorphic to Q , and that differential D is obtained from d by a perturbation d ′ thatlowers the weight grading, as required. Î References [1] J. F. Adams,
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