JOURNAL OF ALGEBRA 202, 455᎐511Ž. 1998 ARTICLE NO. JA977280

Lie Theory of Formal Groups over an Operad

Benoit Fresse*

Laboratoire J. A. Dieudonne,´ Uni¨ersite de Nice, Parc Valrose, 06108 Nice Cedex 02, France

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0. INTRODUCTION

Let k be a field of characteristic zero. Let K be a commutative k-algebra. Let R be a complete commutati¨e Hopf algebra defined over K. Explicitly, if R denotes the augmentation ideal of R, then R s K [ R and n R s lim RrR . Moreover R is equipped with a coassociative coproduct ␥:RªRmˆˆR, where m denotes the completed tensor product over K. Notice that R is nothing but a cogroup in the category of commutative complete algebras. When R is the completion of the symmetric algebra generated by a finitely generated projective K-module V, then R is called a Ž finite dimensional. formal group.If V is freely generated by the variables

x1,..., xn, then the coproduct ␥ is equivalent to a formal group law. That is, ␥ is equivalent to an n-tuple of power series GxŽ.,y in 2n-variables

Ž.Žx,ysx1,..., xn, y1,..., yn., which verify the identities

Ž.i Gxiii Ž,0 .sG Ž0, x .s x , Ž.ii Gx Ž,Gy Ž,z ..sGG Ž Ž x,y .,z .. The tangent space of a formal group is equipped with a canonical Lie algebra structure. Usually, a formal group, defined by the complete Hopf algebra R, is denoted by the letter G. We denote by Lie G its associated Lie algebra. Notice that Lie G is dual to V. In the case of a formal group

law, Lie G is equipped with a canonical basisŽ. the dual basis of x1,..., xn . The structure constants of the Lie bracket are given by the quadratic part of the formal group law GxŽ.,y.

*E-mail: [email protected].

455

0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. 456 BENOIT FRESSE

The main theorem of Lie theory asserts that the Lie algebra functor G ¬ Lie G is an equi¨alence from the category of formal groups to the category of Lie algebras. The notion of a complete algebra, of a cogroup, and hence of a formal group makes sense for any usual kind of algebras, such as associative algebras, Lie algebras, etc. More generally, we can consider formal groups for any kind of algebras defined over an operad. An operad is an algebraic device which encodes a category of algebrasŽ such as the ones given above. . The notion of an operad was introduced by P. May for the needs of the iterated loop space machineryŽ cf.wx M. . It turns out that operads are also particularly useful for organizing the many algebraic structures arising in mathematical physics. Let P be an operad. The associated algebrasŽ. resp. formal groups are called P-algebras Ž.resp. P-formal groups . The aim of this paper is to generalize the main theorem of Lie theory to the case of P-formal groupsŽ cf. theorem 4.3.1. . From a certain point of view, an operad is very similar to a ring. Pursuing the analogy further, we can define the notion of a right P- module, and of a P-linear Lie algebra. We will show the following. We construct a Lie algebra functor, which induces an equi¨alence between the P-formal groups and the P-linear Lie algebras, whose underlying right P- module is in some sense finite dimensional.

Consider the left K-module freely generated by x1,..., xn. As in the classical case, the free P-algebra generated by this module is a kind of power series algebra. Its elements are known as P-power series. In the case P s Com, the operad of commutative algebras, we recover the classical power series. In the case P s As, the operad of associative algebras, we obtain the power series in n non-commutative variables. Finally, we can also define a P-formal group law as a particular instance of a P-formal group. A P-formal group law GxŽ.,y is an n-tuple of P-power series satisfying relationsŽ. i and Ž ii . . In fact, M. Lazard has already extended some parts of Lie theory to the context of analyzers Žcf.wx L1, L2. . An analyzer is an axiomatization of the traditional notion of power series algebra. For instance, the sequence of P-power series algebras forms an analyzer. Lazard proved in particular that a formal group law defined in an analyzer is determined, up to an isomorphism, by its quadratic part. In our language, the isomorphism class of a P-formal group law is determined by its Lie algebra. Lazard used essentially direct calculations of power series. We follow another approach, more conceptual, and close to the usual proof of the main theorem of Lie theory in the classical caseŽ cf.wx C, Se. . Let us recall the idea of this proof. By the process of Cartier duality, the LIE THEORY OF FORMAL GROUPS 457 complete Hopf algebra of a formal group G is dual to UŽ.Lie G , the enveloping algebra of its Lie algebra. Therefore, the main theorem of Lie theory can be deduced from the Milnor᎐Moore theorem. In this paper, we provide a Milnor᎐Moore type theory for P-linear Hopf algebrasŽ see Section 4.Ž , and, as a main tool, we state a Cartier duality type theorem see Theorem 4.3.3. . We now give a detailed summary of the paper. In the first section, we recall the prerequisites on operad theory. Let A be a P-algebra. In this section, in particular, we construct an operad UAP Ž., whose algebras are equivalent to P-algebra morphisms X ª A. This operad is known as the en¨eloping operad of A. In this section, we also define the notion of a complete algebra over an operad. In the second section, we define precisely the notion of a formal group over an operad. We give also some examples of P-formal group laws. Mainly, we provide some explicit expansions of Lie power series, which are L ie-formal group laws, such as the Hausdorff᎐Campbell formula. As pointed out earlier, an operad P is similar to a ring. In Section 3, in some sense, we are doing P-linear algebra. We define the notion of a right P-module. We show that the category of right P-modules is monoidal symmetric, i.e., this category is equipped with a tensor product m similar to the tensor product of k-vector spaces. We show also that this category has an internal hom functor. In the second part of this section, we study the cocommutative coalgebras in the category of right P-modulesŽ known as P-linear coalgebras.. Again, the definition of a P-linear coalgebra is the same as the definition of a classical coalgebra, the right P-module tensor product replacing the classical tensor product. This notion is important for our purpose because we have a kind of duality between P-linear coalge- bras and P-algebras. Let C be a P-linear coalgebra. Let A be a complete P-algebra. A coalgebra-algebra pairing between C and A is a map

²:y,y:C=AªP, which is P-linear in C and k-linear in A. Moreover, we assume that this map makes the coproduct of C adjoint to the product of A. We have also a notion of perfect coalgebra-algebra pairingŽ similar to the classical notion of perfect pairing. . This process plays the role of Cartier duality in our setting. Section 4 is devoted to Lie theory. In the first subsection, we study the Lie algebras in the category of right P-modulesŽ known as P-linear Lie algebras.. Using the analogy between right P-modules and k-vector spaces, we define the enveloping algebra of a P-linear Lie algebra. It is not 458 BENOIT FRESSE difficult to extend the classical Milnor᎐Moore theory to the P-linear framework.Ž In particular, the enveloping algebra of a P-linear Lie alge- bra is a cocommutative Hopf algebra in the category of right P-modules.. Consider a P-formal group G. Let RGŽ.be its underlying P-algebra. One can define QRŽ. G as the module of the indecomposable elements of this P-algebra. In the second subsection, we show that the P-linear dual of QRŽ. G is a P-linear Lie algebra, called the Lie algebra of the formal group G, and denoted by Lie G. There exists a natural perfect coalgebra-algebra pairing between UŽ.Lie G , the enveloping algebra of the P-linear Lie algebra Lie G, and RGŽ.. This pairing is denoted by ²y , y :G. Moreover, in some sense, this pairing makes the coproduct of RGŽ., adjoint to the product of UŽ.ŽLie G cf. Theorem 4.3.3 . . In the third subsection, assuming the existence of ²:y , y G we prove the main theorem of Lie theory. The construction of ²:y , y G is the purpose of the fifth and last section. The main ingredient in this construction is a kind of differential calculus. More precisely, we need the notion of an operad derivation. We relate the Lie algebra of G to the operad derivations of URGP ŽŽ.., the enveloping operad of RGŽ., defined in the first section.

0.1. Notation and Con¨entions Throughout this paper we work over a fixed field k of characteristic 0.

We denote by Modk the category of k-modules. Most algebras are defined over a ground operad P Ž.cf. Subsection 1.1.8 . Except in the first part of the first section, we assume PŽ.0 s 0. We denote by K the associative algebra PŽ.Ž1 cf. Subsection 1.1.8 . . A finite dimensional P-formal group is usually denoted by G. This P-formal group G is specified by a complete P-algebra, denoted by RGŽ., and a coproduct ␥ : RGŽ.ªRG Ž.kRG Ž.Žcf. Definition 2.1.2 . . We de- note by UGP Ž.the completed enveloping operad of RG Ž.Žcf. Subsec- tion 1.2.2. .

We denote the symmetric group by Sn. In the sequel, many k-modules are endowed with a natural Sn-action. We denote this action by ␴ и ¨, where ␴ belongs to Snnand ¨ belongs to an S -module V. Here is the main example where this convention is applied. Let V be an Si-module and W be an S -module. Consider the induced module IndSn V W, where jSij=Sm iqjsn. Let ¨ g V, w g W. By abuse of notation, we denote by ¨ m w the element 1 w kS V W IndSn V W. Thus, if m ¨ m g wxnkmwSij=Sxm s Sij=Sm ␴S, then ␴ и w denotes the action of ␴ on w in IndSn V W, gn ¨ m ¨ m Sij=S m hence, the tensor ␴ w IndSn V W. Some modules are m ¨ m g Sij=S m equipped with extra Sr-actions, which should not be confused with their natural Sn-module structureŽ. see, e.g., Subsection 1.1.5 . LIE THEORY OF FORMAL GROUPS 459

1. COMPLETE ALGEBRAS OVER AN OPERAD

1.1. Operads The notion of a topological operad was introduced by P. May for the needs of homotopy theoryŽ cf.wx M. . In this subsection, we give a short overview of the operad theory. For a more detailed account, we refer to wxGe-J, G-K, J, Lo1, M . 1.1.1. Monoidal Categories. Recall that a monoidal category is a cate- gory C together with an ‘‘associative’’ bifunctor m: C = C ª C, and an object I g C, which is a two sided ‘‘unit’’ for m Žcf.wx ML. . An associati¨e algebra in Ž.C, m , I consists of an object A g C, together with an associa- tive product m: A m A ª A, and a unit e: I ª A. More precisely, m and e verify the equations

m и m m A s m и A m m, m и e m A s m и A m e s A.

Let F denote the category of the k-module endofunctors together with natural transformations. The functor composition (: F = F ª F makes F into a monoidal category. An associative algebra in Ž.F, ( is known as a monad Žcf.wx ML, B-W. . Let S, T g F. We define the tensor product of S, T by setting

ŽS m TV .Ž.[SV Ž.mTV Ž..

Clearly, this provides F with another monoidal structure. Notice that the functor tensor product is symmetric.

1.1.2. S-Modules.AnS-module V is a sequence of Sn-modules V Ž.n , n g N. We denote by S-Mod the category of S-modules. As in the case of graded modules, the notation ¨ g V Ž.i may be abbreviated to ¨ g V. The integer i is called the degree of ¨ and we denote <<¨ s i. An S-module gives rise to a functor TŽ.V , ygF, which is defined by

ϱ T V , E [ V n Emn . Ž. Ž.mSn [ns0

Furthermore, this construction provides a fully faithful functor T: S-Mod ª F. 460 BENOIT FRESSE

1.1.3. S-Module Tensor Product. Let V , W be S-modules. The tensor product of V and W is defined by the formula

V W N [ IndSN V i W j . Ž.Ž.m Sij=SŽ.Ž.m iq[jsN

This bifunctor is associative. In fact, we have

SN Ž.Ž.V 1m VnSN s Ind =иии=S V 11Ž.Ž.i m иии m Vnni . [ii1n i1qиии qinsN

The S-module 1, defined by

k if n 0, 1Ž.n s s ½0 otherwise, is a two sided unit for m. Let us construct a symmetry isomorphism cV , W : V m W ª W m V . Let ␶denote the transpositionŽ. 1 2 g S2 . We denote by ␶ Ž.i, j the block transposition, defined by

␶ Ž.Ž.i, jksiqk, ks1,..., j, ␶Ž.Ž.i,jjqksk, ks1,...,i.

As explained in Convention 0.1, an element of Ž.Ž.V m W N is denoted by ␴и¨mwwith ¨ g V Ž.i , w g W Žj ., ␴ g SN , and i q j s N. We set

cV,WŽ.Ž.␴ и ¨ m w s ␴␶ i, j и w m ¨.

As a conclusion, the S-module tensor product makes the category S-Mod into a symmetric monoidal category.

1.1.4. LEMMA. The functor T: S-Mod ª F commutes with the tensor product. More precisely, we ha¨e a natural isomorphism

T Ž.Ž.Ž.V m W , E ( T V , E m T W , E .

This functor isomorphism, carries the symmetry operator TŽ. cV , W , E to the switching map

T Ž.Ž.Ž.Ž.V , E m T W , E ( T W , E m T V , E .

1.1.5. The Tensor Power of an S-Module. Let V be an S-module. Since the S-module tensor product is symmetric, the symmetric group Sn acts on LIE THEORY OF FORMAL GROUPS 461 the n-fold tensor power V mn by place permutation. In other words, V mn is an Snn-Ž.S-module . One should not confuse the S -action with the S-module structure. mn For s g Sn, we denote the associated S-module morphism by s#: V mn y1 ªV,orbys*, with s# s Žs .Ž*. Let us denote by si1,...,in.the block permutation defined by

иии иии siŽ.1,...,iinsŽ.Ž1.qqi sŽky1.q r [ i1q qi sŽk.y1q r,

rs1,...,isŽk., ks1,...,n.

mn Let ␴ и ¨ 1 m иии m ¨n g V . We have clearly

s*Ž.␴и¨1mиии m ¨ns ␴ и sŽ.<<¨ 1,..., <<¨nsи¨ Ž1.mиии m ¨ sŽn..

1.1.6. S-Module Composition. Let V , W be S-modules. The composi- tion of V and W is the S-module V (W defined by the formula ϱ V W V n W mn . ( [ Ž.Ž.m Sn [ns0 One may check that this bifunctor is associative. Consider the S-module defined by InŽ.sk if n s 1 and In Ž.s0 otherwise. This S-module is obviously a two sided unit for the composition product.

1.1.7. LEMMA. The functor T: S-Mod ª F commutes with the composi- tion product. More precisely, we ha¨e a natural isomorphism

T Ž.V (W , E ( T Ž.V , T Ž.W , E . 1.1.8. Operads.Anoperad P is an associative algebra in the category of S-modules equipped with the composition product. Hence, an operad structure consists of an associative product ␮: P( P ª P and a unit ␩: I ª P. Because of the form of the composition product, the operad structure is equivalent to a unit 1 g PŽ.1 and a collection of equivariant morphisms

PŽ.n m P Ž.i1 m иии m P Žin .ª P Ži1q иии qin ., which satisfy the May axiomsŽ cf.wx M. . The image of ␮ m ␯ 1 m иии m ␯n under the product is denoted by ␮␯Ž.1,...,␯n.Theproduct ␮Ž.1,...,1,␯, 1, . . . , 1 , with ␯ at the ith place, is also denoted by ␮(i ␯. As I is a unit for (, it is equipped with a canonical operad structure. Furthermore, I is an initial object in the category of operads. 462 BENOIT FRESSE

Among the products above, we have

PŽ.1 m P Ž.1 ª P Ž.1. This product makes PŽ.1 into an associative algebra. Throughout this paper, we denote this algebra by K. 1.1.9. Algebras. Because of Lemma 1.1.7, an operad P is equivalent to a monad whose underlying functor is of the form TŽ.P, y . An algebra over this monad is known as a P-algebra. Because of the form of the functor TŽ.P, y ,a P-algebra product TŽ.P,AªAis equivalent to a sequence of Sn-equivariant maps

n PŽ.n m Am ª A, which are associative with respect to the operad product, and which make the operad unit act as the identity. Let ␮ g PŽ.n , a1,...,an gA. The image of ␮ m a1 иии an under the P-algebra product is denoted by ␮Ž.a1,...,an . As for any monad, if V is a k-module, then TŽ.P, V is the free P-algebra generated by V.

1.1.10. EXAMPLES. Many classical algebras are in fact algebras over an operad. For example, there are operads As, Com, L ie, whose algebras are respectively the associative algebrasŽ. without unit , the associative and commutative algebrasŽ. without unit , and the Lie algebras. Let us recall the expansion of the corresponding functors TŽ.Ž.As, y , T Com, y , and TŽ.Lie, y . The free associative algebra is the tensor algebra ϱ n T Ž.Ž.As, V s TV s Vm . [ns1 The free commutative algebra is the symmetric algebra

ϱ mn T Ž.Ž.Ž.Com, V s SV s V Sn. [ns1 In characteristic zero, the free Lie algebra is the primitive part of the tensor algebra equipped with the ‘‘shuffle’’ coproduct

T Ž.L ie, V s Prim TV Ž.. It can also be described as

ϱ w1xmn TŽ.Lie, V s eVn , [ns1 LIE THEORY OF FORMAL GROUPS 463

w1x where ennis an idempotent of kSwxknown as the first Eulerian idempotent Žcf.wx R1, R2, Lo2. .

1.1.11. Operads Defined by Generators and Relations Žcf.wx G-K, 2.1. . The forgetful functor from the category of operads to the category of S-modules possesses a left adjoint known as the free operad. Let V be an S-module. The free operad generated by V is denoted by TŽ.V . This operad is endowed with an S-module morphism ␩: V ª TŽ.V and is characterized by the following universal property. Let P be an operad. If ␩Ј: V ª P is an S-module morphism, then there exists a unique operad morphism ␾: TŽ.V ª P, such that ␩Ј s ␾␩. It is also possible to define the notion of an operad ideal. In fact, a sub-S-module I ; P is an operad ideal if and only if the operad product induces an operad product on the quotient S-module PrI. An operad morphism PrI ª Q is equivalent to an operad morphism P ª Q which vanishes on the ideal I. Given a sub-S-module S ; P, the ideal generated by S is the smallest ideal containing S. We denote this ideal by Ž.S .An operad morphism PrŽ.S ª Q is equivalent to an operad morphism P ª Q which vanishes over S. These definitions enable us to specify operads by generators and rela- tions. The generators are specified by an S-module V , and the relations by a sub-S-module R ; TŽ.V . Suppose that the generators are concen- trated in degree 2, and the relations in degree 3. Explicitly, V Ž.n s 0 unless n s 2, and RŽ.n s 0 unless n s 3. In this case, the operad PsTŽ.Ž.VrRis known to be quadratic. Notice that we can recover V and R from the operad, since V Ž.2 s P Ž.2 and R Ž.3 is the kernel of the natural operad morphism TŽ.Ž.V 3 ª P Ž.3. The operads As, L ie, Com are all quadraticŽ cfwx G-K. .

1.1.12. Algebras o¨er a Quadratic Operad. Let V be a k-module. The endomorphism operad denoted by E ndVVis defined by E ndŽ. n [ mn ModknŽV , V .together with the obvious S -action and operad product. Let A be a P-algebra. Recall that the P-algebra structure can be n specified by a sequence of products PŽ.n m Am ª A. Such a product is mn equivalent to a map PŽ.n ª Modk ŽA , A.. In fact, a P-algebra product TŽ.P,AªAis equivalent to an operad morphism P ª E ndA. Assume that the operad P is quadratic. Let V be the S-module concentrated in degree 2 with V Ž.2 s P Ž.2 . Let R be the S-module concentrated in degree 3 with RŽ3 .s ker ŽT ŽV .Ž3 .ª P Ž3 .. . Since P s TŽ.Ž.VrR, an operad morphism P ª E ndA is equivalent to an S-mod- ule morphism V ª E ndA such that the induced operad morphism TŽ.V ª E ndA vanishes over R. 464 BENOIT FRESSE

1.1.13. EXAMPLE. Let us show how this works for the operad P s L ie. The operad L ie is quadratic. Let V Ž.2 be the signature representation of

S2 . Hence, V Ž.2 is the k-module generated by a single operation ␥, withŽ 1 2.Ž␥sy␥. Let c denote the cycle 1 2 3.g S3. We define RŽ3. to be the sub-S3-module of TŽ.Ž.V 3 generated by the element

2 Ž.Ž.Ž.1 q c q c ␥ (2␥ g T V 3.

We have L ie s TŽ.Ž.V r R . Let L be a k-module. An S-module morphism ␳: V ª E ndL is equiva- lent to the specification of an antisymmetric bracket

wxy,y:LmLªL, which represents the image of ␥. The induced operad morphism ␳: TŽ.V ª E ndL maps ␥ (2 ␥ to the corresponding product in E ndL, i.e., ␳␥Ž.(2 ␳␥ Ž., and hence, to the map

XmYmZ¬X,wxY,Z.

2 Thus,Ž 1 q c q c .␥ (2 ␥ is mapped to

XmYmZ¬X,wxY,ZqY, wZ,X xqZ, wX,Y x.

2 Finally, ␳ cancelsŽ. 1 q c q c ␥ (2 ␥ if and only if

X,wxY,ZqY, wZ,X xqZ, wX,Y xs0, ᭙X,Y, Z g L.

As a conclusion, an operad morphism L ie ª E ndL is equivalent to the specification of an antisymmetric bracket wxy, y : L m L ª L which veri- fies the Jacobi identity.

1.2. Algebra Morphisms as Algebras o¨er an Operad In this subsection, we define operads whose algebras are the objects of some comma categories. 1.2.1. Algebras under a P-Algebra. Let A be a P-algebra. A P-algebra under A Ž.aka an A-algebra is a P-algebra X together with a P-algebra morphism ␩X : A ª X. We denote by ArP-Alg the category of A- algebras. For example, if A s PŽ.0 , then the operad product provides A with a P-algebra structure. In fact, PŽ.0 is the free P-algebra generated by 0. Moreover, a PŽ.0 -algebra is nothing but a P-algebra. As a consequence, PŽ.0 is an initial object in the category of P-algebras. Conversely, the LIE THEORY OF FORMAL GROUPS 465

enveloping operad construction below supplies an operad UAP Ž.Žcalled the en¨eloping operad of A., whose algebras are the A-algebras. More- over, UP Ž.0 s A. The idea of representing the category of A-algebras by an operad goes back towx Ge-J . Independently, the enveloping operad is necessary for the construction of a useful differential calculusŽ. cf. Section 5 .

1.2.2. Construction of the En¨eloping Operad. The canonical embedding Ä4Ä 4 u:1,...,r¨1,...,rqn gives rise to a group embedding Srr¨ S qn. Indeed, any permutation ofÄ4 1, . . . , r extends to a permutation ofÄ 1, . . . , r q n4, which fixes the elements outside the image of u. Consider the S-module

ϱ PAn[ Pr n Amr. wxŽ. Žq .mSr [rs0

The P-algebra product induces an S-module morphism d0: PwTŽ.P, A x ªPwxA. The operad product P( P ª P induces another S-module morphism d1: PwTŽ.P, A xwxª P A . The operad unit induces an S- module morphism s00: PwxA ª P wTŽ.P, A x. We have ds0sds10s PwxA. The operad UAP Ž.is defined as the linear coequalizer

TŽ.d06 TŽ.PTŽ.P,A TŽ.PwxA ªUAPŽ.. TŽ.d1

The pair ŽŽT d01 ., T Žd ..is known to be reflexi¨e, i.e., we have

TŽ.Ž.d00Ts sT Ž.Ž.d 10Ts sTŽ.PwxA.

This property forces the image of TŽ.d01y T Ž.d to be an operad ideal. Hence, the operad product of TŽPwxA .induces an operad product on the quotient UAPŽ.. As mentioned earlier, this operad UAP Ž.is called the enveloping operad of the P-algebra A.

1.2.3. Generators of the En¨eloping Operad. Let us write a concrete definition of UAP Ž.by generators and relations. We denote an element of PŽ.rnAmrby ␮Ž.Ža ,...,a,h ,...,h , where ␮ P r n., qmSr 1 r 1 n g q a1,...,ar gA, and h1,...,hn denote extra-variables. Let ␮ g PŽ.r q n , ␯ 1 gPŽ.i1 ,...,␯rrgP Ž.i , a1,...,aNgA, Nsi1qиии qir. The map d0is given by

␮␯Ž.Ža,...,a ,...,␯ a иии ,...,a иии .,h ,...,h Ž.11 iri11qqiry11q1 i q qiry1qir1 n XX ¬␮Ž.a1,...,ar,h1,...,hn, 466 BENOIT FRESSE

with aX ␯ a ,...,a , 111s Ž.i1 . . X a␯Ž.aиии ,...,a иии . rris1qqiry11q1 i q qiry1qir

The map d1 is given by

␮␯Ž.Ža,...,a ,...,␯ a иии ,...,a иии .,h ,...,h Ž.11 iri11qqiry11q1 i q qiry1qir1 n

¬␮ЈŽ.a1,...,aN,h1,...,hn,

with ␮Ј s ␮␯Ž.1,...,␯r , 1, . . . , 1 . In other words, the operad UAP Ž.is generated by PwxA with relations given by

X X ␮␯Ž.1,...,␯r,1,...,1 Ž.a1,...,aN,h1,...,hns␮Ž.a1,...,ar,h1,...,hn.

In a generator ␮Ž.a1,...,ar ,h1,...,hn , we assume the elements of the algebra and the extra-variables to be in a strict order. But, we may drop

this assumption. The symmetric group Srqn may permute the factors of the tensor

a1 m иии m ar m h1 m иии m hn .

Nevertheless, any tensor has a well ordered tensor in its Srqn-orbit. Hence, any generator with the aiiand the h disordered is equivalent to a

well-ordered generator under the Srqn-action. Finally, the two construc- tions give the same set of generators. Notice also that the enveloping operad is generated by the S-module A[P, which is A [ PŽ.0 in degree 0 and P Žn .in degree n ) 0.

1.2.4. LEMMA. The operad UP Ž. A is an augmented P-operad, i.e., we ha¨e a natural pair of operad morphisms

⑀ P ~ UAPŽ., ␩

such that ⑀␩ s P. Proof. As PŽ.n is a summand of PwxAnŽ., we obtain a canonical S-module morphism

PªPwxAªTŽ.P wxA ªUAPŽ.. We check easily that the composite is an operad morphism. This gives ␩. The projection PwxAnŽ.ªP Ž.ninduces an operad morphism T ŽPwxA . LIE THEORY OF FORMAL GROUPS 467

ª P. It is not hard to check that this morphism induces an operad morphism ⑀: UAP Ž.ªP. The identity ⑀␩ s P is clear. In the case A s PŽ.0, ␩ and ⑀ are indeed inverse isomorphisms. Therefore, UP ŽŽ..P 0 s P.

1.2.5. THEOREM. Let A be a P-algebra. The category of UP Ž. A -algebras is equi¨alent to the category of A-algebras. This theorem is proved by an immediate inspection. To conclude the study of the enveloping operad, let us calculate it in some universal cases. In the case of a free P-algebra, an easy calculation returns the following result.

1.2.6. PROPOSITION. If A is a free P-algebra, say A s TŽ.P, V , then

UAPŽ.sV[PwxV. In fact, in this case, the enveloping operad is universal among the objects equipped with an operad morphism P ª UAP Ž.and a map VªUAPŽ.Ž.0. Let us now consider the case of a P-algebra coproduct. Let A, B be a pair of P-algebras, with coproduct denoted by A k B. Recall that the category of operads is cocompleteŽ cf.wx Ge-J, Theorem 1.13. . Let P ª PЈ, P ª Q be operad morphisms. Form the pushout PЈ kP Q. We have a restriction functor from the category of PЈ-algebrasŽ. resp. Q-algebras to the category of P-algebrasŽ cf.wx Ge-J, 1.6. . This is just as in commutative algebra. A PЈ kP Q-algebra consists of a k-module equipped with a PЈ-algebra and a Q-algebra structure, which restrict to the same P- algebra structure. In particular, a UAP Ž.kP UBP Ž.-algebra is a P-alge- bra X, together with a pair of P-algebra morphisms Ž.A ª X, B ª X . Hence, a UAP Ž.kP UBP Ž.-algebra is nothing but a A k B-algebra. As a consequence, we have proved the following proposition.

1.2.7. PROPOSITION. We ha¨e

UAPŽ.Ž.Ž.kBsUAP kPUBP.

1.2.8. Remark. The presentation of UAP Ž.by generators and relations given above shows that the algebra UAP Ž.Ž.1 is the enveloping algebra defined by V. Ginzburg and M. KapranovŽ cf.wx G-K. . A UAP Ž .Ž1 . -module is known as an A-module. One may show that an A-module is the same as an abelian group in the category of P-algebras over A Ž.see below .

1.2.9. Algebras o¨er a P-Algebra. Let A be a P-algebra. A P-algebra o¨er A is a P-algebra X together with a P-algebra morphism ⑀X : X ª A. We denote by P-AlgrA the category of P-algebras over A. 468 BENOIT FRESSE

A connected A-algebra is an A-algebra X together with an A-algebra morphism ⑀X : X ª A. In other words, X is equipped with two P-algebra morphisms ␩XX: A ª X and ⑀ : X ª A, and ⑀␩ XXsA. A connected PŽ.0 -algebra X is called a connected P-algebra. By the trick of the enveloping operad, any connected algebra is equivalent to a connected algebra over an operad. As an example, the free algebra TŽ.P, V is a connected P-algebra. Let us consider the operad P˜, defined by

0, if n 0, P˜Ž.n s s ½PŽ.n, otherwise. ˜ AP-algebra X gives rise to a connected P-algebra Xq. We set Ž. XqqsP0[Xand we equip X with the following product. We define Ž. ␮ Ž. P0ªXq to be the canonical embedding. Let g P n , a1,...,an g PŽ.0, x1,..., xn gX. We expand ␮Ža11q x ,...,annqx .. We obtain a sum of terms depending at most linearly in each variable aiiand x .By evaluation of the operad products

mr mnyr PŽn .m P Ž.0 m P Ž.1 ª P Žn y r . we eliminate the ai from this expansion. Hence, we obtain a sum of the kind

␮ a ,...,a ␮ x ,..., x , Ž.1nIiiq Ý Ž1r . IsŽ.i1,...,ir where I ranges over the non-void subsets ofÄ4 1, . . . , n . The term

␮Ž.a1,...,an gives the component of the product on P Ž.0 . After evalua- tion, the other terms gives the component of the product on X. Clearly, this construction provides Xq with a P-algebra structure. EMMA 1.2.10. L . The functor X ¬ Xq induces an isomorphism between the category of P˜-algebras and the category of connected P-algebras. Proof. The proof of this lemma is immediate. Just notice that the image of a connected P-algebra X under the inverse functor is ker ⑀X .

1.3. The Reduced Free Algebra 1.3.1. The Wreath Product. Let A be any associative ring. The wreath mn product A X Snnis the k-module ASwxequipped with the following XX mn associative product. Let Ž.Ž␭1,...,␭n;s , ␭1,...,␭nn;sЈ.gASwx. We set

X X X X Ž.Ž.␭1,...,␭n;s и ␭1,...,␭n;sЈ [Ž.␭␭1sŽ1.,...,␭␭nsŽn.;sЈs. LIE THEORY OF FORMAL GROUPS 469

If V is a left A-module, then its nth tensor power is equipped with a left A X Sn-module structure defined by

Ž.Ž.␭1,...,␭n;s и ¨1,...,¨n[Ž.␭1¨sŽ1.,...,␭ns¨ Žn.,

mn for Ž.␭1,...,␭nn;s gAXS and Ž.¨ 1,...,¨ngV . Recall that we denote K s PŽ.1 . The operad products

PŽ.1 m P Žn .ª P Žn ., PŽn .mP Ž.1mиии m P Ž.1 ª P Žn ., make PŽ.n into a K y K X Snn-bimodule. The right K X S -module struc- ture is given by

␮ и Ž.Ž.␭1,...,␭n;s ssи␮␭1,...,␭n, for Ž.␭1,...,␭nn;s gKXS and ␮ g P Ž.n . 1.3.2. The Reduced Free P-Algebra. Let A be a P-algebra. Recall that the P-algebra product

PŽ.1 m A ª A provides A with the structure of a left K-module. In other words, we have a forgetful functor from the category of P-algebras to the category of K-modules. This forgetful functor admits a left adjoint, known as the free P-algebra generated by a K-moduleŽ or the reduced free P-algebra, if one prefers. . Let V be a left K-module. The free P-algebra generated by V is defined by ϱ T P, V P n V mn . Ž.s Ž.mKXSn [is0 Notice that this algebra is connected. In the sequel, TŽ.P, V denotes the free P-algebra or the reduced free P-algebra, whether V denotes a k-module or a left K-module. Notice that the free P-algebra generated by a k-module V is equal to the reduced free P-algebra generated by K m V. For this reason, in the sequel, in general, we consider only the case of left K-modules. 1.3.3. Coproducts. The category of P-algebras is complete and cocom- pleteŽ seewx Ge-J, 1.6. . The coproduct of two P-algebras is denoted by k. In this paper, we deal mainly with free P-algebras or reduced free P-algebras. Let V, W be left K-modules. In this case, we have

T Ž.ŽP, V k T P, W .Žs T P, V [ W .. 470 BENOIT FRESSE

1.4. Complete Algebras and Power Series In this section, we define the notion of a complete algebra over an operad. A reasonable complete algebra should be connected. As a conse- quence, from now on, we assume PŽ.0 s 0. If not, we may replace P by the operad P˜. Of course, one could follow the usual convention dealing with connected P-algebras instead of P˜-algebras. But in our situation, this makes the formalism heavy going.

1.4.1. Ideals.Anideal of a P-algebra A is a submodule of A, say I, such that

␮ ᭙␮ ᭙ ᭙ Ž.a1,...,any11,b gI, gP Žn., a ,...,any1gR, bgI.

Clearly, the P-algebra product TŽ.P, A ª A induces a P-algebra prod- uct on the quotient ArI. Let us denote by An the image of PŽ.k Amk under the [k G nSm k product. Of course, An is an ideal of A, called the nth power of the augmentation ideal of A. 2 The quotient QA s ArA is a left K-module, known as the K-module of the indecomposable elements.

1.4.2. Complete Algebras.AP-algebra is said to be nilpotent when the n product PŽ.n m Am ª A vanishes for n sufficiently large. A complete P-algebra is a P-algebra A together with a sequence of ideals In, n G 1, such that the P-algebra ArInnis nilpotent and A s lim ArI . Notice that the hypothesis ArIn nilpotent implies that the topology defined by the nth powers of the augmentation ideal An is equivalent to the topology defined by the In. As a consequence, any P-algebra mor- phism is continuous.

Let A be a P-algebra together with a sequence of ideals In, n G 1, such that the P-algebra ArInnis nilpotent. If Aˆˆs lim ArI , then A is a complete P-algebra known as the completion of A with respect to the topology given by the In.

1.4.3. The Free Complete P-Algebra. Let V be a left K-module. We denote by PˆŽ.V the free complete P-algebra generated by V. Explicitly, PˆŽ.V is the completion of TŽ.P, V with respect to the powers of the augmentation ideal. Since

n T P, V P i V mi , Ž.s Ž.mKXSi [iGn LIE THEORY OF FORMAL GROUPS 471 we have

ϱ Pˆ V P i V mi. Ž.s Ł Ž.mKXSi is1

Let A be a complete P-algebra. The restriction to V induces a one-one correspondence between P-algebra morphisms PˆŽ.V ª A and K-module morphisms V ª A. 1.4.4. Power Series. Let V be the K-module freely generated by the variables x1,..., xn. In this case, we write PˆˆŽ.Ž.x1,..., xn for P V .An element of PˆŽ.x1,..., xn is called a P-power series. Let ␣ be a multi- ␣ m␣1 m␣n index ␣ s Ž.␣1,...,␣n . We denote by x the tensor x m иии m x . Any element of PˆŽ.x1,..., xn has a unique expansion of the form

␣ Ý␮␣Ž.x . ␣

Hence, the name P-power series. An n-tuple of variables Ž.x1,..., xn may be denoted by a single letter x. For P s Com, we recover the classical power series. For P s As,we obtain the power series in non-commuting variables. Since V is freely generated, and because of the discussion above, a ˆ P-algebra morphism PŽ.x1,..., xn ªA is equivalent to an n-tuple of ␣ elements a1,...,an gA.If fxŽ.sÝ␣␣␮ Žx .gPˆ Ž.x, then its image under the corresponding P-algebra morphism is given by

␣ Ý␮␣Ž.a g A. ␣

␣ n This sum is well defined since ␮␣Ža .g R for ␣ sufficiently large. When AsPˆŽ.y, the elements a1,...,an are P-power series, and this process is known as the substitution of P-power series. 1.4.5. Operads and Analyzers. The usual substitution rules of power series hold over a general operad. The substitution of P-power series makes the sequence of PŽ.x1,..., xn into an analyzer in the sense of LazardŽ cf.wx L1, Chap. I. . In this way, we obtain an equivalence between the notion of an operad and the notion of a multilinear analyzer Žcf.w L1, p. 332x. . But an analyzer is not multilinear in general. Let us recall how to recover the operad P from its analyzer. We have a ˆ natural morphism M: PŽ.n ª P Žx1,..., xn .defined by

MŽ.␮ [ ␮ Žx1,..., xn .. 472 BENOIT FRESSE

This morphism identifies PˆŽ.n with the submodule of PŽ.x1,..., xn generated by the multilinear monomials, i.e., by the monomials of degree 1 in each variable xin. This morphism is S -equivariant since

MŽ.Ž.Ž␴␮ s ␴␮ x1,..., xn .s␮ Žx␴Ž1.,..., x␴Žn. ..

Moreover, let us consider the operation which maps the P-power series

fxŽ.Ž.1,..., xngPˆ x1,..., xn gx,..., x Pˆx ,..., x 11Ž.Ž.i11g 1 i . . gx,..., x Pˆx ,..., x nŽ.Ž.1in g 1 in to the power series obtained by the substitution

fgŽ.Ž x,..., x ,..., gx иии ,..., x иии .. Ž.11 ini11qqiny11q1 i q qin

Clearly, this operation restricts to the composition law of the operad. 1.4.6. The Completed Coproduct. The category of complete P-algebras is equipped with a coproduct. Let A s lim ArIkk, B s lim BrJ be com- plete P-algebras. The coproduct of A and B is the P-algebra lim A k BrŽ.IkkkBqAkJ, i.e., the completion of A k B with respect to the sequence of ideals Ikkk B q A k J ; A k B. When A and B are as- sumed to be complete, A k B denotes their completed coproduct. We denote by ٌ: A k A ª A the folding map, whose restriction to each summand of the coproduct is the identity. Recall that the null space is a zero object in the category of complete P-algebras. We denote by 0 the zero arrow.

1.4.7. The En¨eloping Operad of a Complete P-Algebra. We need to adapt the enveloping operad construction, for working with complete P-algebras, because a complete P-algebra under A is not equivalent to a

UAP Ž.-algebra. Let A s lim ArIk be a complete P-algebra. By using the tensor products

P r n A иии I иии A , ÝŽ.Žq mSkrm m m m . we equip UAnP Ž.Ž.with a canonical topology. We define the completed enveloping operad UAnˆP Ž.Ž., as the completion of UAnP Ž.Ž.with respect to this topology. We equip UAnˆP Ž.Ž.with the induced operad product. LIE THEORY OF FORMAL GROUPS 473

In the sequel, if A is assume to be a complete P-algebra, then the enveloping operad of A denotes the completed one. In the same way, we omit the hat in the notation.

For example, if A s PˆˆŽ.x , then an element of UP ŽP Žxn ..Ž . is a power series like ␣ Ý␮␣Ž.x , h1,...,hn. ␣ ˆ Furthermore, the augmentation ⑀: UP ŽŽ..P x ª P is given by the evalua- tion at x s 0

⑀ Ž.u [ u< xs0.

2. FORMAL GROUPS

2.1. Basic Definitions

2.1.1. DEFINITION.Acogroup object in the category of complete P- algebras is a complete P-algebra R together with a coproduct ␥ : R ª R k R, and an antipode ␫: R ª R such that the following usual identities are satisfied R k 0 и ␥ s 0 k R и ␥ s R,Ž. 2.1.1 ␥kRи␥sRk␥и␥, Ž.2.1.2 иRk␫и␥sٌи␫kRи␥s0.Ž. 2.1.3ٌ A cogroup morphism is a P-algebra morphism which commutes with the coproduct. In the classical case P s Com, a cogroup object is just a commutative complete Hopf algebra.

Notice that the antipode is unique. Let us denote by ␥ 3 s ␥ и ␥ k R s и R k ␥, the 3-fold coproduct. In the same way, we denote ٌ3 s ٌиٌk ␥ RsٌиRkٌ.If␫Јis another antipode, then we have

.␫ s ٌи0k␫и␥sٌи33␫ЈkRk␫и␥ sٌи␫Јk0и␥s␫Ј

2.1.2. DEFINITION Ž.Formal Group . A finite dimensional P-formal group is a cogroup in the category of complete P-algebras, whose underlying P-algebra is freely generated by a finitely generated projective K-module. In the sequel, any P-formal group is tacitly assumed to be finite dimensional.

2.1.3. PROPOSITION. A P-formal group is equi¨alent to a free complete P-algebra R s PˆŽ.V , with V a finitely generated projecti¨eK-module, equipped with a K-linear map ␥ : V ª PˆŽ.V [ V . This map induces a coproduct, also 474 BENOIT FRESSE denoted by ␥. This coproduct is assumed to ¨erify the identities

R k 0 и ␥ s 0 k R и ␥ s R,Ž. 2.1.4 ␥kRи␥sRk␥и␥. Ž.2.1.5

In fact, this proposition is true even if R is not assumed to be a free complete P-algebra. We refer towx F for a detailed demonstration. We just sketch the proof in the free case here. Proof. We have to construct an antipode ␫. For convenience, in a coproduct, we label each copy of V by a variable. The canonical projections provide a P-algebra morphism

ˆˆˆ PŽ.Vxy[VªPŽ.V x[P Ž.V y.

This morphism is surjective, and has a linear splitting given by the sum of the canonical inclusions. The kernel of this morphism, denoted by PˆŽVVxy< . is known as the cross-effect of the functor Pˆˆ. An element of PŽVVxy<.is a monomial with at least one factor belonging to the copy Vxy, resp. V . Thus, we have a canonical splitting

PˆˆˆˆŽ.VxymVsPŽ.V x[P Ž.V y[PŽ.VV xy<.

Because of the unit equationŽ. 2.1.4 , the projection of the coproduct ␥ onto the component PˆˆŽ.Vxy, resp. P Ž.V , is given by the canonical mor- phism V ª PˆŽ.V . We fit this decomposition of ␥ into a step by step approximation process. Running this approximation process, we get a map ␫:VªPˆŽ.Vwhich is a solution to the equation

.иRk␫и␥s0ٌ

In the same way, we get ␫Ј: V ª PˆŽ.V such that ٌи␫ЈkRи␥s0. The argument proving the uniqueness of the antipode shows also ␫ s ␫Ј. Let R be a complete P-algebra equipped with a cogroup structure. 2 Recall that QR s RrR .If VsQR is a finitely generated projective K-module, then, using the methods involved inwx F , one may show R s PˆŽ.V. In other words, R is necessarily a finite dimensional P-formal group.

Assume that V is freely generated by the variables x1,..., xn. In this ˆˆ ˆ case, PŽ.Ž.Vxy[ V s P x, y , and a coproduct ␥ : V ª PŽ.Vxy[ V is equiv- alent to an n-tuple of P-power series GxŽ.,y. As a consequence, a P-formal group is equivalent to a P-formal group law: LIE THEORY OF FORMAL GROUPS 475

2.1.4. DEFINITION.Ann-dimensional P-formal group law is an n-tuple of P-power series in 2n-variables Ž.Žx, y s x1,..., xn, y1,..., yn., say

GxŽ.,ysŽ.Gx1 Ž.,y,...,Gxn Ž.,y gPˆ Ž.x,y, such that

GxiiiŽ.,0 sG Ž.0, x s x , i s 1,...,n,Ž. 2.1.6

GGxiŽ.Ž.Ž.,y,zsGx,Gy Ž.,z , is1,...,n.Ž. 2.1.7 Moreover, a cogroup morphism from an nЈ-dimensional P-formal group law GЈŽ.Ž.xЈ, yЈ to Gx,y, is equivalent to an nЈ-tuple of P-power series ␾Ž.x , such that GЈŽ␾ Žx ., ␾ Žy ..s ␾ ŽGx Ž,y ... The antipode of GxŽ.,y is provided by an n-tuple of P-power series ␫Ž.x. These P-power series verify the equation Gx Ž,␫ Ž..x sG ŽŽ.␫x,x .s 0. In this context, the step by step approximation process is an algebraic version of the local inversion theorem. In the case P s Com, we recover the classical definition of a formal group law. For a comprehensive account of this theory, we refer to the following textbookswx H, Z, Se . 2.1.5. Group Functors. Let R be a cogroup object. The coproduct of R makes P-AlgŽ.R, y into a functor from nilpotent P-algebras to the category of groups.

For example, suppose that R is a free P-algebra, say R s PˆŽ.x1,..., xn . Let GxŽ.,ygPˆ Ž.x,y be the P-formal group law associated to the co- n product. In this case, P y AlgŽ.R, A s A . The group structure is given by

Ža1,...,anG .q Žb1,...,bn .sŽ.Ga1 Ž.,b,...,Gan Ž.,b. Conversely, any reasonable group functor turns out to be given by a P-formal groupŽ. cf. the next theorem for a precise statement . In the classical case P s Com, this process is explained with some details inw Z, Chap. IIx . In fact, in this reference, we may assume P to be any operad. For this reason, we just recall the following definitions and then state the theorem without proof. Let G be a functor from nilpotent P-algebras to the category of groups. This functor is known to be smooth when it preserves surjections. For example, if G s P-AlgŽŽ.Pˆ V , y .for some projective K-module V, then GAŽ.sModK ŽV, A .. Hence, in this case G is smooth. Let us consider K as a 2-nilpotent P-algebra. The group structure of GKŽ.turns out to be abelian. In fact, GKŽ.is equipped with a right K-module structure. For example, if G s P-AlgŽ.R, y for some complete P-algebra R, then GKŽ.sModK ŽQR, K .. 476 BENOIT FRESSE

2.1.6. THEOREM. Let G be a functor from nilpotent P-algebras to the category of groups. Assume that G preser¨es the fibered products. Assume that G preser¨es the direct sums. If G is smooth and GŽ. K is a finitely generated projecti¨eK-module, then G is pro¨ided by a P-formal group. 2.1.7. Notation. Following Convention 0.1, we specify a P-formal group by the single letter G. The underlying complete P-algebra is denoted by RGŽ.. This P-algebra is known as the algebra of regular functions of the P-formal group G.

2.2. Examples In this section, we give examples of L ie-formal group laws. 2.2.1. The Campbell᎐Hausdorff Formula. The Campbell᎐Hausdorff for- mula provides an example of a L ie-formal group law. Let ⌽Ž.x, y be the power series in two non-commuting variables which verifies the equation

exp x exp y s exp ⌽Ž.x, y . $ This power series is clearly an As-formal group law. Recall that L ieŽ. x, y $ is a subspace of AsxŽ.Ž,y cf. Examples 1.1.10. . It is not hard to show that $ ⌽Ž.x,yis primitive, and hence belongs to L ieŽ.Ž x, y cf.w H, Theorem 14.4.14x. . In some sense, this formal group law has a L ie-structure. As a conclusion, the power series ⌽Ž.x, y defines L ie-formal group law. In characteristic zero, this is certainly the main example of a formal group law. It follows from the results of Lazardw L1, Theorem 6.1 and Proposition 7.2x , that any P-formal group law is isomorphic to a P-formal group law built from the Campbell᎐Hausdorff formulaŽ cf. also Subsec- tions 2.2.2 and 2.2.3. . We refer towx Se for an account of this result in the classical case of a Com-formal group law. In the next paragraph, following M. Kontsevich, we give an account of a variant of this construction.

2.2.2. Construction Žcf.wx K, p. 186.Ž . For simplicity, we assume P 1.s Ksk. Suppose that P is a quadratic operadŽ. cf. Subsection 1.1.11 , such as P s As, L ie, Com. Inwx G-K , by analogy with the theory of quadratic algebrasŽ cf.wx P. , V. Ginzburg and M. Kapranov define a duality between quadratic operads, known as the quadratic duality. The quadratic dual ! ! ! operad of P is denoted by P . For example, we have As s As, Com s L ie, ! L ie s Com. Let L be a finite dimensional P! algebra. Let A be a P-algebra. The algebra L m A is endowed with a L ie-algebra structure, which is defined as follows. Recall that P!Ž.2 is the K-linear dual of PŽ.2 equipped with the contragredient representation. Let ␮i denote a basis of PŽ.2 and let U ␮i denote the dual basis. The Lie bracket on L m A is given by the LIE THEORY OF FORMAL GROUPS 477 formula

U wxXma,Ymb[Ý␮iiŽ.X,Ym␮ Ž.a,b,X,YgL,a,bgR. i

Moreover, if A is a nilpotent P-algebra then L m A is a nilpotent L ie-algebra. Thus, the Campbell᎐Hausdorff formula endows L m A with the structure of a group. Clearly, this constructionŽ. with L fixed provides a functor from the category of nilpotent P-algebras to the category of groups, which is denoted by exp L. We have L m R ( ModkŽ.L*, R ( P- AlgŽŽPˆ L*, .R .. Therefore exp L is a P-formal group. 2.2.3. Explicit Calculations. Let us give a power series expansion of the ! associated formal group law in the case P s L ie, P s Com. The process is well knownŽ cf.wx R1, Lo2. . Let x1,..., xn denote a basis of L. Consider a sequence of indices IsŽ.i1,...,ir . We denote the integer r by lIŽ.. Consider an n-tuple of variables X ,..., X . We denote by X the monomial X иии X . 1 nIii1r 2.2.4. PROPOSITION. The L ie-formal group law of exp L has the expansion

1 ⌽ w1x Ž.X, Y s Ý xxIJmeXY lŽI.qlŽJ. IJ. I,JlIŽ.Ž.!lJ!

ŽŽWe identify an element of L m PˆX, Y. with an n-tuple of P-power series by using the basis x1,..., xn..

Proof. Let VxyŽ.resp. V denote the vector space generated by the variables X1,..., Xn Ž.Žresp. Y1,...,Yn . Let TVˆ.denote the completed tensor algebra

ϱ n TVˆ Ž.sŁVm . ns0

Consider the associative algebra

TsTVˆˆŽ.Ž.xy[VmTV xy[V.

The first factor is equipped with the shuffle product, and the second with the concatenation product. Consider the element

S [ Ý w m w g T , where w ranges over the set of monomials in X, Y. 478 BENOIT FRESSE

By a theorem of ReeŽ cf.wx Re. , we have

log S s Ý u m Pu , for some Lie polynomials Pu.Ž The variable u ranges over the set of non-constant monomials in X, Y.. In fact, the map u ¬ Pu is the projec- tion onto L ieŽ. Vxy[ V , provided by the first Eulerian idempotentŽ cf. mnw1x Examples 1.1.10.Ž . More precisely, if u g Vxy[ V ., then P uns eu. ˆ Consider the map ␾: TVŽ.xy[V ªLwhich maps the monomials XYII to 1rnIŽ.!nJ Ž.!иxxIJ, and which cancels the other monomials. Clearly, ␾ is an algebra morphism ŽŽTVˆxy[V .being equipped with the shuffle product. . Therefore

␾ m 1 и log S s logŽ.␾ m 1 и S . The right side of this equation is equal to

logŽ. expŽ.x11m X q иии qxnnm X expŽ.x11m X q иии qxnnm X .

The left side of this equation is the sum

1 Ý xxIJmP u. nIŽ.!nJ Ž.! usXYIJ

Hence, we are done.

3. LINEAR ALGEBRA OVER AN OPERAD

Recall that an operad is an associative algebra for the composition product. Therefore, we have an analogy between rings and operads. Following this analogy, we can do linear algebra over an operad. This is the aim of this section. In the first subsection, we define and study the notion of a right module over an operad P. Mainly, we show that the category of right P-modules is symmetric monoidal, and possesses an internal hom functor. In the second subsection, we make the free complete P-algebras dual to the cofree cocommutative coalgebras in the symmetric monoidal category of right P-modules. This duality plays the role of Cartier duality in our setting.

3.1. Right Modules o¨er an Operad 3.1.1. DEFINITION. Fix an operad P, with product denoted by ␮: P( P ª P and unit denoted by ␩: I ª P. LIE THEORY OF FORMAL GROUPS 479

A right P-module V is an S-module together with an S-module morphism ␳ : V ( P ª V such that ␳ и ␳ ( P s ␳ и V (␮, ␳ и V (␩ s V . As for an operad, this means that we have linear maps

␳ : V Ž.n m P Ž.i1 m иии m P Žin .ª V Ži1q иии qin . which satisfy the May axioms, except that we put V in the first position instead of P. A morphism of right P-modules f: V ª W is a morphism of S-modules which commutes with the P-action, i.e., such that

f Ž.¨ Ž.Ž.Ž.Ž.␮1,...,␮nsf ¨ ␮1,...,␮n, ᭙¨gV n , ᭙␮1,...,␮ngP. A right P-module morphism is also known as a P-linear map. We denote by ModP the category of right P-modules.

3.1.2. EXAMPLE. Let E be a right K-module. The free right P-module generated by E, denoted by EP , is given by the S-module

EnPŽ.[EmK P Ž.n, equipped with the obvious right P-action.Ž Recall that the operad product PŽ.1mP Ž.nªP Ž.nmakes P Ž.n into a left K-module. . Let V be a right P-module. Any right K-module morphism E ª V Ž.1 gives rise to a unique right P-module morphism EP ª V . Let V be a left K-module. Let ModK Ž.V, P be the sequence ModKKŽ.Ž.V, P n s Mod ŽŽ..V, P n . We equip ModKŽ.V, P with the obvi- ous right P-module structure, defined by

f Ž␮1,...,␮n .Ž.¨ [f Ž.Ž¨ ␮1,...,␮n ., for f g ModK Ž.Ž.V, P n , ␮1,...,␮nkgP, ¨gV. The module Mod Ž.V, P is a kind of ‘‘ P-linear dual’’ of V. Suppose V to be a finitely generated projective K-module. In this case, we have a canonical isomorphism

( ModKKŽ.V, K m P Ž.n ª Mod KŽ.V, P Ž.n , and hence, ModK Ž.V, P is a free right P-module. 480 BENOIT FRESSE

3.1.3. PROPOSITION. Let V , W be two right P-modules. There exists a unique right P-module structure on V m W such that ␮ ␮ ␮ ␮ ␮ ␮ Ž.Ž.Ž¨ m w 1,..., iqj[¨ 1,..., ii .Žmw q1,..., iqj .,

᭙¨gVŽ.i,᭙wgW Žj .,᭙␮1,...,␮ngP.

Proof. Consider an arbitrary element ␴ и w IndSN V Ž.i ¨ m g Sij=S m WŽ.j;VmW ŽN .. The equivariance condition on the product forces us to set

Ž.Ž.␴ и ¨ m w ␮1,...,␮n ␴␮<< <<␮ и ␮ ␮ ␮ ␮ [Ž.1,..., n¨Ž.Ž.␴1,..., ␴imw ␴Žiq1.,..., ␴Žiqj.. We let the reader check that this definition is consistent and provides V m W with a right P-module structure.

3.1.4. COROLLARY. The category Ž.ModP , m is symmetric monoidal.

Proof. We have just to check that the twisting map cV , W : V m W ª W m V is P-linear. The other verifications are obvious. We check first

Ž.cV,WŽ.Ž.¨ m w ␮1,...,␮n

s␶Ž.<<<<¨,wwm¨. Ž␮1,...,␮n . ␶␮<<иии <<<<␮ ␮ иии <␮ < ␮ ␮ sŽ.1q q ii, q1q q iqjiwŽ.q1,..., iqj

m¨Ž.␮1,...,␮i

scV,WŽ.Ž.Ž.¨mw␮1,...,␮n,

and the general case follows by Sn-equivariance. 3.1.5. Remark. Recall that I is equipped with an operad structureŽ cf. Subsection 1.1.8. . Furthermore, a right I-module structure is nothing more

than a right S-module structure, and the monoidal categoriesŽ. ModI , m and Ž.S-Mod, m are clearly isomorphic. 3.1.6. Shifted Modules. Let V be an S-module. Let n be a non- negative integer. We define an S-module V wxn , known as the shifted module. We set

VwxnrŽ.[V Žnqr ..

Consider the embeddingÄ4Ä 1, . . . , r ¨ 1,...,nqr 4mapping 1, . . . , r onto n q 1,...,nqr. This embedding provides a group embedding Srn¨ S qr, and thus the Sr-module structure of V wxnrŽ.sV Žnqr .. LIE THEORY OF FORMAL GROUPS 481

Let ¨ g V wxnrŽ.sV Žnqr .. The integers 1, . . . , n are called the n first entries of ¨ and n q 1,...,nqr the last r entries of ¨. Hence, in our definition, Sr acts on the last r entries of ¨. Notice that V wxn is an Snn-Ž.S-module . In fact, S acts on the n first entries of ¨ g V wxnrŽ.. This Snr-action commutes obviously with the S - module structure, since the supports of these actions are disjoint. 3.1.7. Right P-Actions on Shifted Modules.IfVis a right P-module, then V wxn is equipped with a right P-module structure. Let ¨ g V wxnrŽ.. We provide V wxn with a right P-module structure by letting P act on the last r entries of ¨. More precisely, the image of ¨ under the product

VwxnrŽ.mP Ži1 .mиии m P Žir .ª V wxni Ž1qиии qir .

mn is given by ¨Ž1,␮1,...,␮r .ŽgV nqi1 qиии qir .s V wxniŽ.1 qиии qir . The operad P may also act on the i first entries of ¨. Thus we obtain maps

VwxnrŽ.mP Ži1 .mиии m P Žinnn .ª V wi q иии qir xŽ..

These maps are P-linear by associativity of the right P-module product. As for classical graded modules, we use the shifted module in the construction of an internal Hom.

3.1.8. THEOREM. Let V , W be right P-modules. The bifunctor

HOMŽ.Ž.V , W n [ Mod PŽ.V , W wxn is an internal hom in the monoidal category Ž.ModP , m . Explicitly, we ha¨ea canonical isomorphism

Mod PŽ.U m V , W ( Mod PŽ.U , HOM Ž.V , W .

Proof. First, we make HOMŽ.V , W into a right P-module. Let f g HOMŽ.Ž.ŽV , W n s Mod P V , W wxn .. Recall that f consists in a sequence of maps f: V Ž.r ª W Žn q r ., which commute with the right P-module product. Let ␮11g PŽ.i ,...,␮nngP Ž.i . We define

fŽ.Ž.Ž.␮1,...,␮ngHOM V , W i1q иии qin

sMod PŽ.V , W wxi1q иии qin as follows. Let ¨ g V Ž.r . We set

f Ž.Ž.Ž.Ž.␮1,...,␮n¨ [f ¨ ␮1,...,␮n,1,...,1 . 482 BENOIT FRESSE

Once again, fŽ.␮1,...,␮n is obviously P-linear, because of the associativ- ity of the right P-module product. Clearly, this product provides HOMŽ.V , W with a right P-module structure.

Let U, V , W g ModP . It remains to show that

Mod PŽ.U m V , W ( Mod PŽ.U , HOM Ž.V , W .

An arrow f g S-ModŽ.U m V , W is given by a sequence f: Ž.Ž.U m V r ª WŽ.r, or equivalently, by a sequence of Sij= S -equivariant arrows

fi,j:UŽ.imV Žj .ªW Žiqj ..

Now, the map fi, j corresponds to a map g : U i Mod V j , W i j . i,jSŽ.ª jŽ.Ž.Ž.q Moreover, f is P-linear if and only if

Ž.1 for each i, for each u g U Ž.i , the sequence guii Ž .[ Žgu,jj Ž ..: VŽ.j ªWwxijŽ.is P-linear, i.e., gi: U Ž.i ª Mod P ŽV , W wxi .,

Ž.2 the sequence g [ Ž.gii is P-linear, i.e., g g Mod PŽU, HOMŽ..V , W . This completes the proof of the theorem.

3.1.9. COROLLARY. Let V , W be S-modules. The bifunctor HOMŽ.Ž.V , W i [ S-ModŽ.V , W wxi is an internal hom in the monoidal categoryŽ. S-Mod, m . 3.1.10. Composition. By some general nonsense arguments we have associative composition laws

HOMŽ.V , W m HOM Ž.U , V ª HOM Ž.U , W . Indeed, let g g HOMŽ.Ž.V , W i , f g HOM Ž.Ž.U, V j . Then g и f g HOMŽ.Ž.U, W i q j is given by the sequence of maps fg UŽ.rªV Žjqr .ªW Žiqjqr ..

In particular, if V g ModP , then ENDŽ.V s HOM ŽV , V .is an associa- tive algebra in the categoryŽ. ModP , m .

3.2. Coalgebra-Algebra Duality 3.2.1. Coalgebras. In this section, we consider coalgebras in the sym- metric monoidal category of right P-modules. A coalgebra is a right P-module C, equipped with a coassociative coproduct ⌬: C ª C m C and LIE THEORY OF FORMAL GROUPS 483 an augmentation: ⑀: C ª 1. Explicitly, ⌬ and ⑀ satisfy the usual identities ⌬ m C и⌬sCm⌬и⌬, Cm⑀и⌬s⑀mCи⌬sC.

The coalgebra C is known to be cocommutati¨e when ⌬ s cC, C и⌬. The coalgebra C is known to be connected when ⑀ induces an isomorphism from CŽ.0 into k. In the sequel, any coalgebra is assumed to be connected. In this case, we have CŽ.0 s k и 1 and ⌬1 s 1 m 1. Furthermore, the coproduct of c g CNŽ.has the expansion i i i ⌬c s c m 1 q Ý ␴ и cŽ1.Žm c 2.q 1 m c, Ž.c

ii i with ␴ g Sn, cŽ1.Žg CpŽ.,c2.gCqŽ., for some p, q - n such that p q q sn. Here, we use Sweedler’s notation. More generally, given c g C,an i i i expression like ÝŽc.Ž␴ и c 1.Žm иии m c n.denotes the n-fold coproduct of c. Sometimes we may omit the summation index i. Using Sweedler’s nota- tion, the cocommutativity of the coproduct may be written

ÝÝ␴иcŽ1.Žmc2.Žs␴␶ Ž.<<<

3.2.2. Cofree Coalgebras. A right P-module V is known to be con- nected when V Ž.0 s 0. There exists an obvious forgetful functor from the category of cocommutative coalgebras to the category of connected right P-modules. This functor has a right adjoint known as the cofree cocommu- tati¨e coalgebra. We denote by CŽ.V the cofree cocommutative coalgebra cogenerated by V . This coalgebra is endowed with a P-linear map ␲:CŽ.VªPand is characterized by the following universal property. Let C be a cocommutative coalgebra; let ␲ Ј: C ª V be a P-linear map. There exists a unique coalgebra morphism ␾: C ª CŽ.V such that ␲␾ s ␲Ј. Let ¨ 1,...,¨nNgV , let ␴ g S , with N s <<¨ 1q иии q <<¨n. We denote by mn ␴иŽ.¨1,...,¨n the tensor ␴ и ¨ 1 m иии m ¨n g V . Recall that the sym- mn metric group Sn acts on V by place permutation, and that this action is given by

s*Ž.␴иŽ.¨1,...,¨ns␴иsŽ.<<¨1,..., <<¨nsиŽ.¨ Ž1.,...,¨sŽn..

We denote by ŽV mn.Snthe right P-module of invariant tensors. Consider the right P-module ϱ n Sn CŽ.V s ŽV m .. [ns0 484 BENOIT FRESSE

We equip CŽ.V with the coproduct defined on each summand by the canonical inclusion

nSSSnpqpq Ž.Ž.Ž.Vm;VmmVm,

i ii mnSn with p q q s n. Hence, if Ýi ␴ и Ž¨Ž1.Ž,...,¨ n..Žg V ., then we have

⌬␴iиi,..., i ž/ÝŽ.¨Ž1.Ž¨ n. i ␴iиi i i i sÝŽ.Ž.¨Ž1.Ž,...,¨ p.Žm ¨ pq1.Ž,...,¨ pqq.. pqqsn

3.2.3. THEOREM. The coalgebra CŽ.V is the cofree cocommutati¨e coal- gebra cogenerated by V . The uni¨ersal arrow ␲ : CŽ.V ª V is gi¨en by the projection of CŽ.V onto the summand V .

Proof. Let ␲ Ј: C ª V be a right P-module morphism. First, we check the existence of a coalgebra morphism ␾: C ª CŽ.V such that ␲ Ј s ␲␾. Consider the map

i i i ␾ Ž.c [ Ý␴ и ž/␲ ЈŽ.cŽ1.Ž,...,␲Ј Ž.c n.. n

Notice that ␾ is well-defined, because C is connected and V Ž.0 s 0. The map ␾ is obviously P-linear. Moreover, we have

i i i ⌬␾ Ž.c s Ý␴ и ž/␲ ЈŽ.cŽ1.Ž,...,␲Ј Žc p. . pqqsn ␲Јci,...,␲Ј ci . mž/Ž.Žpq1.Ž Ž.pqq.

Because of the coassociativity of the coproduct, this last expression is exactly ␾ m ␾ и⌬Ž.c . Hence, ␾ is a coalgebra morphism. It remains to show the uniqueness of such a factorization. Let us call n S ŽVm . n the component of order n of CŽ.V . Let ␺ : C ª CŽ.V be i ii another factorization of ␲ Ј. Write ␺ Ž.c s Ýn ␴ и Ž¨ 1,...,¨n.. Since ␺ is a coalgebra morphism, we have

n n ⌬ ␺ Ž.c s ␺ m иии m ␺ и⌬ Ž.c , where ⌬n denotes the n-fold coproduct. On one hand, the multilinear componentŽ i.e., the component of order one in each factor of the tensor LIE THEORY OF FORMAL GROUPS 485 product. of the left side of this equation is

i i i Ý ␴ и ¨ 1 m иии m ¨n . i

On the other hand, the multilinear component of the right side is

i i i Ý␴ и ␲ ЈŽ.cŽ1.Žm иии m ␲ Ј Ž.c n., i because the restriction of ␺ to V is given by ␲ Ј by hypothesis. Hence, the component of order n of ␺ Ž.c agrees with that of ␾Ž.c . 3.2.4. Shuffle Coproduct on the Symmetric Algebra. For the needs of the Poincare´᎐Birkhoff᎐Witt theorem, we supply another realization of the cofree cocommutative coalgebra. We denote by ŽV mn. the coinvariants Sn of the tensor power under the action of the symmetric group. Consider the right P-module

ϱ mn SŽ.V s ŽV .Sn. [ns0

Let X1,..., XnNgV and let ␴ g S , with N s <

s*Ž.␴иX1иии Xns ␴ sXŽ.<<1,..., <

Let S h p, q denote the set of p, q-shuffles, i.e., the set of permutations Ž.иии Ž . Ž .иии Ž . sgSpqq such that s 1 - - spand spq1- - spqq.We equip SŽ.V with the coproduct defined by

⌬Ž.␴иX1иии Xns ÝÝ␴ sXŽ.<<1,..., <

3.2.5. THEOREM. The coalgebra morphism ␾: SŽ.V ª C Ž.V induced by the projection onto the linear component of SŽ.V is a coalgebra isomor- phism. Proof. By an immediate calculation, we obtain

␾␴Ž.иX1иии Xns Ý␴ sXŽ.<<1,..., <

Hence, ␾ is clearly an isomorphism, with an inverse isomorphism given by

1 y1 ii i ii i ␾␴иXŽ1.Ž,..., X n.Žs ␴ иX1.Žиии X n., ž/ÝÝŽ.n! ii

ii i mnSn for Ýi ␴ и Ž.Ž.XŽ1.Ž,..., X n.g V . 3.2.6. Coalgebra-Algebra Pairings. Let C be a coalgebra; let R be a complete P-algebra. We equip the S-module Ž.Ž.Ž.C m Rn[CnmR with the obvious right P-action. A coalgebra-algebra pairing between C and R is a P-linear map

B: C m R ª P, which makes the coproduct of C adjoint to the product of R. More precisely, we have

i i i BcŽ.,␮Ž.r1,...,rnsÝ␴ и␮ž/BcŽ.Ž1.,r1,...,Bc Ž.Žn.,rn,

᭙␮gPŽ.n,᭙cgC,᭙r1,...,rngR.

Notice that PŽ.0 s 0 implies BŽ.1, r s 0, ᭙r. 3.2.7. The Canonical Coalgebra-Algebra Pairing. Let V be a left K- module. Let us consider the cofree coalgebra CŽŽ..ModK V, P . We define a canonical coalgebra-algebra pairing

ˆ ²:y,y:CŽ.ModKŽ.V, P m P Ž.V ª P as follows. Let c be any homogeneous element of order n of i ii ii CŽŽ..ModKiV, P . Hence, c s Ý ␴ и ŽXŽ1.Ž,..., X n.Ž., with X 1.Ž,..., X n.g

ModK Ž.V, P . Let r s ␮ Ž¨ 1,...,¨m .Ž.gPˆV . We set

␴iiи␮²:²:X,,..., X i, ,ifnm, ¡ÝŽ.Ž1.¨1Žn.¨ns ²:c,rs~i ¢0, otherwise.

i ii This pairing is well defined since Ýi ␴ и Ž XŽ1.Ž,..., X n..is Sn-invariant. It is straightforward to check that it makes the coproduct of CŽŽ..ModK V, P adjoint to the product of PˆŽ.V .

Since SŽŽ..ModKKV, P is isomorphic to C ŽŽ..Mod V, P , we obtain an isomorphic pairing between SŽŽ..Ž.ModK V, P and PˆV . As a conse- quence, in the next three lemmas, we can equivalently replace the coalge- bra CŽŽ..ŽŽ..ModKKV, P by S Mod V, P . LIE THEORY OF FORMAL GROUPS 487

Let us write a concrete formula for this pairing ˆ ²:y,y:SŽ.ModKŽ.V, P m P Ž.V ª P.

Let c be any homogeneous element of order n of SŽŽ..ModK V, P . Hence, csÝ␴иX1 иии Xn,with X1,..., XnKgMod Ž.V, P .Let rs ␮Ž.Ž.¨1,...,¨m gPˆV . We have

␴иs␮²:²:X,,..., X , ,ifnm, ¡ ÝŽ.Ž.sŽ1.¨1sŽn.¨ns ²:c,rs~sgSn ¢0, otherwise.

3.2.8. LEMMA. Let ²:y , y : C m Pˆ Ž.V ª P be a coalgebra-algebra pairing. There exists a unique coalgebra morphism ⑀CK: C ª CŽŽ..Mod V, P , such that

²:²:⑀CŽ.c,rsc,r,᭙cgC,᭙rgPˆŽ.Ž.V. 3.2.1

Proof. By restriction of such a pairing to V, we obtain a P-linear map

␲ Ј: C ª ModKŽ.V, P . For the universal coalgebra-algebra pairing, this P-linear map is the canonical projection ␲ : CŽŽ..Ž.ModKKV, P ª Mod V, P . The identity

²:²:⑀CŽ.c,¨sc,¨,᭙¨gV is equivalent to ␲⑀CKs ␲Ј. Since CŽŽ..Mod V, P is cofree, there exists a unique coalgebra morphism ⑀CC, such that ␲⑀ s ␲Ј. It remains to show that ⑀C verifies Eq.Ž. 3.2.1 . Let ¨ 1,...,¨n gV. We have

²:⑀CŽ.c,␮ Ž¨1,...,¨n .

sÝ␴␮Ž.²:²: ⑀CŽ.cŽ1., ¨ 1,..., ⑀C Ž.cŽn.,¨n Ž.c

sÝ␴␮ Ž.²:²:cŽ1., ¨ 1,..., cŽn.,¨n Ž.c

s²:c,␮Ž.¨1,...,¨n. Thus, we are done.

3.2.9. Remark. For the coalgebra C s SŽŽ..ModK V, P , this map is obviously the canonical isomorphism SŽŽ..ŽŽ..ModKKV, P ª C Mod V, P . 488 BENOIT FRESSE

3.2.10. LEMMA. Assume V to be a finitely generated projecti¨eK-module. Let

²:y,y:CŽ.ModKŽ.V, P m R ª P be a coalgebra-algebra pairing. There exists a unique K-linear map ␩R: R ª PˆŽ.V such that

²:²c,rsc,␩RKŽ.r :, ᭙cgCŽ.Mod Ž.V, P , ᭙r g R.

Proof. Since V is finitely generated and projective, we have

SN Ind S=иии=SKMod Ž.V, PŽ.i1иии ModKn Ž.V, PŽ.i ii1n m m ( mnSN ModKSmnV , Ind =иии=S PŽ.i1 иии P Ž.in . ª ž/ii1nm m Thus

mn ( mn mn ModKKŽ.V, P ª Mod mn ŽV , P .. And finally

Sn Mod V, P mn ( Mod V mn , Pmn . Ž.KKŽ. ª XSnŽ.

In particular, we have

Sn Mod V, P mn n ( Mod V mn , K S . Ž.KKŽ.Ž.ª XSnnŽ.X

In effect, Pmn is both an nth power and a right P-module. As a mn consequence P Ž.n is a K X Sn-bimodule, which turns out to be isomor- phic to K X Sn. Let us equip Mod ŽV mn, K S . R with the K-module structure K X SnnX m induced by that on R.If V is a finitely generated projective K-module, mn then V is a finitely generated projective K X Sn-module. Thus, we have an isomorphism

Mod R, P n V mn KKŽ.Ž.m XSn

(Mod Mod V mn, K S R, P n . ªKKŽ.XSnnŽ.X m Ž.

Therefore, there exists a unique K-linear map

␩ : R P n V mn nKª Ž.m XSn LIE THEORY OF FORMAL GROUPS 489 such that

␴ i и X i ,..., Xi ,r ¦;ÝŽ.Ž1.Žn. i

␴iиXi,..., Xi ,␩ r , 3.2.2 s¦;ÝŽ.Ž1.Žn.RŽ. Ž . i for any tensor ␴ i и X i ,..., Xi Mod V mn , K S C Mod V, P n . ÝŽ.Ž1.Žn.g KXSnKnŽ.X : Ž.Ž.Ž. i

mn Moreover, since V is a finitely generated projective K X Sn-module, we have ( Mod V mn , K S Pmn N Mod V mn , Pmn N . KXSnKnnŽ.X m XSKŽ.ª XS nŽ.Ž. As a consequence, ϱ C Mod V, P Mod V mn , K S Pmn. Ž.KKŽ.s XSnKnnŽ.X m XS [ns0 It follows that Eq.Ž. 3.2.2 is true for any tensor

i i i Ý␴ и Ž.XŽ1.Ž,..., X n.gCŽ.ModKŽ.V, P . i This completes the proof of the lemma.

3.2.11. LEMMA. The K-linear map supplied by the pre¨ious lemma ␩R: R ªPˆŽ.VisaP-algebra morphism. ˆ Proof. Since the P-algebra product ␭R: PŽ.R ª R is a P-algebra morphism, by restriction, the coalgebra-algebra pairing

²:y,y:CŽ.ModKŽ.V, P m R ª P provides a coalgebra-algebra pairing between CŽŽ..Ž.ModK V, P and PˆR . We have

²:c,␩␮RŽ.Ž.r1,...,rn

s²:c,␮Ž.r1,...,rn

i i i sÝ␴и␮ž/²:²:cŽ1.,r1,..., cŽn.,rn i

ii i sÝ␴и␮ž/²:²:cŽ1.,␩RŽ.r1,..., cŽn.,␩RnŽ.r i

s²:c,␮␩Ž.RŽ.r1,...,␩Rn Ž.r . 490 BENOIT FRESSE

And hence, for any c g C, x g PˆŽ.R , we have

²:c,␩RRи␭Ž.xs²:c,␭RRиPˆŽ␩ .Ž.x.

As a consequence, by the previous lemma, we have ␩RRи ␭ s ␭ Rи PˆŽ.␩ R, i.e., ␩R is a P-algebra morphism. 3.2.12. COROLLARY. Let V, W be finitely generated projecti¨eK-modules. The adjunction relation pro¨ides a one-one correspondence between the set of coalgebra morphisms

coAlg PŽ.CŽ.Ž.ModKKŽ.W, P , C Mod Ž.V, P and the set of P-algebra morphisms

Pˆˆˆ-AlgŽ.PŽ.V , P ŽW ..

3.2.13. Perfect Coalgebra-Algebra Pairings. Let B: C m PˆŽ.V ª P be a coalgebra-algebra pairing. This pairing is said to be perfect, when V is a finitely generated projective K-module, and when the associated coalgebra morphism ⑀CK: C ª Mod Ž.V, P is an isomorphism. Obviously, the previ- ous corollary remains valid for any pair of perfect pairings, in place of the ˆ standard ones CŽŽ..Ž.ModK V, P m P V ª P. Let C12, C be cocommutative coalgebras. The tensor product C12m C is a realization of the product in the category coAlgP . The next lemma follows as obvious nonsense. In this lemma, we consider the canonical arrows

pp12 C1122¤CmCªC, ii ˆˆ12 ˆ PŽ.V1122ªP ŽV[V .¤P Ž.V.

ˆˆ 3.2.14. LEMMA. Let B11: C m PŽ.V 1ª P, B 22: C m PŽ.V 2ª P be coalgebra-algebra pairings. There exists a unique coalgebra-algebra pairing between C12m C and PˆŽ.V 12[ V which makes the canonical projections of the product p12, p adjoint to the canonical embeddings i12, i . Explicitly

²:c,ia11Ž.sBpc 1Ž. 1 Ž.,a 1, ᭙cgC1mC 2,᭙a 1gPˆŽ.V 1,

²:c,ia22Ž.sBpc 2Ž. 2 Ž.,a 2, ᭙cgC1mC 2,᭙a 2gPˆŽ.V 2.

Furthermore, this pairing is perfect when B12 and B are perfect. LIE THEORY OF FORMAL GROUPS 491

4. CARTIER DUALITY AND THE MAIN THEOREM OF LIE THEORY

This section is devoted to the Lie theory of formal groups over an operad. In the first subsection, we study the P-linear Lie algebrasŽ the Lie algebras in the symmetric monoidal category of right P-modules. . In a second subsection, we show that the tangent space of a P-formal group is a P-linear Lie algebra. In the third subsection, we show that the tangent space induces an equivalence of categories between P-formal groups and some finite dimensional P-linear Lie algebras.

4.1. P-Linear Lie Algebras

4.1.1. DEFINITION. A Lie algebra inŽ. ModP , m is called a P-linear Lie algebra.A P-linear Lie algebra consists in a right P-module L equipped with a P-linear map ␥ : L m L ª L, which is antisymmetric and satisfies the Jacobi relation. Explicitly, we have

␥ q ␥ и cL,Ls 0,Ž. A 2 ␥и1m␥иŽ.1qc#qc#s0,Ž. J where c denotes the standard 3-cycleŽ. 1 2 3 g S3. Let X, Y g L.We denote by wxX, Y the image of X m Y under ␥. RelationsŽ. A and Ž. J are equivalent to the relations

wxX,Yq␶Ž.<<<

AL. Let X, Y g A. We have

wxX,YsmXŽ.Ž,Y y␶<<<

4.1.3. The P-Linear En¨eloping Algebra. Let V be any right P-mod- ule. We denote by TŽ.V the tensor algebra generated by V , defined as the right P-module

ϱ n T Ž.V s V m , [ns0 equipped with the concatenation product. As for classical modules, TŽ.V is the free associative algebra generated by V . Let L be a P-linear Lie algebra. The P-linear en¨eloping algebra of L, denoted by ULŽ., is the quotient of the tensor algebra generated by L,by the ideal generated by the relations

XmYy␶Ž.<<<

This ideal is clearly a sub- P-module of TLŽ., hence UL Ž.is a P-linear associative algebra. The canonical P-linear map L ª TLŽ.induces a Lie algebra morphism L ª ULŽ.. Moreover, any Lie algebra morphism L ª ALextends clearly to a unique associative algebra morphism ULŽ.ªA. 4.1.4. Hopf Algebras in the Right P-Modules.AHopf algebra in

Ž.ModP , m consists of a P-linear associative algebra H, equipped with a coassociative coproduct ⌬: H ª H m H, and an augmentation ⑀: H ª 1. Moreover, the coproduct and the augmentation are assumed to be algebra morphisms. We do not require the existence of an antipode in this definition. Recall that we tacitly assume H to be connected. Hence, as in the classical case, the antipode always exists. An element of H, say X, is known to be primiti¨e when ⌬ X s X m 1 q 1 m X. We denote by Prim H the right P-module of the primitive ele- ments. The P-linear enveloping algebra is equipped with a canonical Hopf algebra structure. As in the classical theory, we define a Lie algebra morphism ⌬: L ª ULŽ.mUL Ž., by setting

⌬ X s X m 1 q 1 m X. More generally, the usual tricks for classical enveloping algebras apply to P-linear enveloping algebras. For instance ULŽ.is endowed with an increasing filtration, defined by

n F ULŽ.[Span ŽX1иии Xn, for X1,..., XngL.. LIE THEORY OF FORMAL GROUPS 493

4.1.5. THEOREM ŽPoincare´᎐Birkhoff᎐Witt . . Let ␾: SL Ž.ªUL Ž. be the P-linear map defined by 1 ␾␴Ž.иX иии X [ ␴ и sXŽ.<<,..., <

This map is a coalgebra isomorphism. Proof. The proof follows exactly the classical oneŽ cf.w Q1, Appendix Bx. , except that we must replace the classical Koszul signs by the appropri- ate permutations. Therefore, we only sketch it. The map ␾ is clearly well defined. In order to show that it is a coalgebra morphism, we expand ␾ m ␾ и⌬Ž.Ž.X1 иии Xn and ⌬␾ X1 иии Xn . Using the fact that any permutation can be written either as s и t12[ t or as ts, where sis a p, q-shuffle, t1 g Sp, t2 g Sqp, t g S qq, we see that these two expressions are equal. It remains to check that ␾ is an isomorphism. This can be done as in wxQ1, Appendix B . Another method consists in showing that gr ␾ is an isomorphism Ž.␾ preserves obviously the filtration by using the basis of the k-module ULŽ.given bywx St, Theorem 11.3 . Further, results on classical enveloping algebras may be generalized to our framework. In fact we can readily translate Appendix B ofwx Q1 to the context of right P-modules. In particular, we want to mention the follow- ing Milnor᎐Moore theorem. This theorem remains valid in a characteristic free contextŽ cf.wx St. . 4.1.6. THEOREM Ž.Milnor᎐Moore . The en¨eloping algebra functor in- duces an equi¨alence of categories between the category of P-linear Lie algebras and the category of connected cocommutati¨e Hopf algebras. Its in¨erse is gi¨en by the primiti¨e elements functor. 4.1.7. Relationship with the Notion of a Lie Algebra in P. To conclude, we explain how to relate the notion of a P-linear Lie algebra to the notion of a Lie algebra in P due to V. Ginzburg and M. KapranovŽ cf.w G-K, 2.2.12x. . For simplicity, we assume K s k. Recall that a Lie algebra bracket on V is equivalent to an operad morphism L ie ª E ndV . Similarly, a Lie algebra in P consists of a k-module V together with an operad morphism Lie ª E ndVVVm P, where ŽE nd m P .Ž.n s E nd Ž. n m P Ž.n is equipped with the obvious operad product. If P s Com, then PŽ.n s k, and we have E ndVVm Com s E nd . As a consequence, a Lie algebra in Com is nothing but a Lie algebra.

Let L s VP . As for Lie algebrasŽ. cf. Example 1.1.13 , the operad morphism L ie ª E ndV m P provides an S2-anti-equivariant bracket wxy,y:VmVªVmPŽ.2.By P-linearity, this bracket induces an anti- 494 BENOIT FRESSE symmetric bracket wxy, y : L m L ª L. By a straightforward calculation Ž.similar to Example 1.1.13 , we show that this bracket satisfies the Jacobi relation. Conversely, let L be a free right P-module equipped with a P-linear Lie algebra structure. By restriction, we obtain wxy, y : LŽ.1 m LŽ.1ªL Ž.2sL Ž.1mP Ž.2 . It turns out that this bracket induces an operad morphism

L ie ª E ndLŽ1. m P.

Clearly, these two constructions are inverse to each other, and give an equivalence between Lie algebras in P and the P-linear Lie algebras, whose underlying right P-module is free. Assume that P is a quadratic operad. In this case, byw G-K, Theorem 2.2.13x , a structure of Lie algebra in P is equivalent to a P!-algebra product. Hence, the category of P-linear Lie algebras is equivalent to the category of P!-algebras. Let L be a P!-algebra. The underlying right

P-module of the associated P-linear Lie algebra is L P . Let us write the formula of the Lie bracket. Recall that the tensor product of a P-algebra and a P!-algebra is equipped with a Lie bracket. In fact, this Lie algebra structure is obtained ! by restriction of structure through an operad morphism l: L ie ª P m P. U ! Let ␮iibe a basis of PŽ.2 . Let us denote by ␮ the dual basis of P Ž.2 . Let ␥ be the generator of L ieŽ.2 . According to the definition of the Lie bracket inŽ. 2.2.2 , we define the operad morphism l by

U lŽ.␥ s Ý␮iim ␮ .

Therefore, if X1122m ␯ , X m ␯ g L m P, then we have

U wxX1122m␯,Xm␯sÝ␮iŽ.Ž.X12,Xm␮␯i12,␯ .

4.2. The Lie Algebra of a P-Formal Group 4.2.1. The Tangent Space of a P-Algebra. Let A be a complete P-alge- bra; its tangent space is defined to be the right P-module ModK Ž.QA, P . Namely, the tangent space of A is the P-linear dual of QA, the K-module of the indecomposable elements of R. Consider the algebra of the regular functions of a P-formal group G.In this section, we show that the tangent space of this algebra is equipped with a natural Lie bracket. Furthermore, this bracket makes it into a P-linear Lie algebra. We denote by Lie G the resulting P-linear Lie algebra, known as the Lie algebra of G. First, we interpret the tangent space as a right P-module of derivations. LIE THEORY OF FORMAL GROUPS 495

4.2.2. Bimodule o¨er an Operad. Let Q be an operadŽ we do not assume QŽ.0 s 0.A . Q-bimodule is an S-module endowed with a collec- tion of products

QŽ.n m Q Ž.i1 m иии m M Žikn .m иии m Q Ži .ª M Ži1q иии qin .,

MŽ.nmQ Ž.i1 mиии m Q Žin .ª M Ži1q иии qin ..

These products are supposed to satisfy the May axioms, in which one occurrence of the operad Q is replaced by M. This definition is equivalent towx Ma, Definition 1.3 .

4.2.3. Remark. As for algebras, the S-module Mq[ Q [ M is equipped with a natural operad structure. Moreover, Mq is an abelian group in the category of operads over Q. The functor M ª Mq is an equivalence of categories from the Q-bimodules to the abelian groups in the category of operads over Q.

4.2.4. Operad Deri¨ations.AP-operad Q is an operad together with a fixed operad morphism P ª Q ŽŽ..we do not assume Q 0 s 0 . Let M denote a Q-bimodule. A P-linear deri¨ation from Q to M is an S-module morphism d: Q ª M wxn , whose restriction to P vanishes, and which satisfies the Leibniz identity. We denote the integer n by <

mn dŽ.␮␯Ž.Ž.1,...,␯rs d␮ Ž.1 ,␯1,...,␯riqÝ⑀ и␮␯Ž.1,...,d␯ir,...,␯ , i

᭙␮gQŽ.r,᭙␯1,...,␯rgQ,Ž. 4.2.1

where ⑀i denotes the block permutation

⑀isŽ.1иии i q 1 Ž.<<

In particular, the derivation commutes with the left and the right P-action up to a permutation. In the case <

4.2.5. LEMMA. LetAbea P-algebra. Let M be a UP Ž. A -bimodule. Let d:AªMŽ.n satisfy the Leibniz relation

dŽ.␮Ž.Ža1,...,ansÝ␮ a1,...,dain,...,a ., i

᭙a1,...,angA.Ž. 4.2.2

The map d extends to a unique P-linear deri¨ation d: UAPŽ.ªMwxn.

4.2.6. COROLLARY. By restriction through the augmentation ⑀: UAPŽ.ª P,the ground operad P is equipped with a UP Ž. A -bimodule structure. We ha¨e a canonical isomorphism

ModKŽ.QA, P s Der PŽ.UAP Ž.,P.

Proof. In effect, an element of ModK Ž.QA, P is a K-linear map 2 d: A ª P which vanishes over A . This is equivalent to the Leibniz identityŽ. 4.2.2 .

In particular, when G is a P-formal group, we have Lie G s Der P ŽŽ..ŽUGP ,P. Recall that UGP Ž.denotes the completed enveloping operad of RGŽ... In the next lemma, we provide a linear extension process for derivations. Using this lemma, we equip any module of P-linear derivations with an external Lie bracket. In turn, by composition with the coproduct of the formal group, we obtain the Lie bracket of Lie G.

4.2.7. LEMMA. Let Q, QЈ be P-operads. Let QЉ be a Q-operad. Let Ѩ:QЈªQЉwxnbeaP-linear deri¨ation. By embedding Q into the first summand of Q kP QЈ, we pro¨ide Q kP QЈ with the structure of a Q- operad. There exists a unique Q-linear deri¨ation

QkPѨ:QkP QЈªQkP QЉwxn, whose restriction to the second summand is gi¨en by Ѩ followed by the canonical embedding into the second summand. Proof. This is an immediate consequence of the form of the operad coproduct and of the Leibniz formula.

Let Q be a P-operad. Let d1122: Q ª Pwxn , d : Q ª P wxn be P-linear derivations. By the previous lemma, we have a P-linear derivation

QkPd2:QkPQªQwxn2. LIE THEORY OF FORMAL GROUPS 497

Recall that we have a P-linear composition product

HOMŽ.Q, P m HOM ŽQ kPQ, Q .ª HOM ŽQ kPQ, P ..

We denote by d12ଙ d : Q kPQ ª Pwxn12q n the P-linear map defined as the composite d1 и Ž.Q kP d2 . Notice that this product is associative by uniqueness of the linear extension of a P-linear derivation.

The external bracket of d12and d is the P-linear map

d12ଙdy␶Ž.n 1221,nиdଙd:QkPQªPwxn12qn. As in classical commutative algebra, the next lemma is an immediate exercice.

4.2.8. LEMMA. The external bracket of d12 and d is a P-linear deri¨ation. 4.2.9. Construction of the Lie Algebra Bracket. Let X, Y be elements of the Lie algebra Lie G. We consider them as derivations X, Y: UGPŽ.ªP. We define their Lie bracket by

wxX,Y[Ž.XଙYy␶Ž.<<<

As a consequence of the previous lemma, wxX, Y g Lie G. 4.2.10. PROPOSITION. The bracket defined abo¨e endows Lie G with the structure of a P-linear Lie algebra.

Proof. This bracket is clearly antisymmetric. Let X, Y, Z g Lie G.We have

X,wxY,Z

sŽXଙYଙZ .иUP Ž.␥3y Ž23 .Ž<<<<<

yŽ.Ž.Ž123 <<<<<

qŽ.Ž13 <<<<<

We complete the proof of the Jacobi identity by a brute force verification.

Moreover, it is not difficult to show that the tangent map of a formal group morphism commutes with the Lie brackets. Therefore, we have

4.2.11. COROLLARY. The map G ¬ Lie G induces a functor from the category of formal groups to the category of P-linear Lie algebras. 4.2.12. The Case of a P-Formal Group Law. Let us give an expression of the Lie bracket in the case of a formal group law GxŽ.,y. Let 498 BENOIT FRESSE

VsKx1 [ иии [ Kxn. Let X1,..., XnKgMod Ž.V, K be a dual basis of x1,..., xn. Thus Lie G s Ž.KX1 [ иии [ KXn P . Because of the unit equationŽ. 2.1.6 , we have the following P-power series expansion

k,l GxiiiikŽ.,ysxqyqÝ␥ Žx,yl .Žqdeg G 3, . k,l k,l with ␥i g PŽ.2 . By a straightforward inspection, we obtain k,ll,k wxXkl,XsÝX imŽ.␥ iy␶и␥ i. i

ŽŽWe identify ModKKV, K .m PŽ2.Ž with ModKV, PŽ2.... In the classical case P s Com, we have Lie GnŽ .sLie G Ž.1 m Com Ž n .s Lie G Ž.1. In the classical theory, the Lie bracket is defined on Lie GŽ.1 . Up to some shift of degrees, it is equal to the Lie bracket defined above.

4.3. The Main Theorem of Lie Theory We now state the main theorem of this paper.

4.3.1. THEOREM. Consider the category of finite dimensional P-formal groups. Consider the category of P-linear Lie algebras, whose underlying right

P-module is of the form ModK Ž.V, P , for a certain finitely generated projec- ti¨eK-module V. The Lie algebra functor induces an equi¨alence between these categories. We deduce this statement from the next theorem. Before, we recall some obvious facts on the cartesian product of two formal groups.

4.3.2. DEFINITION. The cartesian product of G12and G is the P-for- mal group defined as follows. We set

RGŽ.Ž.Ž.12=G sRG 1kRG 2, and we equip this P-algebra with the coproduct

␥12q␥ 6 RGŽ.12kRG Ž. RG Ž. 1122kRG Ž.kRG Ž.kRG Ž.

T6 RGŽ.1212kRG Ž.kRG Ž.kRG Ž., where T switches the second and the third summand.

The Lie algebra LieŽ.G12= G splits as the direct sum of the Lie algebras Lie G12and Lie G . Furthermore, the biproduct diagram

ii126 6

Lie G Lie G Lie G 6 Lie G . 11226 [ pp12 LIE THEORY OF FORMAL GROUPS 499 is the ‘‘tangent diagram’’ of

␫12␫ 6 6

RG6 RG RG RG , Ž.1122 Ž.k Ž.6 Ž. ␲12␲

where ␲ 12, ␲ are the canonical embeddings, and ␫1Ž.resp. ␫ 2is the P-algebra morphism whose restriction to the 1st summand is the identity Ž.resp. 0 and whose restriction to the 2nd summand is 0 Ž resp. the identity . . Of course, these maps are morphisms of P-formal groups. We have a natural isomorphism

UŽ.LieŽ.Ž.Ž.G12= G ( U Lie G1m U Lie G2.

In fact, if u g UŽLie ŽG12= G .., then u s ipu1122Ž.Ž.Žиip u. We keep the same notation for the image of a Lie algebra morphism under the enveloping algebra functor..

4.3.3. THEOREM. Let G be a finite dimensional P-formal group. There exists a natural coalgebra-algebra pairing

²:y,yG:UŽ.Ž.Lie G m RG ªP, endowed with the following properties.

Ž.1 Recall that by definition Lie G s ModK ŽŽ..QR G , P ; ModKŽŽ.RG,P ..The restriction of the pairing ²y , y :G to Lie G m RGŽ. is gi¨en by the P-linear duality pairing. Ž.2 This pairing is perfect. Ž.3 This pairing makes the product of U ŽLie G . adjoint to the coproduct of RŽ. G . More precisely, we ha¨e

²:²u¨,rGGsum¨,␥r :=G.

The construction of the pairing, together with the proof of assertionsŽ. 1 andŽ. 3 of this theorem is the purpose of the next section. Statement Ž. 2 turns out to be independent from the construction of the pairing. In fact, it is a consequence of the other properties. More precisely, we have the lemma

4.3.4. LEMMA. Consider a coalgebra-algebra pairing

²:y,yG:UŽ.Ž.Lie G m RG ªP.

If this pairing satisfies assertion Ž.1 of the pre¨ious theorem, thenitis necessarily perfect. 500 BENOIT FRESSE

Proof. This lemma is a consequence of the Poincare´᎐Birkhoff᎐Witt theoremŽ. 4.1.5 . By this theorem, we have a coalgebra isomorphism ␾:SŽ.Ž.Lie G ª U Lie G defined by 1 ␾␴Ž.иX иии X [ ␴ и sXŽ.<<,..., <

Let us denote by ⑀G the map provided by Lemma 3.2.8. Recall that UŽ.Lie G is equipped with a canonical filtration, denoted by FUrŽ.Lie G

Ž.cf. Subsection 4.1.4 . We want to compare gr ␾ with gr ⑀G. Of course, we need to check that ⑀G preserves the filtration. We need some insight into this map. Recall from the proof of Lemma 3.2.8 that the composite

⑀G 6 UŽLie GS .Ž.ModKK Ž.V, P ª Mod Ž.V, P maps u to the P-linear form XuŽ.[ ²u,y :G. Therefore 1 ii i ⑀GŽ.u[Ý␴иXuŽ.Ž1.Žиии Xu Ž.n.. nn!

r Let u g FUŽ.Lie G . We can assume u s X1 иии Xr, with X1,..., Xr g ii ModK Ž.V, P . Assume n ) r. Any tensor in the expansion ÝŽu.Ž␴ и u 1. i mиии m uŽn. contains at least one 1 as a factor. As a consequence 1 ⑀Ž.u[␴iiиXu иии Xu i. G Ýn!Ž.Ž1.Ž Ž.n. nFr

r That is, ⑀GGmaps u into FSVŽ.. In other words, ⑀ preserves the filtration. Furthermore, by an immediate induction, we show that the r-fold coproduct of u is equal to

Ý␴Ž.<

1 r gr ⑀GŽ.u s sXŽ.<<1,..., <

sX1 иии Xr .

Hence gr ⑀GGи gr ␾ s 1, and as a consequence, gr ⑀ is an isomorphism. LIE THEORY OF FORMAL GROUPS 501

As an immediate consequence of the theorem above and of Corollary 3.2.12, the Lie algebra functor is full and faithful. Indeed propertyŽ. 1 readily implies that a formal group morphism is adjoint to its tangent map. It remains to prove the next lemma and the proof of the main theorem will be complete.

4.3.5. LEMMA. The Lie algebra functor is essentially surjecti¨e.

Proof. Let L be a P-linear Lie algebra, with L s ModK Ž.V, P , where V denotes a finitely generated projective K-module. By the Poincare´᎐ Birkhoff᎐Witt theoremŽ. 4.1.5 , we have a natural coalgebra isomorphism SLŽ.ªUL Ž.. As a consequence, we have a perfect coalgebra-algebra pairing

²:y,y:ULŽ.mPˆ Ž.V ªP.

By Lemma 3.2.14, this pairing induces a perfect coalgebra-algebra pairing between ULŽ.mUL Ž.and Pˆ ŽV [ V .. Now, let us consider ␥ : PˆˆŽ.V ª P ŽV [ V ., the dual of the algebra product ULŽ.mUL Ž.ªUL Ž.. Since this product is associative, by uniqueness of the dual, we deduce that ␥ is associative. By the same argument, we obtain the relation 1 k 0 и ␥ s 0 k 1 и ␥ s 1. Therefore, ␥ equips PˆŽ.V with the structure of a P-formal group. Let us denote this P-formal group by G. It remains to check that L is the Lie algebra of G. Since ULŽ.mPˆˆ Ž.V ªPand U ŽLie G .m P Ž.V ª P are both perfect coalgebra-algebra pairings, the coalgebras ULŽ.and UŽ.Lie G are isomorphic. Moreover, the isomorphism between UL Ž.and UŽ.Lie G respects the pairings. Since in the both cases, the algebra product is dual to the coproduct ␥, this isomorphism is a Hopf algebra isomor- phism. By the Milnor᎐Moore theoremŽ. 4.1.6 , we obtain L ( Lie G. 4.3.6. An Alternati¨e Proof of the Main Theorem of Lie Theory.As mentioned inwx G-K , in the case PŽ.1 s k, the main theorem of Lie theory can be deduced from the Lazard result. Let us give more precision on this alternative demonstration. Let P be a quadratic operad. The Construction 2.2.2 provides a functor from the category of P!-algebras into the category of P-formal groups. We have denoted this functor by L ¬ Exp L. We have soon mentioned that the category of P!-algebras is equivalent to the category of P-linear Lie algebrasŽ. cf. Subsection 4.1.7 . Let us extend the Construction 2.2.2 to P-linear Lie algebras, for a general operad P.

Let L s ModkŽ.V, P be equipped with a P-linear Lie algebra struc- ture. If A is a P-algebra, then LŽ.1 m A is equipped with a natural Lie 502 BENOIT FRESSE algebra structure. Indeed, let X12, X g LŽ.1 . Expand their bracket as

wxX12,XsÝYiim␥, with Yiig LŽ.1 , and ␥ g P Ž.2 . Let a12, a g A. We set

wxX1122ma,XmasÝYiim␥Ž.a12,a. As a consequence, using the Campbell᎐Hausdorff formula, we obtain a similar construction L ¬ exp L, for L a P-linear Lie algebra. Moreover, L is the Lie algebra of the P-formal group exp L. By the Lazard resultswx L1, Sect. 7 , this functor L ¬ exp L is essentially surjective. It is clearly faithful. It remains to show that this functor is full. This is not contained in the Lazard result. In fact, within the analyzer formalism, it is difficult to define the notion of a morphism between two formal groups which do not have the same dimension. Nevertheless, by the next lemma, a P-formal group law morphism is completely determined by its tangent morphism. The assertion follows.

4.3.7. LEMMA. Consider an nЈ-tuple of P-power series ␾Ž.x , which pro- ¨ides a P-formal group law morphism from GЈŽ.xЈ, yЈ to G Ž. x, y . The homogeneous components of ␾Ž.x are determined by the linear one. Proof. We write

␣ ␾iiŽ.x s Ý␮ ,␣ Žx ., i s 1,...,nЈ. ␣

Fix a multi-index ␣. We denote by <<␣ , the integer ␣1 q иии q␣n. Assume that the components of ␾Ž.x of degree less than <<␣ are known. Žr. Let us denote by GxŽ.1,..., xr the rth iteration of the formal group law. Explicitly Ž1. G Ž.x11s x , Ž2. GŽ.Ž.x12,xsGx 12,x , Ž3. GŽ.Ž.x123,x,xsGGŽ.Ž. x12,x ,x 3sGx 1,Gx Ž. 23,x , etc. Let r s <<␣ . The multilinear part of the equation Žr. Žr. GЈ Ž.␾ Ž.x1,...,␾ Ž.xrs␾Ž.G Žx1,..., xr . provides an equation between the degree r component of ␾Ž.x and its components of lower degree.

More precisely, recall that xkkdenotes a grouping of variables x s Ž.xk,1,..., xk,n . In the equation

XŽr. Žr. GiŽ.␾Ž.x1,...,␾ Ž.xris␾ Ž.G Žx1,..., xr ., LIE THEORY OF FORMAL GROUPS 503

we consider the monomials of degree at least one in each variable xk, j, with ␣1 variables x?,1, ␣ 2 variables x?,2, etc. We obtain an equation which involves only ␮i, ␣ and the ␮?, ␤ with <<␤ - r. The operation ␮i, ␣ occurs only on the right side of this equation in monomials like

cwiи ␮ ,␣Ž.w , where cw is a non-negative integer and w ranges over the set of tensors of degree at least one in each variable xk, j, with ␣1?,12variables x , ␣ variables x?, 2 , etc. In this equation, we make the substitution x1, j s иии s ␣ xr, jjs x , for j s 1,...,n. Finally, we have written ␮ i,␣Ž x . in terms of the components of lower degree of ␾Ž.x .

5. DIFFERENTIAL CALCULUS

5.0. Warning

In the sequel, the dot и denotes either the product in ENDŽURP Ž .. Žor any product which maps to this one. , or the result of the evaluation

ENDŽ.URPŽ.mURP Ž.ªURP Ž..

For example, let d g DerŽŽ.Ž..URP ,URP . Let u g URP Ž.. We denote by d и u the image of d m u under the evaluation map

DerŽ.URPŽ.,URP Ž.mURP Ž.ªEND Ž.URP Ž.mURP Ž.ªURP Ž.. Assume <

5.1. Differential Operators Throughout this section, V denotes a fixed finitely generated projective K-module. We denote by R the algebra PˆŽ.V . In the previous section, we have defined the notion of an operad derivation. In this subsection we specializes this notion to the case of the operad URP Ž.. We denote by Der R the right URPŽ.-module Der ŽURP Ž.,URP Ž... 5.1.1. Deri¨ation with Respect to a Tangent Vector.LetXg ModK Ž.Ž.ŽV, P n i.e., X belongs to the tangent space of R .. We denote by ѨX :URP Ž.ªURnP Ž.wxthe unique P-linear derivation, whose restriction to V is the composite

X ␩ V ª PŽ.n ª URnP Ž .Ž..

ŽŽRecall that UP PˆŽV ..is a kind of free P-operad, cf. Proposition 1.2.6.. 504 BENOIT FRESSE

Since V is a finitely generated projective K-module, these derivations generate Der R as a right URP Ž.-module. Let V be the free K-module generated by the variables x1,..., xn. Let X,..., X Mod Ž.V, K denote the dual basis. In this case, Ѩ is the 1 nKg Xi obvious derivation, which replaces each occurrence of xi in a monomial of UP ŽŽ..Pˆ xby an extra-variable. Explicitly k $ Ѩ и ␮ x ,..., x ,h ,...,h ␮ x ,...,h ,..., x ,h ,...,h . XiiŽ.1 ir1 nis Ýž/11ir2nq1 isik

Moreover, Der PˆˆŽ.x is the free right UP ŽP Ž..x -module generated by Ѩ ,...,Ѩ . XX1 n 5.1.2. Differential Operators. The algebra of differential operators is the quotient of the tensor algebra TŽ.Der R by the two sided ideal generated by the Schwartz and the Leibniz relations. We denote this algebra by OpŽ.R .

Recall that TŽ.Der R is a right URP Ž.-module. Let d g Der R. Let pgTŽ.Der R . Let u1,...,un gURP Ž.. The Leibniz relation is given by

md dpŽ.Ž. u1,...,uns dp Ž.1 , u1,...,uniqÝ⑀ иpuŽ.1,...,dиuin,...,u , i where ⑀i is the permutationŽ.Ž 1 иии i q 1 <<

ѨѨXYs␶Ž.<<<

To conclude, OpŽ.R is a right URP Ž.-module, but the product in OpŽ.R is only P-linear.

5.1.3. Hopf Algebra Structure of OpŽ.R . Consider the map Der R ª OpŽ.R m Op Ž.R defined by Ѩ ¬ Ѩ m 1 q 1 m Ѩ. This map induces a unique associative algebra morphism

OpŽ.R ª Op Ž.R m Op Ž.R , which makes OpŽ.R into a cocommutative Hopf algebra. In the next lemma, we show that Der R is a sub Lie algebra of OpŽ.R . In fact, it is not hard to show OpŽ.R s U ŽDer R .. But we do not need this fact.

5.1.4. PROPOSITION. If Ѩ12and Ѩ are deri¨ations, then wxѨ12, Ѩ is a deri¨ation.

Proof. Since Der R is generated by ModK Ž.V, K , we can assume Ѩ1 s ѨX Ž.u,Ѩ2 sѨYK Ž.¨, with X, Y g Mod Ž.V, K , and u, ¨ g URPŽ.. Because LIE THEORY OF FORMAL GROUPS 505 of the Leibniz formula, we have

ѨѨ12sŽ.Ž.Ž.ѨѨXYu,¨ qѨѨ YŽ. Xи¨(1u,

ѨѨ21sŽ.Ž.Ž.ѨѨYX¨,uqѨѨ XŽ. Yиu(1¨.

And, by the Schwartz identity, we obtain ␶ Ž<<<

wxѨ12,ѨsѨѨYXŽ.Ž.Ž.и¨(1uyѨѨXY Ž.иu(1¨.

By definition, we have Der R : ENDŽŽ..URP . We study this map in order to produce an algebra morphism OpŽ.R ª END ŽURP Ž... 5.1.5. LEMMA. Consider the ordered set I Ž.i ,...,i . Let d ,...,d s 1 nii1 n be an I-indexed sequence of deri¨ations. We denote by dI the composite dиии d ENDŽŽ..UR.Let ␮ P Ž.n , u ,...,u URŽ.. ii1 ng P g 1 rg P LetŽ. I1,...,Ir be a partition of I. This partition corresponds to a shuffle s . We denote by ␴ the block shuffle Ž I1,...,Ir.ŽI1,...,Ir.

sd<<,..., <

Let ⑀ g S2 r , be the shuffle defined by

⑀ Ž.k s 2k y 1, k s 1,...,r, ⑀Ž.rqks2k, ks1,...,r.

Let c be the integer Ý <

⑀Ž.c1,...,cr,<

We ha¨e

dIи ␮Ž.u1,...,ur ␴ 1и⑀и␮dиu,...,d иu . sÝŽIrr,...,I .Ž[I1,...,Ir. Ž.I11 Irr I1qиии qIrsI

The sum ranges o¨er the set of partitions of I. An element of the partition is equipped with the induced order. Proof. This theorem is easier to prove than to state. Its proof is performed by a self-running induction.

5.1.6. LEMMA. Let X, Y g ModK Ž.V, K . In END ŽŽ..URP ,we ha¨e ѨѨXYs␶иѨѨ YX. 506 BENOIT FRESSE

Proof. The previous lemma enables us to prove by induction that

ŽѨѨXY.и␮Ž.¨1,...,¨r,h1,...,hnYXsŽ␶иѨѨ.и␮Ž.¨1,...,¨r,h1,...,hn, for ␮ g PŽ.r q n , ¨ 1,...,¨r gV.

5.1.7. THEOREM. The canonical embedding Der R ¨ ENDŽŽ..URP in- duces an algebra morphism

ev: OpŽ.R ª ENDŽ.URP Ž..

This morphism turns out to be UP Ž. R -linear. Proof. It remains to check that the induced morphism cancels the Leibniz relations. This is an immediate consequence of the Leibniz for- mula for derivations.

Let p g OpŽ.R , u g URP Ž.. As for derivations, we denote by p и u the result of the evaluation evŽ.Ž.pu. As a consequence of the previous lemma, we are able to rewrite Lemma 5.1.5 using the coproduct of OpŽ.R . This is done in the next statement. This formula is also known as the Leibniz᎐Hormander¨ formula.

5.1.8. COROLLARY. Let p g OpŽ.R . Let ␮ g P Ž.n , u1,...,urgURPŽ.. We ha¨e

i i i pи␮Ž.u1,...,ursÝ␴ [1и⑀Ž.< pŽ1.Ž<<,..., p r.<<, u1 <,...,

i i и␮Ž.pŽ1.иu1,..., pŽr.иur, with ⑀ g S2 r defined as in Lemma 5.1.5.

5.2. In¨ariant Deri¨ations 5.2.1. DEFINITION. Fix a P-formal group G. A derivation Ѩ is said to be right in¨ariant when

UPŽ.␥ 6 UGPŽ. UGPŽ.kPUGP Ž.

Ž. 6 Ѩ6 UGP kPѨ

UPŽ.␥wxn 6 UGnPŽ.wx UGPŽ.kUGnP Ž.wx

In the free case, UGP Ž.sUP ŽPˆ Ž..x, and any derivation has a unique expansion Ѩ Ѩ ux,h,...,h . s ÝXiiŽ.Ž.1 n i LIE THEORY OF FORMAL GROUPS 507

Furthermore, UP ŽŽ..Pˆ x kP Ѩ is the derivation Ѩ Ѩ uy,h,...,h Der U Pˆ x, y . yYis ÝiŽ.Ž.1ng PŽ. Ž. i Therefore, a derivation is right invariant if and only if its components verify the equations

uGxiŽ.Ž.,y,h1,...,hn ѨGx,y иuy,h,...,h , i 1,...,n. sÝŽ.Yij Ž.j Ž1 n . s j

The canonical augmentation ⑀: UGP Ž.ªPprovides a UGP Ž.-linear map

Der RGŽ.sDerŽ.Ž.UGP Ž.,UGP Ž.ªDer UGP Ž.,PsLie G.

We denote this map by ⑀#. For example, let X g ModK Ž.V, P be a tangent vector. We have ⑀#ѨX s X. And

⑀#Ž.Ž.ѨXŽ.Ž.Ž.u1,...,unsX ⑀ u1,...,⑀ un, for u1,...,ungUGPŽ.. In the free case, recall that ⑀ is given by the evaluation x s 0. There- fore Ž.Ž.ŽŽ..⑀#Ѩ fxsѨfx

Proof. We construct the inverse morphism. Let Ѩ 0g DerŽŽ..UGP ,P be a tangent vector. Consider the derivation which is the composite of the

UGP Ž.-linear extension of Ѩ 0 and of the P-formal group coproduct. Explicitly

Ѩ и u [ Ž.UGPŽ.kPѨ0иŽ.UPŽ.␥u.

ŽŽThis is a derivation since UP ␥ ..is an operad morphism. By uniqueness of the linear extension of an operad derivation, we have

UGPŽ.kPŽ.Ž.UGP Ž.kPѨ0sUGPŽ.kPUGP Ž.kPѨ0. Therefore

Ž.UGPŽ.kPѨиŽ.UP Ž.␥u

sŽ.Ž.Ž.UGPŽ.kPUGP Ž.kPѨ0иUPŽ.1k␥и␥u

sŽ.Ž.Ž.UGPŽ.kPUGP Ž.kPѨ0иUPŽ.␥k1и␥u. 508 BENOIT FRESSE

The linear extension of an operad derivation is functorial by uniqueness. Hence, we obtain

Ž.UGPŽ.kPѨиŽ.UP Ž.␥u

sŽ.UPŽ.␥kPPиŽUGPŽ.kPѨ0 .Ž.иŽUPŽ.␥u

sUPŽ.␥иѨиu.

Finally, Ѩ is a right invariant derivation. We deduce ⑀#Ѩ s Ѩ from the unit relation

UPŽ.⑀ kPUGP Ž .иUP Ž.␥s1.

5.2.3. LEMMA. The map ⑀# induces a Lie algebra morphism between the sub-right- P-module of right in¨ariant deri¨ations and Lie G.

Proof. Let Ѩ12, Ѩ be right invariant derivations. For any u g UGPŽ., we have

⑀#Ž.ѨѨ12 иu

s⑀#Ž.Ѩ12иѨиus⑀# Ž.Ѩ 1иŽ.UGP Ž.kP Ž⑀#Ѩ2 .иŽ.UP Ž.␥u.

Therefore

⑀#Ž.ŽѨѨ12 s ⑀#Ѩ1ଙ⑀#Ѩ 2 .Ž.иUP␥ and the lemma follows.

5.2.4. COROLLARY. The map ⑀# induces a Lie algebra isomorphism between the submodule of right in¨ariant deri¨ations and Lie G. As a consequence, we have a canonical algebra morphism

OG: UŽ.Lie G ª OpŽ.RG Ž..

Furthermore, OG is a Hopf algebra morphism, because Der RGŽ.is primitive in OpŽŽ..RG .

5.2.5. LEMMA. Consider a pair of formal groups G12, G . We ha¨ean associati¨e algebra isomorphism

OpŽ.Ž.ŽRGŽ.12mOp RG Ž.ªOp RG Ž 1=G2 . .,

gi¨en by

p121122m p ¬ ␫ Ž.p и ␫ Ž.p , LIE THEORY OF FORMAL GROUPS 509

where ␫12, ␫ are the images of the canonical embeddings under the functor OpŽ.y .

Proof. By definition, OpŽŽRG12=G ..is generated by the P-linear derivations. Therefore, by an immediate induction, we obtain

␫11Ž.pи␫ 22 Ž.ps␶Ž.<<<

᭙p1122gOpŽ.RGŽ.,᭙pgOp Ž.RGŽ..

It follows that p121122m p ¬ ␫ Ž.p и ␫ Ž.p is an algebra morphism. This morphism is injective, since we have

␲␫11Ž.Ž.p 1и␫ 2 Ž.p 2sp 1,

␲␫21Ž.Ž.p 1и␫ 2 Ž.p 2sp 2.

By another immediate induction, we obtain

p s ␫␲111Ž.Ž.Ž.p и␫␲ 2 2 Ž.p 2.

This proves the surjectivity of our morphism.

5.2.6. LEMMA. The following diagram is commutati¨e

O O GG11m 6

UŽ.Ž. Lie G12m U Lie G OpŽŽ R G1 ..m Op ŽŽ R G 2 ..

6 (6 ( O G12=G 6 UŽŽ Lie G12= G .. OpŽŽ R G12= G ..

Proof. We have just to check that the diagram commutes for a genera- tor of UŽ.Ž.Lie G12m U Lie G , and hence, for an element of Lie G1[ Lie G2 . But in this case, this is straightforward.

5.2.7. LEMMA. For u12, u g UŽ.Lie G , ¨ g UGP Ž.,we ha¨e

⑀Ž.OuG=GŽ.Ž.Ž.12muиŽ.ŽUP␥¨s⑀OuuG12 и¨ ..

Proof. Let p1 s OuGŽ.12,psOuG Ž.21. Since U ŽLie G .is generated by Lie G as an algebra, using Lemma 5.2.3, we obtain by a straightforward induction

p2и ¨ s Ž.Ž.UGPŽ.kP⑀иŽ.␫22 Žp .иUP Ž.␥¨. 510 BENOIT FRESSE

By another induction, we obtain

pp12и¨sŽ.UGPŽ.kP⑀и␫11 Žp .и␫ 2 Žp 2 .и Ž.UP Ž.␥¨. Therefore

⑀ Ž.Ž.Ž.Ž.pp12и¨ s⑀␫Ž.1p 1и␫ 2p 2иŽ.UP␥¨ . By the previous lemma, we are done. 5.2.8. Construction of the Duality Pairing. We are now in position to construct the duality pairing ²:y , y G of Theorem 4.3.3. Let u g ULŽ.P ,rgRG Ž.. Hence, r g URGŽŽ... We set

²:u,rG[⑀Ž.OuGŽ.иr.

The Leibniz᎐Hormander¨ formulaŽ.²: Corollary 5.1.8 implies that y , y G is a coalgebra-algebra pairing. AssertionŽ. 1 of the theorem follows by construction of OG , and assertionŽ. 3 is an immediate consequence of the previous lemma.

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