Lie Theory of Formal Groups over an Operad

Transcription

1 JOURNAL OF ALGEBRA 202, ARTICLE NO. JA Le Theory of Formal Groups over a Operad Beot Fresse* Laboratore J. A. Deudoe, Uerste de Nce, arc Valrose, Nce Cedex 02, Frace Commucated by Susa Motgomery Receved Jauary 7, INTRODUCTION Let k be a feld of characterstc zero. Let K be a commutatve k-algebra. Let R be a complete commutate Hopf algebra defed over K. Explctly, f R deotes the augmetato deal of R, the R K R ad R lm RR. Moreover R s equpped wth a coassocatve coproduct :RRR, ˆ where ˆ deotes the completed tesor product over K. Notce that R s othg but a cogroup the category of commutatve complete algebras. Whe R s the completo of the symmetrc algebra geerated by a ftely geerated projectve K-module V, the R s called a Ž fte dmesoal. formal group. If V s freely geerated by the varables x 1,..., x, the the coproduct s equvalet to a formal group law. That s, s equvalet to a -tuple of power seres GŽ x, y. 2-varables Ž x, y. Ž x,..., x, y,..., y., whch verfy the dettes 1 1 Ž. GŽ x,0. GŽ 0, x. x, Ž. GŽx,GŽ y, z.. GG Ž Ž x,y,z... The taget space of a formal group s equpped wth a caocal Le algebra structure. Usually, a formal group, defed by the complete Hopf algebra R, s deoted by the letter G. We deote by Le G ts assocated Le algebra. Notce that Le G s dual to V. I the case of a formal group law, Le G s equpped wth a caocal bass Ž the dual bass of x,..., x. 1. The structure costats of the Le bracket are gve by the quadratc part of the formal group law GŽ x, y.. *E-mal: fresse@math.uce.fr $25.00 Copyrght 1998 by Academc ress All rghts of reproducto ay form reserved.

2 456 BENOIT FRESSE The ma theorem of Le theory asserts that the Le algebra fuctor G Le G s a equalece from the category of formal groups to the category of Le algebras. The oto of a complete algebra, of a cogroup, ad hece of a formal group makes sese for ay usual kd of algebras, such as assocatve algebras, Le algebras, etc. More geerally, we ca cosder formal groups for ay kd of algebras defed over a operad. A operad s a algebrac devce whch ecodes a category of algebras Žsuch as the oes gve above.. The oto of a operad was troduced by. May for the eeds of the terated loop space machery Žcf. M.. It turs out that operads are also partcularly useful for orgazg the may algebrac structures arsg mathematcal physcs. Let be a operad. The assocated algebras Ž resp. formal groups. are called -algebras Ž resp. -formal groups.. The am of ths paper s to geeralze the ma theorem of Le theory to the case of -formal groups Žcf. theorem From a certa pot of vew, a operad s very smlar to a rg. ursug the aalogy further, we ca defe the oto of a rght - module, ad of a -lear Le algebra. We wll show the followg. We costruct a Le algebra fuctor, whch duces a equalece betwee the -formal groups ad the -lear Le algebras, whose uderlyg rght - module s some sese fte dmesoal. Cosder the left K-module freely geerated by x 1,..., x. As the classcal case, the free -algebra geerated by ths module s a kd of power seres algebra. Its elemets are kow as -power seres. I the case Com, the operad of commutatve algebras, we recover the classcal power seres. I the case As, the operad of assocatve algebras, we obta the power seres o-commutatve varables. Fally, we ca also defe a -formal group law as a partcular stace of a -formal group. A -formal group law GŽ x, y. s a -tuple of -power seres satsfyg relatos Ž. ad Ž.. I fact, M. Lazard has already exteded some parts of Le theory to the cotext of aalyzers Žcf. L1, L2.. A aalyzer s a axomatzato of the tradtoal oto of power seres algebra. For stace, the sequece of -power seres algebras forms a aalyzer. Lazard proved partcular that a formal group law defed a aalyzer s determed, up to a somorphsm, by ts quadratc part. I our laguage, the somorphsm class of a -formal group law s determed by ts Le algebra. Lazard used essetally drect calculatos of power seres. We follow aother approach, more coceptual, ad close to the usual proof of the ma theorem of Le theory the classcal case Žcf. C, Se.. Let us recall the dea of ths proof. By the process of Carter dualty, the

3 LIE THEORY OF FORMAL GROUS 457 complete Hopf algebra of a formal group G s dual to UŽ Le G., the evelopg algebra of ts Le algebra. Therefore, the ma theorem of Le theory ca be deduced from the MlorMoore theorem. I ths paper, we provde a MlorMoore type theory for -lear Hopf algebras Žsee Secto 4., ad, as a ma tool, we state a Carter dualty type theorem Žsee Theorem We ow gve a detaled summary of the paper. I the frst secto, we recall the prerequstes o operad theory. Let A be a -algebra. I ths secto, partcular, we costruct a operad U Ž A., whose algebras are equvalet to -algebra morphsms X A. Ths operad s kow as the eelopg operad of A. I ths secto, we also defe the oto of a complete algebra over a operad. I the secod secto, we defe precsely the oto of a formal group over a operad. We gve also some examples of -formal group laws. Maly, we provde some explct expasos of Le power seres, whch are L e-formal group laws, such as the HausdorffCampbell formula. As poted out earler, a operad s smlar to a rg. I Secto 3, some sese, we are dog -lear algebra. We defe the oto of a rght -module. We show that the category of rght -modules s moodal symmetrc,.e., ths category s equpped wth a tesor product smlar to the tesor product of k-vector spaces. We show also that ths category has a teral hom fuctor. I the secod part of ths secto, we study the cocommutatve coalgebras the category of rght -modules Žkow as -lear coalgebras.. Aga, the defto of a -lear coalgebra s the same as the defto of a classcal coalgebra, the rght -module tesor product replacg the classcal tesor product. Ths oto s mportat for our purpose because we have a kd of dualty betwee -lear coalgebras ad -algebras. Let C be a -lear coalgebra. Let A be a complete -algebra. A coalgebra-algebra parg betwee C ad A s a map ² :, :CA, whch s -lear C ad k-lear A. Moreover, we assume that ths map makes the coproduct of C adjot to the product of A. We have also a oto of perfect coalgebra-algebra parg Žsmlar to the classcal oto of perfect parg.. Ths process plays the role of Carter dualty our settg. Secto 4 s devoted to Le theory. I the frst subsecto, we study the Le algebras the category of rght -modules Žkow as -lear Le algebras.. Usg the aalogy betwee rght -modules ad k-vector spaces, we defe the evelopg algebra of a -lear Le algebra. It s ot

4 458 BENOIT FRESSE dffcult to exted the classcal MlorMoore theory to the -lear framework. ŽI partcular, the evelopg algebra of a -lear Le algebra s a cocommutatve Hopf algebra the category of rght -modules.. Cosder a -formal group G. Let RG be ts uderlyg -algebra. Oe ca defe QRŽ G. as the module of the decomposable elemets of ths -algebra. I the secod subsecto, we show that the -lear dual of QRŽ G. s a -lear Le algebra, called the Le algebra of the formal group G, ad deoted by Le G. There exsts a atural perfect coalgebra-algebra parg betwee UŽ Le G., the evelopg algebra of the -lear Le algebra Le G, ad RG. Ths parg s deoted by ², : G. Moreover, some sese, ths parg makes the coproduct of RG, adjot to the product of UŽ Le G. Ž cf. Theorem I the thrd subsecto, assumg the exstece of ², : G we prove the ma theorem of Le theory. The costructo of ², : G s the purpose of the ffth ad last secto. The ma gredet ths costructo s a kd of dfferetal calculus. More precsely, we eed the oto of a operad dervato. We relate the Le algebra of G to the operad dervatos of U ŽRŽ G.., the evelopg operad of RG, defed the frst secto Notato ad Coetos Throughout ths paper we work over a fxed feld k of characterstc 0. We deote by Modk the category of k-modules. Most algebras are defed over a groud operad Ž cf. Subsecto Except the frst part of the frst secto, we assume Ž We deote by K the assocatve algebra Ž.Ž 1 cf. Subsecto A fte dmesoal -formal group s usually deoted by G. Ths -formal group G s specfed by a complete -algebra, deoted by RŽ G,. ad a coproduct : RG RG RG Ž cf. Defto We deote by U Ž G. the completed evelopg operad of RG Ž cf. Subsecto We deote the symmetrc group by S. I the sequel, may k-modules are edowed wth a atural S -acto. We deote ths acto by, where belogs to S ad belogs to a S-module V. Here s the ma example where ths coveto s appled. Let V be a S-module ad W be a S -module. Cosder the duced module Id S V W, where j SSj j. Let V, w W. By abuse of otato, we deote by w S the elemet 1 w ks VWId ks S S SV W. Thus, f j j S, the w deotes the acto of o w Id S S S V W, j hece, the tesor w Id S S S V W. Some modules are j equpped wth extra Sr-actos, whch should ot be cofused wth ther atural S -module structure Ž see, e.g., Subsecto

5 LIE THEORY OF FORMAL GROUS Operads 1. COMLETE ALGEBRAS OVER AN OERAD The oto of a topologcal operad was troduced by. May for the eeds of homotopy theory Žcf. M.. I ths subsecto, we gve a short overvew of the operad theory. For a more detaled accout, we refer to Ge-J, G-K, J, Lo1, M Moodal Categores. Recall that a moodal category s a category C together wth a assocatve bfuctor : C C C, ad a object I C, whch s a two sded ut for Žcf. ML.. A assocate algebra Ž C,, I. cossts of a object A C, together wth a assocatve product m: A A A, ad a ut e: I A. More precsely, m ad e verfy the equatos m m A m A m, m e A m A e A. Let F deote the category of the k-module edofuctors together wth atural trasformatos. The fuctor composto : F F F makes F to a moodal category. A assocatve algebra Ž F,. s kow as a moad Žcf. ML, B-W.. Let S, T F. We defe the tesor product of S, T by settg Ž S T.Ž V. SŽ V. TŽ V.. Clearly, ths provdes F wth aother moodal structure. Notce that the fuctor tesor product s symmetrc S-Modules. A S-module V s a sequece of S -modules V Ž., N. We deote by S-Mod the category of S-modules. As the case of graded modules, the otato V Ž. may be abbrevated to V. The teger s called the degree of ad we deote. A S-module gves rse to a fuctor TŽ V,. F, whch s defed by T Ž V, E. V Ž. S E. 0 Furthermore, ths costructo provdes a fully fathful fuctor T: S-Mod F.

6 460 BENOIT FRESSE S-Module Tesor roduct. Let V, W be S-modules. The tesor product of V ad W s defed by the formula V W N Id S N V Ž. W Ž j.. jn S S j Ths bfuctor s assocatve. I fact, we have V V N Id S N V Ž. V Ž.. 1 S S N The S-module 1, defed by k f 0, 1Ž. ½ 0 otherwse, s a two sded ut for. Let us costruct a symmetry somorphsm c V, W : V W W V. Let deote the trasposto Ž 1 2. S. We deote by Ž, j. 2 the block trasposto, defed by Ž, j.ž k. k, k1,..., j, Ž, j.ž jk. k, k1,...,. As explaed Coveto 0.1, a elemet of Ž V W.Ž N. s deoted by wwth V Ž., w W Ž j., S, ad j N. We set cv, WŽ w. Ž, j. w. As a cocluso, the S-module tesor product makes the category S-Mod to a symmetrc moodal category LEMMA. The fuctor T: S-Mod F commutes wth the tesor product. More precsely, we hae a atural somorphsm T Ž V W, E. T Ž V, E. T Ž W, E.. Ths fuctor somorphsm, carres the symmetry operator TŽ c, E. V, W to the swtchg map T Ž V, E. T Ž W, E. T Ž W, E. T Ž V, E.. N The Tesor ower of a S-Module. Let V be a S-module. Sce the S-module tesor product s symmetrc, the symmetrc group S acts o

7 LIE THEORY OF FORMAL GROUS 461 the -fold tesor power V by place permutato. I other words, V s a S -Ž S-module.. Oe should ot cofuse the S-acto wth the S-module structure. For s S, we deote the assocated S-module morphsm by s: V V,orbys*, wth s Žs 1.*. Let us deote by s,..., 1 the block permutato defed by sž 1,...,.Ž sž1. sžk1. r. 1 sžk.1 r, r1,..., sžk., k1,...,. Let V. We have clearly 1 s* s,...,. 1 1 sž1. sž S-Module Composto. Let V, W be S-modules. The composto of V ad W s the S-module V W defed by the formula V W Ž V Ž. W. S. 0 Oe may check that ths bfuctor s assocatve. Cosder the S-module defed by Ik f 1 ad I0 otherwse. Ths S-module s obvously a two sded ut for the composto product LEMMA. The fuctor T: S-Mod F commutes wth the composto product. More precsely, we hae a atural somorphsm T Ž V W, E. T Ž V, T Ž W, E Operads. A operad s a assocatve algebra the category of S-modules equpped wth the composto product. Hece, a operad structure cossts of a assocatve product : ad a ut : I. Because of the form of the composto product, the operad structure s equvalet to a ut 1 Ž. 1 ad a collecto of equvarat morphsms Ž. Ž 1. Ž. Ž 1., whch satsfy the May axoms Žcf. M.. The mage of 1 uder the product s deoted by,..., 1. The product Ž 1,...,1,, 1,..., 1., wth at the th place, s also deoted by. As I s a ut for, t s equpped wth a caocal operad structure. Furthermore, I s a tal object the category of operads.

8 462 BENOIT FRESSE Amog the products above, we have Ž 1. Ž 1. Ž 1.. Ž. Ths product makes 1 to a assocatve algebra. Throughout ths paper, we deote ths algebra by K Algebras. Because of Lemma 1.1.7, a operad s equvalet to a moad whose uderlyg fuctor s of the form TŽ,.. A algebra over ths moad s kow as a -algebra. Because of the form of the fuctor TŽ,., a -algebra product TŽ,A. A s equvalet to a sequece of S -equvarat maps Ž. A A, whch are assocatve wth respect to the operad product, ad whch make the operad ut act as the detty. Let Ž., a 1,...,a A. The mage of a1 a uder the -algebra product s deoted by Ž a,...,a. 1. As for ay moad, f V s a k-module, the TŽ, V. s the free -algebra geerated by V EXAMLES. May classcal algebras are fact algebras over a operad. For example, there are operads As, Com, L e, whose algebras are respectvely the assocatve algebras Ž wthout ut., the assocatve ad commutatve algebras Ž wthout ut., ad the Le algebras. Let us recall the expaso of the correspodg fuctors TŽ As,., TŽ Com,., ad TŽ Le,.. The free assocatve algebra s the tesor algebra T Ž As, V. TŽ V. V. 1 The free commutatve algebra s the symmetrc algebra T Ž Com, V. SŽ V. Ž V. S. 1 I characterstc zero, the free Le algebra s the prmtve part of the tesor algebra equpped wth the shuffle coproduct It ca also be descrbed as T Ž L e, V. rm TŽ V.. 1 TŽ Le, V. e V, 1

9 LIE THEORY OF FORMAL GROUS where e s a dempotet of ks kow as the frst Eulera dempotet Žcf. R1, R2, Lo Operads Defed by Geerators ad Relatos Žcf. G-K, The forgetful fuctor from the category of operads to the category of S-modules possesses a left adjot kow as the free operad. Let V be a S-module. The free operad geerated by V s deoted by TŽ V.. Ths operad s edowed wth a S-module morphsm : V TŽ V. ad s characterzed by the followg uversal property. Let be a operad. If : V s a S-module morphsm, the there exsts a uque operad morphsm : TŽ V., such that. It s also possble to defe the oto of a operad deal. I fact, a sub-s-module I s a operad deal f ad oly f the operad product duces a operad product o the quotet S-module I. A operad morphsm I Q s equvalet to a operad morphsm Q whch vashes o the deal I. Gve a sub-s-module S, the deal geerated by S s the smallest deal cotag S. We deote ths deal by Ž S.. A operad morphsm Ž S. Q s equvalet to a operad morphsm Q whch vashes over S. These deftos eable us to specfy operads by geerators ad relatos. The geerators are specfed by a S-module V, ad the relatos by a sub-s-module R TŽ V.. Suppose that the geerators are cocetrated degree 2, ad the relatos degree 3. Explctly, V Ž. 0 uless 2, ad RŽ. 0 uless 3. I ths case, the operad TŽ V. Ž R. s kow to be quadratc. Notce that we ca recover V ad R from the operad, sce V Ž. 2 Ž. 2 ad RŽ. 3 s the kerel of the atural operad morphsm TŽ V.Ž 3. Ž 3.. The operads As, L e, Com are all quadratc Žcf G-K Algebras oer a Quadratc Operad. Let V be a k-module. The edomorphsm operad deoted by E d s defed by Ed Ž. V V Ž Mod V, V. k together wth the obvous S-acto ad operad product. Let A be a -algebra. Recall that the -algebra structure ca be specfed by a sequece of products A A. Such a product s Ž equvalet to a map Mod A, A. k. I fact, a -algebra product TŽ, A. A s equvalet to a operad morphsm Ed A. Assume that the operad s quadratc. Let V be the S-module cocetrated degree 2 wth V Ž. 2 Ž. 2. Let R be the S-module cocetrated degree 3 wth RŽ 3. keržtž V.Ž 3. Ž 3... Sce TŽ V. Ž R., a operad morphsm E da s equvalet to a S-module morphsm V E d such that the duced operad morphsm TŽ V. A EdA vashes over R.

10 464 BENOIT FRESSE EXAMLE. Let us show how ths works for the operad L e. The operad L e s quadratc. Let V Ž. 2 be the sgature represetato of S. Hece, V Ž. 2 s the k-module geerated by a sgle operato, wth Ž Let c deote the cycle Ž S. We defe RŽ 3. 3 to be the sub-s -module of TŽ V.Ž 3. geerated by the elemet 3 Ž 1 c c 2. 2 TŽ V.Ž 3.. We have L e TŽ V. Ž R.. Let L be a k-module. A S-module morphsm : V E d let to the specfcato of a atsymmetrc bracket L s equva-, : LLL, whch represets the mage of. The duced operad morphsm : TŽ V. EdL maps 2 to the correspodg product E d L,.e.,, ad hece, to the map 2 XYZ X, Y, Z. Thus, Ž1 c c 2. 2 s mapped to XYZ X, Y, Z Y, Z, X Z, X,Y. Ž 2. Fally, cacels 1 c c f ad oly f 2 X, Y, Z Y, Z, X Z, X,Y 0, X, Y, Z L. As a cocluso, a operad morphsm L e EdL s equvalet to the specfcato of a atsymmetrc bracket, : L L L whch verfes the Jacob detty Algebra Morphsms as Algebras oer a Operad I ths subsecto, we defe operads whose algebras are the objects of some comma categores Algebras uder a -Algebra. Let A be a -algebra. A -algebra uder A Ž aka a A-algebra. s a -algebra X together wth a -algebra morphsm X : A X. We deote by A-Alg the category of A- algebras. For example, f A Ž. 0, the the operad product provdes A wth a -algebra structure. I fact, Ž. 0 s the free -algebra geerated by 0. Moreover, a Ž. 0 -algebra s othg but a -algebra. As a cosequece, 0 Ž. s a tal object the category of -algebras. Coversely, the

11 LIE THEORY OF FORMAL GROUS 465 evelopg operad costructo below supples a operad U Ž A. Ž called the eelopg operad of A., whose algebras are the A-algebras. Moreover, U Ž. 0 A. The dea of represetg the category of A-algebras by a operad goes back to Ge-J. Idepedetly, the evelopg operad s ecessary for the costructo of a useful dfferetal calculus Ž cf. Secto Costructo of the Eelopg Operad. The caocal embeddg u: 1,..., r41,...,r4 gves rse to a group embeddg Sr S r. Ideed, ay permutato of 1,..., r4 exteds to a permutato of 1,..., r 4, whch fxes the elemets outsde the mage of u. Cosder the S-module r AŽ. Ž r. S r A. r0 The -algebra product duces a S-module morphsm d : TŽ, A. 0 A. The operad product duces aother S-module morphsm d : TŽ, A. A 1. The operad ut duces a S- module morphsm s : A TŽ, A. 0. We have d0s0ds 1 0 A. The operad U Ž A. s defed as the lear coequalzer TŽ d. TŽ d 0 1. TŽ TŽ, A.. TŽ A. UŽ A.. Ž. The par T d, T d s kow to be reflexe,.e., we have 0 1 TŽ d. TŽ s. TŽ d. TŽ s. TŽ A Ths property forces the mage of TŽ d. TŽ d. 0 1 to be a operad deal. Hece, the operad product of TŽ A. duces a operad product o the quotet U Ž A.. As metoed earler, ths operad U Ž A. s called the evelopg operad of the -algebra A Geerators of the Eelopg Operad. Let us wrte a cocrete defto of U Ž A. by geerators ad relatos. We deote a elemet of r r A by Ž a,...,a,h,...,h., where Ž r. S 1 r 1, r a,...,a A, ad h,...,h deote extra-varables. Let Ž r. 1 r 1, 1 Ž.,..., Ž. 1 r r, a 1,...,aN A, N1 r. The map d0 s gve by Ž 1 1 r 1 1. Ž a,...,a.,..., Ž a,...,a.,h,...,h 1 1 r1 1 r1 r a,...,a,h,...,h, 1 r 1

12 466 BENOIT FRESSE wth a 1 1Ž a 1,...,a 1.,... The map d 1 a Ž a,...,a.. r r 1 r11 1r1r s gve by 1Ž a 1,...,a.,...,rŽ a 1,...,a.,h 1,...,h 1 1 r1 1 r1 r Ž a 1,...,a N,h 1,...,h., wth,..., Ž, 1,..., 1.. I other words, the operad U Ž A. 1 r s geerated by A wth relatos gve by Ž,...,,1,...,1 a,...,a,h,...,h a,...,a 1 r.ž 1 N 1. Ž 1 r,h 1,...,h.. I a geerator Ž a,...,a,h,...,h. 1 r 1, we assume the elemets of the algebra ad the extra-varables to be a strct order. But, we may drop ths assumpto. The symmetrc group Sr may permute the factors of the tesor a a h h. 1 r 1 Nevertheless, ay tesor has a well ordered tesor ts Sr-orbt. Hece, ay geerator wth the a ad the h dsordered s equvalet to a well-ordered geerator uder the Sr-acto. Fally, the two costruc- tos gve the same set of geerators. Notce also that the evelopg operad s geerated by the S-module A, whch s A Ž. 0 degree 0 ad Ž. degree LEMMA. The operad U Ž A. s a augmeted -operad,.e., we hae a atural par of operad morphsms such that. UŽ A., roof. As s a summad of A, we obta a caocal S-module morphsm A TŽ A. UŽ A.. We check easly that the composte s a operad morphsm. Ths gves. Ž. The projecto A duces a operad morphsm T A

13 LIE THEORY OF FORMAL GROUS 467. It s ot hard to check that ths morphsm duces a operad morphsm : U Ž A.. The detty s clear. I the case A Ž. 0, ad are deed verse somorphsms. Therefore, U Ž Ž THEOREM. Let A be a -algebra. The category of U Ž A. -algebras s equalet to the category of A-algebras. Ths theorem s proved by a mmedate specto. To coclude the study of the evelopg operad, let us calculate t some uversal cases. I the case of a free -algebra, a easy calculato returs the followg result ROOSITION. If A s a free -algebra, say A T, V, the UŽ A. V V. I fact, ths case, the evelopg operad s uversal amog the objects equpped wth a operad morphsm U Ž A. ad a map VU Ž A.Ž 0.. Let us ow cosder the case of a -algebra coproduct. Let A, B be a par of -algebras, wth coproduct deoted by A B. Recall that the category of operads s cocomplete Žcf. Ge-J, Theorem Let, Q be operad morphsms. Form the pushout Q. We have a restrcto fuctor from the category of -algebras Ž resp. Q-algebras. to the category of -algebras Žcf. Ge-J, Ths s just as commutatve algebra. A Q-algebra cossts of a k-module equpped wth a -algebra ad a Q-algebra structure, whch restrct to the same - algebra structure. I partcular, a U Ž A. U Ž B. -algebra s a -algebra X, together wth a par of -algebra morphsms Ž A X, B X.. Hece, a U Ž A. U Ž B. -algebra s othg but a A B-algebra. As a cosequece, we have proved the followg proposto ROOSITION. We hae UŽ AB. UŽ A. UŽ B Remark. The presetato of U Ž A. by geerators ad relatos gve above shows that the algebra U Ž A.Ž 1. s the evelopg algebra defed by V. Gzburg ad M. Kapraov Ž cf. G-K.. A U Ž A.Ž 1. -module s kow as a A-module. Oe may show that a A-module s the same as a abela group the category of -algebras over A Ž see below Algebras oer a -Algebra. Let A be a -algebra. A -algebra oer A s a -algebra X together wth a -algebra morphsm X : X A. We deote by -AlgA the category of -algebras over A.

14 468 BENOIT FRESSE A coected A-algebra s a A-algebra X together wth a A-algebra morphsm X : X A. I other words, X s equpped wth two -algebra morphsms X: A X ad X: X A, ad XX A. A coected 0 Ž. -algebra X s called a coected -algebra. By the trck of the evelopg operad, ay coected algebra s equvalet to a coected algebra over a operad. As a example, the free algebra TŽ, V. s a coected -algebra. Let us cosder the operad, defed by 0, f 0, Ž. ½ Ž., otherwse. A -algebra X gves rse to a coected -algebra X. We set X 0Xad Ž. we equp X wth the followg product. We defe 0X Ž. to be the caocal embeddg. Let Ž., a 1,...,a 0, Ž. x,..., x X. We expad Ž a x,...,a x We obta a sum of terms depedg at most learly each varable a ad x.by evaluato of the operad products r r 0 1 r we elmate the a from ths expaso. Hece, we obta a sum of the kd Ý Ž a,...,a. Ž x,..., x., 1 I 1 r IŽ 1,...,r. where I rages over the o-vod subsets of 1,..., 4. The term Ž a,...,a. gves the compoet of the product o Ž After evaluato, the other terms gves the compoet of the product o X. Clearly, ths costructo provdes X wth a -algebra structure LEMMA. The fuctor X X duces a somorphsm betwee the category of -algebras ad the category of coected -algebras. roof. The proof of ths lemma s mmedate. Just otce that the mage of a coected -algebra X uder the verse fuctor s ker X The Reduced Free Algebra The Wreath roduct. Let A be ay assocatve rg. The wreath product A S s the k-module A S equpped wth the followg Ž. assocatve product. Let,..., ;s,,..., ;s A S. We set 1 1,..., ;s,..., ;s,..., ; ss sž1. sž.

15 LIE THEORY OF FORMAL GROUS 469 If V s a left A-module, the ts th tesor power s equpped wth a left A S -module structure defed by Ž 1,..., ;s. Ž 1,...,. Ž 1 sž1.,..., sž.., for 1,..., ;s AS ad 1,..., V. Recall that we deote K Ž. 1. The operad products Ž 1. Ž. Ž., Ž. Ž 1. Ž 1. Ž., make Ž. to a K K S-bmodule. The rght K S-module struc- ture s gve by Ž 1,..., ;s. sž 1,...,., for,..., ;s KS ad The Reduced Free -Algebra. Let A be a -algebra. Recall that the -algebra product Ž 1. A A provdes A wth the structure of a left K-module. I other words, we have a forgetful fuctor from the category of -algebras to the category of K-modules. Ths forgetful fuctor admts a left adjot, kow as the free -algebra geerated by a K-module Žor the reduced free -algebra, f oe prefers.. Let V be a left K-module. The free -algebra geerated by V s defed by T Ž, V. Ž. KS V. 0 Notce that ths algebra s coected. I the sequel, TŽ, V. deotes the free -algebra or the reduced free -algebra, whether V deotes a k-module or a left K-module. Notce that the free -algebra geerated by a k-module V s equal to the reduced free -algebra geerated by K V. For ths reaso, the sequel, geeral, we cosder oly the case of left K-modules Coproducts. The category of -algebras s complete ad cocomplete Žsee Ge-J, The coproduct of two -algebras s deoted by. I ths paper, we deal maly wth free -algebras or reduced free -algebras. Let V, W be left K-modules. I ths case, we have T Ž, V. T Ž, W. T Ž, V W..

16 470 BENOIT FRESSE 1.4. Complete Algebras ad ower Seres I ths secto, we defe the oto of a complete algebra over a operad. A reasoable complete algebra should be coected. As a cosequece, from ow o, we assume Ž If ot, we may replace by the operad. Of course, oe could follow the usual coveto dealg wth coected -algebras stead of -algebras. But our stuato, ths makes the formalsm heavy gog Ideals. A deal of a -algebra A s a submodule of A, say I, such that Ž a 1,...,a 1,b. I, Ž., a 1,...,a1R, bi. Clearly, the -algebra product TŽ, A. A duces a -algebra product o the quotet AI. k Let us deote by A the mage of k k S k A uder the product. Of course, A s a deal of A, called the th power of the augmetato deal of A. The quotet QA AA 2 s a left K-module, kow as the K-module of the decomposable elemets Complete Algebras. A -algebra s sad to be lpotet whe the product A A vashes for suffcetly large. A complete -algebra s a -algebra A together wth a sequece of deals I, 1, such that the -algebra AI s lpotet ad A lm AI. Notce that the hypothess AI lpotet mples that the topology defed by the th powers of the augmetato deal A s equvalet to the topology defed by the I. As a cosequece, ay -algebra mor- phsm s cotuous. Let A be a -algebra together wth a sequece of deals I, 1, such that the -algebra AI s lpotet. If Aˆ lm AI, the Aˆ s a complete -algebra kow as the completo of A wth respect to the topology gve by the I The Free Complete -Algebra. Let V be a left K-module. We deote by ˆŽ V. the free complete -algebra geerated by V. Explctly, ˆŽ V. s the completo of TŽ, V. wth respect to the powers of the augmetato deal. Sce T Ž, V. Ž. KS V,

17 LIE THEORY OF FORMAL GROUS 471 we have ˆ Ž V. Ł Ž. KS V. 1 Let A be a complete -algebra. The restrcto to V duces a oe-oe correspodece betwee -algebra morphsms ˆŽ V. A ad K-module morphsms V A ower Seres. Let V be the K-module freely geerated by the varables x,..., x. I ths case, we wrte ˆŽ x,..., x. for ˆŽ V A elemet of ˆŽ x,..., x. 1 s called a -power seres. Let be a mult- 1 dex,...,. We deote by x the tesor x x 1. Ay elemet of ˆŽ x,..., x. has a uque expaso of the form 1 Ý x. Hece, the ame -power seres. A -tuple of varables Ž x,..., x. 1 may be deoted by a sgle letter x. For Com, we recover the classcal power seres. For As, we obta the power seres o-commutg varables. Sce V s freely geerated, ad because of the dscusso above, a -algebra morphsm ˆŽ x,..., x. 1 A s equvalet to a -tuple of elemets a,...,a A. If fž x. Ý Ž x. ˆ Ž x. 1, the ts mage uder the correspodg -algebra morphsm s gve by Ý a A. Ž. Ths sum s well defed sce a R for suffcetly large. Whe A ˆŽ y., the elemets a 1,...,a are -power seres, ad ths process s kow as the substtuto of -power seres Operads ad Aalyzers. The usual substtuto rules of power seres hold over a geeral operad. The substtuto of -power seres makes the sequece of Ž x,..., x. 1 to a aalyzer the sese of Lazard Žcf. L1, Chap. I.. I ths way, we obta a equvalece betwee the oto of a operad ad the oto of a multlear aalyzer Žcf. L1, p But a aalyzer s ot multlear geeral. Let us recall how to recover the operad from ts aalyzer. We have a atural morphsm M: Ž. ˆ Ž x,..., x. defed by 1 M x,..., x. 1

18 472 BENOIT FRESSE Ths morphsm detfes ˆŽ. wth the submodule of Ž x,..., x. 1 geerated by the multlear moomals,.e., by the moomals of degree 1 each varable x. Ths morphsm s S-equvarat sce M Ž x 1,..., x. Ž x Ž1.,..., x Ž... Moreover, let us cosder the operato whch maps the -power seres fž x,..., x ˆ 1. Ž x 1,..., x. g x,..., x ˆ 1Ž 1. Ž x 1,..., x g Ž x,..., x. ˆŽ x,..., x. 1 1 to the power seres obtaed by the substtuto Ž f g Ž x,..., x.,..., g Ž x,..., x Clearly, ths operato restrcts to the composto law of the operad The Completed Coproduct. The category of complete -algebras s equpped wth a coproduct. Let A lm AI k, B lm BJk be com- plete -algebras. The coproduct of A ad B s the -algebra lm A BŽ I BAJ. k k,.e., the completo of A B wth respect to the sequece of deals Ik B A Jk A B. Whe A ad B are as- sumed to be complete, A B deotes ther completed coproduct. We deote by : A A A the foldg map, whose restrcto to each summad of the coproduct s the detty. Recall that the ull space s a zero object the category of complete -algebras. We deote by 0 the zero arrow The Eelopg Operad of a Complete -Algebra. We eed to adapt the evelopg operad costructo, for workg wth complete -algebras, because a complete -algebra uder A s ot equvalet to a U Ž A. -algebra. Let A lm AI k be a complete -algebra. By usg the tesor products Ý Sr k Ž r. Ž A I A., we equp U Ž A.Ž. wth a caocal topology. We defe the completed evelopg operad Uˆ Ž A.Ž., as the completo of U Ž A.Ž. wth respect to ths topology. We equp Uˆ Ž A.Ž. wth the duced operad product.

19 LIE THEORY OF FORMAL GROUS 473 I the sequel, f A s assume to be a complete -algebra, the the evelopg operad of A deotes the completed oe. I the same way, we omt the hat the otato. For example, f A ˆŽ x., the a elemet of U Ž ˆŽ x..ž. s a power seres lke Ý x Ž, h 1,...,h.. Furthermore, the augmetato : U Ž ˆŽ x.. s gve by the evaluato at x 0 Ž u. u x Basc Deftos 2. FORMAL GROUS DEFINITION. A cogroup object the category of complete - algebras s a complete -algebra R together wth a coproduct : R R R, ad a atpode : R R such that the followg usual dettes are satsfed R 0 0 R R, Ž RR, Ž RR0. Ž A cogroup morphsm s a -algebra morphsm whch commutes wth the coproduct. I the classcal case Com, a cogroup object s just a commutatve complete Hopf algebra. Notce that the atpode s uque. Let us deote by R 3 R, the 3-fold coproduct. I the same way, we deote 3 RR.Ifs aother atpode, the we have 0R DEFINITION Ž Formal Group.. A fte dmesoal -formal group s a cogroup the category of complete -algebras, whose uderlyg -algebra s freely geerated by a ftely geerated projectve K-module. I the sequel, ay -formal group s tactly assumed to be fte dmesoal ROOSITION. A -formal group s equalet to a free complete -algebra R ˆŽ V., wth V a ftely geerated projectek-module, equpped wth a K-lear map : V ˆŽ V V.. Ths map duces a coproduct, also

20 474 BENOIT FRESSE deoted by. Ths coproduct s assumed to erfy the dettes R 0 0 R R, Ž RR. Ž I fact, ths proposto s true eve f R s ot assumed to be a free complete -algebra. We refer to F for a detaled demostrato. We just sketch the proof the free case here. roof. We have to costruct a atpode. For coveece, a coproduct, we label each copy of V by a varable. The caocal projectos provde a -algebra morphsm ˆŽ V V. ˆŽ V. ˆŽ V.. x y x y Ths morphsm s surjectve, ad has a lear splttg gve by the sum of the caocal clusos. The kerel of ths morphsm, deoted by ˆŽ V x V y. s kow as the cross-effect of the fuctor. ˆ A elemet of ˆŽV V. x y s a moomal wth at least oe factor belogg to the copy V x, resp. V y. Thus, we have a caocal splttg ˆ V V ˆŽ V. ˆ V ˆ V V. Ž x y. x Ž y. Ž x y. Because of the ut equato Ž , the projecto of the coproduct oto the compoet ˆŽ V., resp. ˆŽ V. x y, s gve by the caocal mor- phsm V ˆŽ V.. We ft ths decomposto of to a step by step approxmato process. Rug ths approxmato process, we get a map : V ˆŽ V. whch s a soluto to the equato R0. I the same way, we get : V ˆŽ V. such that R0. The argumet provg the uqueess of the atpode shows also. Let R be a complete -algebra equpped wth a cogroup structure. Recall that QR RR 2. If VQR s a ftely geerated projectve K-module, the, usg the methods volved F, oe may show R ˆŽ V.. I other words, R s ecessarly a fte dmesoal -formal group. Assume that V s freely geerated by the varables x 1,..., x. I ths case, ˆŽ V V. ˆŽ x, y., ad a coproduct : V ˆŽ V V. x y x y s equv- alet to a -tuple of -power seres GŽ x, y.. As a cosequece, a -formal group s equvalet to a -formal group law:

21 LIE THEORY OF FORMAL GROUS DEFINITION. A -dmesoal -formal group law s a -tuple of -power seres 2-varables Ž x, y. Ž x,..., x, y,..., y., say such that 1 1 GŽ x, y. Ž G Ž x, y.,...,g Ž x, y.. ˆ Ž x, y., 1 GŽ x,0. GŽ 0, x. x, 1,...,, Ž G GŽ x, y., z G x,gž y, z., 1,...,. Ž Moreover, a cogroup morphsm from a -dmesoal -formal group law GŽ x, y. to GŽ x, y., s equvalet to a -tuple of -power seres Ž x., such that GŽŽ x., Ž y.. ŽGŽ x, y... The atpode of GŽ x, y. s provded by a -tuple of -power seres Ž x.. These -power seres verfy the equato GŽx,Ž x.. GŽŽ x., x. 0. I ths cotext, the step by step approxmato process s a algebrac verso of the local verso theorem. I the case Com, we recover the classcal defto of a formal group law. For a comprehesve accout of ths theory, we refer to the followg textbooks H, Z, Se Group Fuctors. Let R be a cogroup object. The coproduct of R makes -AlgŽ R,. to a fuctor from lpotet -algebras to the category of groups. For example, suppose that R s a free -algebra, say R ˆŽ x,..., x. 1. Let GŽ x, y. ˆ Ž x, y. be the -formal group law assocated to the co- product. I ths case, Alg R, A A. The group structure s gve by Ž a,...,a. Ž b,...,b. G Ž a, b.,...,g Ž a, b.. 1 G 1 1 Coversely, ay reasoable group fuctor turs out to be gve by a -formal group Ž cf. the ext theorem for a precse statemet.. I the classcal case Com, ths process s explaed wth some detals Z, Chap. II. I fact, ths referece, we may assume to be ay operad. For ths reaso, we just recall the followg deftos ad the state the theorem wthout proof. Let G be a fuctor from lpotet -algebras to the category of groups. Ths fuctor s kow to be smooth whe t preserves surjectos. For example, f G -AlgŽ ˆ Ž V.,. for some projectve K-module V, the GŽ A. Mod Ž V, A. K. Hece, ths case G s smooth. Let us cosder K as a 2-lpotet -algebra. The group structure of GK turs out to be abela. I fact, GK s equpped wth a rght K-module structure. For example, f G -AlgŽ R,. for some complete -algebra R, the GK Mod Ž QR, K.. K

22 476 BENOIT FRESSE THEOREM. Let G be a fuctor from lpotet -algebras to the category of groups. Assume that G preseres the fbered products. Assume that G preseres the drect sums. If G s smooth ad GŽ K. s a ftely geerated projecte K-module, the G s proded by a -formal group Notato. Followg Coveto 0.1, we specfy a -formal group by the sgle letter G. The uderlyg complete -algebra s deoted by RG. Ths -algebra s kow as the algebra of regular fuctos of the -formal group G Examples I ths secto, we gve examples of L e-formal group laws The CampbellHausdorff Formula. The CampbellHausdorff formula provdes a example of a L e-formal group law. Let Ž x, y. be the power seres two o-commutg varables whch verfes the equato exp x exp y exp Ž x, y.. Ths power seres s clearly a As-formal group law. Recall that L e Ž x, y. s a subspace of As Ž x, y. Ž cf. Examples It s ot hard to show that Ž x, y. s prmtve, ad hece belogs to L ež x, y. Žcf. H, Theorem I some sese, ths formal group law has a L e-structure. As a cocluso, the power seres Ž x, y. defes L e-formal group law. I characterstc zero, ths s certaly the ma example of a formal group law. It follows from the results of Lazard L1, Theorem 6.1 ad roposto 7.2, that ay -formal group law s somorphc to a -formal group law bult from the CampbellHausdorff formula Žcf. also Subsectos ad We refer to Se for a accout of ths result the classcal case of a Com-formal group law. I the ext paragraph, followg M. Kotsevch, we gve a accout of a varat of ths costructo Costructo Žcf. K, p For smplcty, we assume Ž 1. Kk. Suppose that s a quadratc operad Ž cf. Subsecto , such as As, L e, Com. I G-K, by aalogy wth the theory of quadratc algebras Žcf.., V. Gzburg ad M. Kapraov defe a dualty betwee quadratc operads, kow as the quadratc dualty. The quadratc dual operad of s deoted by!. For example, we have As! As, Com! L e, L e! Com. Let L be a fte dmesoal! algebra. Let A be a -algebra. The algebra L A s edowed wth a L e-algebra structure, whch s defed as follows. Recall that! Ž 2. s the K-lear dual of Ž 2. equpped wth the cotragredet represetato. Let deote a bass of Ž. 2 ad let deote the dual bass. The Le bracket o L A s gve by the

23 LIE THEORY OF FORMAL GROUS 477 formula Ý Xa,Yb Ž X,Y. Ž a, b., X,YL, a, br. Moreover, f A s a lpotet -algebra the L A s a lpotet L e-algebra. Thus, the CampbellHausdorff formula edows L A wth the structure of a group. Clearly, ths costructo Ž wth L fxed. provdes a fuctor from the category of lpotet -algebras to the category of groups, whch s deoted by exp L. We have L R Mod Ž L*, R. k - AlgŽˆ Ž L*., R.. Therefore exp L s a -formal group Explct Calculatos. Let us gve a power seres expaso of the assocated formal group law the case L e,! Com. The process s well kow Žcf. R1, Lo2.. Let x 1,..., x deote a bass of L. Cosder a sequece of dces IŽ,...,.. We deote the teger r by lž I. 1 r. Cosder a -tuple of varables X 1,..., X. We deote by XI the moomal X X. 1 r ROOSITION. The L e-formal group law of exp L has the expaso 1 1 Ž X, Y. Ý xixjelži.lžj. XIY J. lž I.!lŽ J.! I, J ŽWe detfy a elemet of L ˆŽ X, Y. wth a -tuple of -power seres by usg the bass x 1,..., x.. roof. Let V Ž resp. V. x y deote the vector space geerated by the varables X,..., X Ž resp. Y,...,Y.. Let TˆŽ V. 1 1 deote the completed tesor algebra Cosder the assocatve algebra ˆT V V. Ł 0 TTˆ Ž V V. Tˆ Ž V V.. x y x y The frst factor s equpped wth the shuffle product, ad the secod wth the cocateato product. Cosder the elemet S Ý w w T, where w rages over the set of moomals X, Y.

24 478 BENOIT FRESSE Ž. By a theorem of Ree cf. Re, we have log S Ý u u, for some Le polyomals. Ž u The varable u rages over the set of o-costat moomals X, Y.. I fact, the map u u s the projec- to oto L ež V V., provded by the frst Eulera dempotet Ž x y cf.. 1 Examples More precsely, f u Vx V y, the u e u. Cosder the map : TVV ˆ x y Lwhch maps the moomals XY I I to 1Ž I!J!x. Ix J, ad whch cacels the other moomals. Clearly, s a algebra morphsm ŽTVV ˆ x y beg equpped wth the shuffle product.. Therefore 1 log S logž 1 S.. The rght sde of ths equato s equal to log expž x X x X. expž x X x X The left sde of ths equato s the sum Hece, we are doe. 1 x x. Ž I.!Ž J.! Ý I J u uxy I J 3. LINEAR ALGEBRA OVER AN OERAD Recall that a operad s a assocatve algebra for the composto product. Therefore, we have a aalogy betwee rgs ad operads. Followg ths aalogy, we ca do lear algebra over a operad. Ths s the am of ths secto. I the frst subsecto, we defe ad study the oto of a rght module over a operad. Maly, we show that the category of rght -modules s symmetrc moodal, ad possesses a teral hom fuctor. I the secod subsecto, we make the free complete -algebras dual to the cofree cocommutatve coalgebras the symmetrc moodal category of rght -modules. Ths dualty plays the role of Carter dualty our settg Rght Modules oer a Operad DEFINITION. Fx a operad, wth product deoted by : ad ut deoted by : I.

25 LIE THEORY OF FORMAL GROUS 479 A rght -module V s a S-module together wth a S-module morphsm such that : V V V, V V. As for a operad, ths meas that we have lear maps : V Ž. Ž 1. Ž. V Ž 1. whch satsfy the May axoms, except that we put V the frst posto stead of. A morphsm of rght -modules f: V W s a morphsm of S-modules whch commutes wth the -acto,.e., such that f Ž,...,. fž,...,., V Ž.,,..., A rght -module morphsm s also kow as a -lear map. We deote by Mod the category of rght -modules EXAMLE. Let E be a rght K-module. The free rght -module geerated by E, deoted by E, s gve by the S-module E Ž. E Ž., equpped wth the obvous rght -acto. ŽRecall that the operad product Ž 1. Ž. Ž. makes Ž. to a left K-module.. Let V be a rght -module. Ay rght K-module morphsm E V Ž. 1 gves rse to a uque rght -module morphsm E V. Let V be a left K-module. Let Mod Ž V,. K be the sequece Mod Ž V,.Ž. Mod ŽV, Ž... We equp Mod Ž V,. K K K wth the obvous rght -module structure, defed by f Ž 1,...,. fž 1,...,., for f Mod Ž V,.Ž.,,...,, V. The module Mod Ž V,. K 1 k s a kd of -lear dual of V. Suppose V to be a ftely geerated projectve K-module. I ths case, we have a caocal somorphsm Mod Ž V, K. Ž. Mod V, Ž., K K K ad hece, Mod V, s a free rght -module. K K

26 480 BENOIT FRESSE ROOSITION. Let V, W be two rght -modules. There exsts a uque rght -module structure o V W such that Ž w.ž 1,...,j. Ž 1,...,. wž 1,..., j., V Ž., w WŽ j., 1,...,. S roof. Cosder a arbtrary elemet w Id N V Ž. S S j WŽ j. V WŽ N.. The equvarace codto o the product forces us to set Ž w.ž,...,. 1,..., Ž,...,. wž,..., Ž1. Žj. We let the reader check that ths defto s cosstet ad provdes V W wth a rght -module structure COROLLARY. The category Mod, s symmetrc moodal. roof. We have just to check that the twstg map c : V W V, W W V s -lear. The other verfcatos are obvous. We check frst c Ž w. Ž,...,. V, W 1. 1 Ž, w. w Ž,...,., wž,...,. 1 1 j 1 j Ž,...,. 1 c Ž w.ž,...,., V, W 1 ad the geeral case follows by S -equvarace Remark. Recall that I s equpped wth a operad structure Žcf. Subsecto Furthermore, a rght I-module structure s othg more tha a rght S-module structure, ad the moodal categores Ž Mod I,. ad Ž S-Mod,. are clearly somorphc Shfted Modules. Let V be a S-module. Let be a oegatve teger. We defe a S-module V, kow as the shfted module. We set V Ž r. V Ž r.. Cosder the embeddg 1,..., r4 1,...,r4 mappg 1,..., r oto 1,...,r. Ths embeddg provdes a group embeddg Sr S r, ad thus the S -module structure of V Ž r. V Ž r.. r

27 LIE THEORY OF FORMAL GROUS 481 Let V Ž r. V Ž r.. The tegers 1,..., are called the frst etres of ad 1,...,r the last r etres of. Hece, our defto, Sr acts o the last r etres of. Notce that V s a S- Ž S-module.. I fact, S acts o the frst etres of V Ž r.. Ths S-acto commutes obvously wth the Sr- module structure, sce the supports of these actos are dsjot Rght -Actos o Shfted Modules. If V s a rght -module, the V s equpped wth a rght -module structure. Let V Ž r.. We provde V wth a rght -module structure by lettg act o the last r etres of. More precsely, the mage of uder the product V Ž r. Ž. Ž. V Ž. 1 r 1 r s gve by Ž1,,...,. V Ž. V Ž. 1 r 1 r 1 r. The operad may also act o the frst etres of. Thus we obta maps V Ž r. Ž. Ž. V Ž r.. 1 These maps are -lear by assocatvty of the rght -module product. As for classcal graded modules, we use the shfted module the costructo of a teral Hom THEOREM. Let V, W be rght -modules. The bfuctor HOMŽ V, W.Ž. Mod Ž V, W. s a teral hom the moodal category Ž Mod,.. Explctly, we hae a caocal somorphsm Mod Ž U V, W. Mod Ž U, HOMŽ V, W... roof. Frst, we make HOMŽ V, W. to a rght -module. Let f HOMŽ V, W.Ž. Mod ŽV, W.. Recall that f cossts a sequece of maps f: V Ž r. W Ž r., whch commute wth the rght -module product. Let Ž.,..., Ž.. We defe 1 1 fž 1,...,. HOMŽ V, W.Ž 1. as follows. Let V Ž r.. We set Mod Ž V, W. 1 f,..., f,...,,1,...,1. 1 1

28 482 BENOIT FRESSE Oce aga, fž,...,. 1 s obvously -lear, because of the assocatvty of the rght -module product. Clearly, ths product provdes HOMŽ V, W. wth a rght -module structure. Let U, V, W Mod. It remas to show that Mod Ž U V, W. Mod Ž U, HOMŽ V, W... A arrow f S-ModŽ U V, W. s gve by a sequece f: Ž U V.Ž r. WŽ r., or equvaletly, by a sequece of S S -equvarat arrows Now, the map f, j f, j: UŽ. V Ž j. WŽ j.. correspods to a map g, j: U Ž. ModS j Ž V Ž j., W Ž j... Moreover, f s -lear f ad oly f Ž. 1 for each, for each u UŽ., the sequece g Ž u. Žg Ž u.., j j: VŽ j. WŽ j. s -lear,.e., g : UŽ. Mod ŽV, W., Ž 2. the sequece g Ž g. s -lear,.e., g Mod Ž U, HOMŽ V, W... Ths completes the proof of the theorem COROLLARY. Let V, W be S-modules. The bfuctor HOMŽ V, W.Ž. S-ModŽ V, W. s a teral hom the moodal category S-Mod, Composto. By some geeral osese argumets we have assocatve composto laws HOMŽ V, W. HOMŽ U, V. HOMŽ U, W.. Ideed, let g HOMŽ V, W.Ž., f HOMŽ U, V.Ž j.. The g f HOMŽ U, W.Ž j. s gve by the sequece of maps f UŽ r. V Ž jr. WŽ jr.. I partcular, f V Mod, the ENDŽ V. HOMŽ V, V. s a assocatve algebra the category Ž Mod, Coalgebra-Algebra Dualty Coalgebras. I ths secto, we cosder coalgebras the symmetrc moodal category of rght -modules. A coalgebra s a rght -module C, equpped wth a coassocatve coproduct : C C C ad g j

29 LIE THEORY OF FORMAL GROUS 483 a augmetato: : C 1. Explctly, ad satsfy the usual dettes C C, CCC. The coalgebra C s kow to be cocommutate whe cc, C. The coalgebra C s kow to be coected whe duces a somorphsm from CŽ. 0 to k. I the sequel, ay coalgebra s assumed to be coected. I ths case, we have CŽ. 0 k 1 ad Furthermore, the coproduct of c CŽ N. has the expaso Ý Ž1. Ž2. c c c 1 c c 1 c, wth S, c C p, c Cq, Ž1. Ž2. for some p, q such that p q. Here, we use Sweedler s otato. More geerally, gve c C, a expresso lke ÝŽc. cž1. cž. deotes the -fold coproduct of c. Sometmes we may omt the summato dex. Usg Sweedler s otato, the cocommutatvty of the coproduct may be wrtte Ý Ý c c c, c c c. Ž1. Ž2. Ž1. Ž2. Ž2. Ž1. Ž c. Ž c Cofree Coalgebras. A rght -module V s kow to be coected whe V Ž There exsts a obvous forgetful fuctor from the category of cocommutatve coalgebras to the category of coected rght -modules. Ths fuctor has a rght adjot kow as the cofree cocommutate coalgebra. We deote by CŽ V. the cofree cocommutatve coalgebra cogeerated by V. Ths coalgebra s edowed wth a -lear map : CŽ V. ad s characterzed by the followg uversal property. Let C be a cocommutatve coalgebra; let : C V be a -lear map. There exsts a uque coalgebra morphsm : C CŽ V. such that. Let,..., V, let S, wth N 1 N 1. We deote by 1,..., the tesor 1 V. Recall that the symmetrc group S acts o V by place permutato, ad that ths acto s gve by s* Ž,...,. s,..., Ž,...,.. We deote by ŽV the rght -module 1 1 sž1. sž.. S the rght -module of varat tesors. Cosder S C Ž V. Ž V.. 0

30 484 BENOIT FRESSE We equp CŽ V. wth the coproduct defed o each summad by the caocal cluso S p Sp q Sq Ž V. Ž V. Ž V., Ž. Ž wth p q. Hece, f Ý,..., V. S Ž1. Ž., the we have ž Ý Ž Ž1. Ž../,..., Ý Ž1. Ž p. Ž p1. Ž pq. pq,...,,..., THEOREM. The coalgebra CŽ V. s the cofree cocommutate coalgebra cogeerated by V. The uersal arrow : CŽ V. V s ge by the projecto of CŽ V. oto the summad V. roof. Let : C V be a rght -module morphsm. Frst, we check the exstece of a coalgebra morphsm : C CŽ V. such that. Cosder the map Ý ž Ž Ž1.. Ž Ž../ c c,..., c. Notce that s well-defed, because C s coected ad V Ž The map s obvously -lear. Moreover, we have Ý ž Ž Ž1.. Ž Ž p../ c c,..., c pq ž Ž Ž p1.. Ž Ž pq../ c,..., c. Because of the coassocatvty of the coproduct, ths last expresso s exactly Ž c.. Hece, s a coalgebra morphsm. It remas to show the uqueess of such a factorzato. Let us call Ž V. S the compoet of order of CŽ V.. Let : C CŽ V. be Ž aother factorzato of. Wrte c Ý,...,. 1. Sce s a coalgebra morphsm, we have Ž c. Ž c., where deotes the -fold coproduct. O oe had, the multlear compoet Ž.e., the compoet of order oe each factor of the tesor

31 LIE THEORY OF FORMAL GROUS 485. product of the left sde of ths equato s. Ý 1 O the other had, the multlear compoet of the rght sde s Ý Ž cž1.. Ž cž.., because the restrcto of to V s gve by by hypothess. Hece, the compoet of order of Ž c. agrees wth that of Ž c Shuffle Coproduct o the Symmetrc Algebra. For the eeds of the ocarebrkhoffwtt theorem, we supply aother realzato of the Ž cofree cocommutatve coalgebra. We deote by V. S the covarats of the tesor power uder the acto of the symmetrc group. Cosder the rght -module SŽ V. Ž V. S. 0 Let X,..., X V ad let S, wth N X X 1 N 1.Wede- ote by X1 X the mage of the tesor X1 X uder the quotet map. By defto, we have s* X X s X,..., X X X X X. 1 1 sž1. sž. 1 Let S h p, q deote the set of p, q-shuffles,.e., the set of permutatos ss such that sž. 1 sž p. ad sž p1. sž pq. pq.we equp S Ž V. wth the coproduct defed by Ý Ý X X s X,..., X X X 1 1 sž1. sž p. pq ssh p, q XsŽ p1. X sž pq THEOREM. The coalgebra morphsm : SŽ V. CŽ V. duced by the projecto oto the lear compoet of SŽ V. s a coalgebra somorphsm. roof. By a mmedate calculato, we obta Ž 1. Ý Ž 1. Ž sž1. sž.. X X s X,..., X X,..., X. ss

32 486 BENOIT FRESSE Hece, s clearly a somorphsm, wth a verse somorphsm gve by ž Ý / Ý! 1 1 X,..., X X X, Ž1. Ž. Ž1. Ž. S for Ý Ž X,..., X. Ž V. Ž1. Ž Coalgebra-Algebra args. Let C be a coalgebra; let R be a complete -algebra. We equp the S-module Ž C R.Ž. CR wth the obvous rght -acto. A coalgebra-algebra parg betwee C ad R s a -lear map B: C R, whch makes the coproduct of C adjot to the product of R. More precsely, we have Ý ž / B c,ž r 1,...,r. B cž1., r 1,...,B cž., r, Ž., cc, r 1,...,rR. Notce that Ž. 0 0 mples BŽ 1, r. 0, r The Caocal Coalgebra-Algebra arg. Let V be a left K- module. Let us cosder the cofree coalgebra CŽMod Ž V,.. K. We defe a caocal coalgebra-algebra parg ²,:: CŽ Mod Ž V,.. ˆ Ž V. K as follows. Let c be ay homogeeous elemet of order of Ž. Ž. C ModK V,. Hece, c Ý X Ž1.,..., X Ž., wth X Ž1.,..., XŽ. Mod Ž V,.. Let r Ž,...,. ˆŽ V.. We set K 1 m ² c, r: 0, Ý Ž Ž1. 1 Ž.. ² X, :,..., ² X, :, f m, otherwse. Ž Ths parg s well defed sce Ý X,..., X. Ž1. Ž. s S-varat. It s straghtforward to check that t makes the coproduct of CŽMod Ž V,.. K adjot to the product of ˆŽ V.. Sce SŽMod Ž V,.. s somorphc to CŽMod Ž V,.. K K, we obta a somorphc parg betwee SŽMod Ž V,.. ad ˆŽ V. K. As a cosequece, the ext three lemmas, we ca equvaletly replace the coalgebra CŽMod Ž V,.. by SŽMod Ž V,... K K