ANNALS OF MATHEMATICS
Lie theory for nilpotent L -algebras 1 By Ezra Getzler
SECOND SERIES, VOL. 170, NO. 1 July, 2009
anmaah
Annals of Mathematics, 170 (2009), 271–301
Lie theory for nilpotent L -algebras 1 By EZRA GETZLER
Dedicated to Ross Street on his sixtieth birthday
Abstract
The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor from L -algebras concentrated in degree > n to n-groupoids. (We actually construct the1 nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont’s proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula. In the case of abelian L algebras (i.e., chain complexes), the functor is the Dold-Kan simplicial set. 1
1. Introduction
Let A be a differential graded (dg) commutative algebra over a field K of characteristic 0. Let be the simplicial dg commutative algebra over K whose n-simplices are the algebraic differential forms on the n-simplex n. Sullivan [Sul77, 8] introduced a functor
A Spec .A/ dAlg.A; / 7! D from dg commutative algebras to simplicial sets; here, dAlg.A; B/ is the set of morphisms of dg algebras from A to B. (Sullivan uses the notation A for this func- h i tor.) This functor generalizes the spectrum, in the sense that if A is a commutative
This paper was written while I was a guest of R. Pandharipande at Princeton University, with partial support from the Packard Foundation. I received further support from the IAS, and from NSF Grant DMS-0072508.
271 272 EZRAGETZLER algebra, Spec .A/ is the discrete simplicial set Spec.A/ Alg.A; K/; D where Alg.A; B/ is the set of morphisms of algebras from A to B. If E is a flat vector bundle on a manifold M , the complex of differential forms ..M; E/; d/ is a dg module for the dg Lie algebra .M; End.E//; denote the action by . To a one-form ˛ 1.M; End.E// is associated a covariant derivative 2 d .˛/ .M; E/ 1.M; E/: r D C W ! C The equation 2 d˛ 1 Œ˛; ˛ r D C 2 shows that is a differential if and only if ˛ satisfies the Maurer-Cartan equation r d˛ 1 Œ˛; ˛ 0: C 2 D This example, and others such as the deformation theory of complex manifolds of Kodaira and Spencer, motivates the introduction of the Maurer-Cartan set of a dg Lie algebra g [NR66]: MC.g/ ˛ g1 ı˛ 1 Œ˛; ˛ 0 : D f 2 j C 2 D g There is a close relationship between the Maurer-Cartan set and Sullivan’s functor Spec .A/, which we now explain. The complex of Chevalley-Eilenberg cochains C .g/ of a dg Lie algebra g is a dg commutative algebra whose underlying graded commutative algebra is the graded symmetric algebra S.gŒ1_/; here, gŒ1 is the shifted cochain complex .gŒ1/i gi 1, and gŒ1 is its dual. D C _ If g is a dg Lie algebra and is a dg commutative algebra, the tensor product complex g carries a natural structure of a dg Lie algebra, with bracket ˝ Œx a; y b . 1/ a y Œx; y ab: ˝ ˝ D j jj j
PROPOSITION 1.1. Let g be a dg Lie algebra whose underlying cochain com- plex is bounded below and finite-dimensional in each degree. Then there is a natural identification between the n-simplices of Spec .C .g// and the Maurer-Cartan elements of g n. ˝ Proof. Under the stated hypotheses on g, there is a natural identification MC.g / dAlg.C .g/; / ˝ Š for any dg commutative algebra . Indeed, there is an inclusion dAlg.C .g/; / Alg.C .g/; / Alg.S.gŒ1 /; / .g /1: D _ Š ˝ LIE THEORY FOR NILPOTENT L -ALGEBRAS 273 1
It is easily seen that a morphism in Alg.C .g/; / is compatible with the differen- tials on C .g/ and if and only if the corresponding element of .g /1 satisfies ˝ the Maurer-Cartan equation.
Motivated by this proposition, we introduce for any dg Lie algebra the simpli- cial set MC .g/ MC.g /: D ˝ According to rational homotopy theory, the functor g MC .g/ induces a cor- 7! respondence between the homotopy theories of nilpotent dg Lie algebras over ޑ concentrated in degrees . ; 0 and nilpotent rational topological spaces. The 1 simplicial set MC .g/ has been studied in great detail by Hinich [Hin97]; he calls it the nerve of g and denotes it by †.g/. However, the simplicial set MC .g/ is not the subject of this paper. Suppose that g is a nilpotent Lie algebra, and let G be the simply-connected Lie group associated to g. The nerve N G of G is substantially smaller than MC .g/, but they are homotopy equivalent. In this paper, we construct a natural homotopy equivalence
(1-1) N G, MC .g/; ! as a special case of a construction applicable to any nilpotent dg Lie algebra. To motivate the construction of the embedding (1-1), we may start by compar- ing the sets of 1-simplices of N G and of MC .g/. The Maurer-Cartan equation on g 1 is tautologically satisfied, since g 1 vanishes in degree 2; thus ˝ 1 ˝ MC1.g/ gŒtdt. Let ˛ .G; g/ be the unique left-invariant one-form whose Š 2 value ˛.e/ TeG g at the identity element e G is the natural identification W ! 2 between the tangent space TeG of G at e and its Lie algebra g. Consider the path space P G Mor.ށ1;G/ .0/ e D f 2 j D g of algebraic morphisms from the affine line ށ1 to G. There is an isomorphism 1 between P G and the set MC1.g/, induced by associating to a path ށ G W ! the one-form ˛. There is a foliation of P G, whose leaves are the fibres of the evaluation map .1/, and whose leaf space is G. Under the isomorphism between P G 7! and MC1.g/, this foliation is simple to characterize: the tangent space to the leaf containing ˛ MC1.g/ is the image under the covariant derivative 2 0 1 g g T˛ MC1.g/ r W ˝ 1 ! ˝ 1 Š of the subspace x g 0 x.0/ x.1/ 0 : f 2 ˝ 1 j D D g 274 EZRAGETZLER
The exponential map exp g G is a bijection for nilpotent Lie algebras; equiva- W ! lently, each leaf of this foliation of MC1.g/ contains a unique constant one-form. The embedding N1G, MC1.g/ is the inclusion of the constant one-forms into ! MC1.g/. What is a correct analogue in higher dimensions for the condition that a one- form on 1 is constant? Dupont’s explicit proof of the de Rham theorem [Dup76], 1 [Dup78] relies on a chain homotopy s . This homotopy induces 1 0 W ! maps sn g g , and we impose the gauge condition sn˛ 0, which W ˝ n ! ˝ n D when n 1 is the condition that ˛ is constant. The main theorem of this paper D shows that the simplicial set (1-2) .g/ ˛ MC .g/ s ˛ 0 D f 2 j D g is isomorphic to the nerve N G. The key to the proof of this isomorphism is the verification that .g/ is a Kan complex, that is, that it satisfies the extension condition in all dimensions. In fact, we give explicit formulas for the required extensions, which yield in particular a new approach to the Campbell-Hausdorff formula. The definition of .g/ works mutatis mutandi for nilpotent dg Lie algebras; we argue that .g/ is a good generalization to the differential graded setting of the Lie group associated to a nilpotent Lie algebra. For example, when g is a nilpotent dg Lie algebra concentrated in degrees Œ0; /, the simplicial set .g/ is isomorphic 1 to the nerve of the Deligne groupoid Ꮿ.g/. Recall the definition of this groupoid (cf. [GM88]). Let G be the nilpotent Lie group associated to the nilpotent Lie algebra g0 g. This Lie group acts on MC.g/ by the formula
n X1 ad.X/ .ı˛X/ (1-3) eX ˛ ˛ : D .n 1/Š n 0 D C The Deligne groupoid Ꮿ.g/ of g is the groupoid associated to this group action. There is a natural identification between 0.MC .g// and 0.Ꮿ.g// MC.g/=G. D Following Kodaira and Spencer, we see that this groupoid may be used to study the formal deformation theory of such geometric structures as complex structures on a manifold, holomorphic structures on a complex vector bundle over a complex manifold, and flat connections on a real vector bundle. In all of these cases, the dg Lie algebra g controlling the deformation the- ory is concentrated in degrees Œ0; /, and the associated formal moduli space is 1 0.MC .g//. On the other hand, in the deformation theory of Poisson structures on a manifold, the associated dg Lie algebra, known as the Schouten Lie algebra, is concentrated in degrees Œ 1; /. Thus, the theory of the Deligne groupoid does not 1 apply, and in fact the formal deformation theory is modeled by a 2-groupoid. (This LIE THEORY FOR NILPOTENT L -ALGEBRAS 275 1
2-groupoid was constructed by Deligne [Del94], and, independently, by Getzler [Get02, 2].) The functor .g/ allows the construction of a candidate Deligne `-groupoid, if the nilpotent dg Lie algebra g is concentrated in degrees . `; /. We 1 present the theory of `-groupoids in Section 2, following Duskin [Dus79], [Dus01] closely. It seemed most natural in writing this paper to work from the outset with a generalization of dg Lie algebras called L -algebras. We recall the definition of L -algebras in Section 4; these are similar1 to dg Lie algebras, except that they 1 have a graded antisymmetric bracket Œx1; : : : ; x , of degree 2 k, for each k. In k the setting of L -algebras, the definition of a Maurer-Cartan element becomes 1 X1 1 ı˛ Œ˛; : : : ; ˛ 0: C kŠ D k 2 „ ƒ‚ … D k times Given a nilpotent L -algebra g, we define a simplicial set .g/, whose n-simplices 1 are Maurer-Cartan elements ˛ g n such that sn˛ 0. We prove that .g/ is 2 ˝ D a Kan complex, and that the inclusion .g/, MC .g/ is a homotopy equivalence, ! by a method similar to that of [Kur62, 2]. The Dold-Kan functor K .V / [Dol58], [Kan58] is a functor from positively graded chain complexes (or equivalently, negatively graded cochain complexes) to simplicial abelian groups. The set of n-simplices of Kn.V / is the abelian group n (1-4) Kn.V / Chain.C . /; V / D of morphisms of chain complexes from the complex C .n/ of normalized simpli- cial chains on the simplicial set n to V . Eilenberg-Mac Lane spaces are obtained when the chain complex is concentrated in a single degree [EML53]. The functor .g/ is a nonabelian analogue of the Dold-Kan functor K .V /: if g is an abelian dg Lie algebra and concentrated in degrees . ; 1, there is a 1 natural isomorphism between .g/ and K .gŒ1/, since (1-4) has the equivalent form 0 n Kn.V / Z .C . / V; d ı/; D ˝ C n where C . / is the complex of normalized simplicial cochains on the simplicial set n. The functor has many good features: it carries surjective morphisms of nilpotent L -algebras to fibrations of simplicial sets, and carries a large class of weak equivalences1 of L -algebras to homotopy equivalences. And of course, it yields generalizations of1 the Deligne groupoid, and of the Deligne 2-groupoid, for L -algebras. It shares with MC an additional property: there is an action of the 1 symmetric group Sn 1 on the set of n-simplices n.g/ making into a functor from L -algebras toC symmetric sets, in the sense of [FL91]. In order to simplify 1 276 EZRAGETZLER the discussion, we have not emphasized this point, but this perhaps indicates that the correct setting for `-groupoids is the category of symmetric sets.
2. Kan complexes and `-groupoids
Kan complexes are a natural nonabelian analogue of chain complexes: just as the homology groups of chain complexes are defined by imposing an equivalence relation on a subset of the chains, the homotopy groups of Kan complexes are defined by imposing an equivalence relation on a subset of the simplices. Recall the definition of the category of simplicial sets. Let be the category of finite non-empty totally ordered sets. This category has a skeleton whose objects are the ordinals Œn .0 < 1 < < n/; this skeleton is generated by the D face maps d Œn 1 Œn for 0 k n, which are the injective maps k W ! Ä Ä i if i < k; d .i/ k D i 1 if i k; C and the degeneracy maps s Œn Œn 1 for 0 k n 1, which are the surjective k W ! Ä Ä maps i if i k; s .i/ Ä k D i 1 if i > k: A simplicial set X is a contravariant functor from to the category of sets. This amounts to a sequence of sets Xn X.Œn/ indexed by the natural numbers D n 0; 1; 2; : : : , and maps 2 f g ık X.dk/ Xn Xn 1 for 0 k n; D W ! Ä Ä k X.sk/ Xn 1 Xn for 0 k n; D W ! Ä Ä satisfying certain relations. (See [May92] for more details.) A degenerate simplex is one of the form i x; a nondegenerate simplex is one that is not degenerate. Simplicial sets form a category; we denote by sSet.X ;Y / the set of morphisms between two simplicial sets X and Y . n n 1 The geometric n-simplex is the convex hull of the unit vectors ek in ޒ C : n n 1 .t0; : : : ; tn/ Œ0; 1 t0 tn 1 : D f 2 C j C C D g n 1 Its kC1 faces of dimension k are the convex hulls of the nonempty subsets of C e0; : : : ; en of cardinality k 1. f g C The n-simplex n is the representable simplicial set n . ; Œn/. Thus, D the nondegenerate simplices of n correspond to the faces of the geometric simplex n n . By the Yoneda lemma, sSet. ;X / is naturally isomorphic to Xn. Let Œk denote the full subcategory of whose objects are the simplices Œi i k , and let sk be the restriction of a simplicial set from op to Œkop. f j Ä g k LIE THEORY FOR NILPOTENT L -ALGEBRAS 277 1
The functor skk has a right adjoint coskk, called the k-coskeleton, and we have n coskk.skk.X//n sSet.skk. /; X /: D n n n 1 n For 0 i n, let ƒi be the union of the faces dkŒ for k i. Ä Ä n ¤ An n-horn in X is a simplicial map from ƒi to X , or equivalently, a sequence of elements n .x0; : : : ; xi 1; ; xi 1; : : : ; xn/ .Xn 1/ C 2 such that @j xk @k 1xj for 0 j < k n. D Ä Ä Definition 2.1. A map f X Y of simplicial sets is a fibration if the maps W ! n n X sSet.ƒ ;X / n Y i n i sSet.ƒi ;Y / n W ! defined by n i .x/ .@0x; : : : ; @i 1x; ;@i 1x; : : : ; @nx/ f .x/ D C are surjective for all n > 0 and 0 i n. A simplicial set X is a Kan complex if Ä Ä the map from X to the terminal object 0 is a fibration. A Kan complex is minimal if the face map @i Xn Xn 1 factors through W ! n for all n > 0 and 0 i n. i Ä Ä A groupoid is a small category with invertible morphisms. Denote the sets of objects and morphisms of a groupoid G by G0 and G1, the source and target maps by s G1 G0 and t G0 G1, and the identity map by e G0 G1. The nerve W ! W ! W ! N G of a groupoid G is the simplicial set whose 0-simplices are the objects G0 of G, and whose n-simplices for n > 0 are the composable chains of n morphisms in G: n NnG Œg1; : : : ; gn .G1/ sgi tgi 1 : D f 2 j D C g The face and degeneracy maps are defined using the product and the identity of the groupoid:
8Œg2; : : : ; gn if k 0; < D @kŒg1; : : : ; gn Œg1; : : : ; gkgk 1; : : : ; gn if 0 < k < n; D : C Œg1; : : : ; gn 1 if k n; D 8Œetg1; g1; : : : ; gn 1 if k 0; < D kŒg1; : : : ; gn 1 Œg1; : : : ; gk 1; etgk; gk; : : : ; gn 1 if 0 < k < n; D : Œg1; : : : ; gn 1; esgn 1 if k n: D The following characterization of the nerves of groupoids was discovered by Grothendieck; we sketch the proof.
PROPOSITION 2.2. A simplicial set X is the nerve of a groupoid if and only if n n the maps i Xn sSet.ƒi ;X / are bijective for all n > 1. W ! 278 EZRAGETZLER
Proof. The nerve of a groupoid is a Kan complex; in fact, it is a very special 2 kind of Kan complex for which the maps i are not just surjective, but bijective. The unique filler of the horn . ; g; h/ is the 2-simplex Œh; h 1g, the unique filler of the horn .g; ; h/ is the 2-simplex Œh; g, and the unique filler of the horn .g; h; / 1 is the 2-simplex Œhg ; g. Thus, the uniqueness of fillers in dimension 2 exactly captures the associativity of the groupoid and the existence of inverses. The nerve of a groupoid is determined by its 2-skeleton, in the sense that
(2-5) N G cosk2.sk2.N G//: Š 2 n It follows from (2-5) and the bijectivity of the maps i that the maps i are bijective for all n > 1. n Conversely, given a Kan complex X such that i is bijective for n > 1, we can construct a groupoid G such that X N G: Gi Xi for i 0; 1, s @1 G1 G0, Š D D D W ! t @0 G1 G0, and e 0 G0 G1. D W ! D W ! Denote by x0; : : : ; xk 1; ; xk 1; : : : ; xn the unique n-simplex that fills the h C i horn n .x0; : : : ; xi 1; ; xi 1; : : : ; xn/ sSet.ƒi ; X/: C 2 Given a pair of morphisms g1; g2 G1 such that sg1 tg2, define their composition 2 D by the formula
g1g2 @1 g2; ; g1 : D h i Given three morphisms g1; g2; g3 G1 such that sg1 tg2 and sg2 tg3, the 2 D D 3-simplex x Œg1; g2; g3 X3 satisfies D 2 g1.g2g3/ @1@2x @1@1x .g1g2/g3 D D D I hence composition in G1 is associative. Here is a picture of the 3-simplex x:
e0 b B b b B b g1g2g3 B b b B b b B b g B g g b e3 1 B 1 2 B B B g2g3 B B g3 B B B e1 h hh hhhhhh B hhhhhhB g2 e2 LIE THEORY FOR NILPOTENT L -ALGEBRAS 279 1
The inverse of a morphism g G1 is defined by the formulas 2 1 g @0 ; etg; g @2 g; esg; : D h i D h i ` To see that these two expressions are equal, call them respectively g and g , and use associativity: g ` g `.gg / .g `g/g g : D D D It follows easily that .g 1/ 1 g, that g 1g esg and gg 1 etg, and that D D D sg 1 tg and tg 1 sg. D D It is clear that
sh @1@2 g; ; h @1@1 g; ; h s.gh/ D h i D h i D and that tg @0@0 g; ; h @0@1 g; ; h t.gh/: D h i D h i D We also see that g @11Œg @1Œg; esg g.esg/ D D D @10Œg @1Œetg; g .etg/g: D D D Thus, G is a groupoid. Since sk2.X / sk2.N G/, we conclude by (2-5) that Š X N G. Š Duskin has defined a sequence of functors …` from the category of Kan complexes to itself, which give a functorial realization of the Postnikov tower. (See [Dus79] and [Gle82], and for a more extended discussion, [Bek04].) Let be ` the equivalence relation of homotopy relative to the boundary on the set X` of `-simplices. Then sk`.X /= ` is a well-defined `-truncated simplicial set, and there is a map of truncated simplicial sets
sk`.X / sk`.X /= `; ! and by adjunction, a map of simplicial sets
X cosk`.sk`.X /= `/: ! Define …`.X / to be the image of this map. Then the functor …` is an idempotent monad on the category of Kan complexes. If x0 X0, we have 2 i .…`.X /; x0/ if i `; i .X ; x0/ Ä D 0 if i > `:
Thus …`.X / is a realization of the Postnikov `-section of the simplicial set X . For example, …0.X / is the discrete simplicial set 0.X /, and …1.X / is the nerve of the fundamental groupoid of X . It is interesting to compare …`.X / to other realizations of the Postnikov tower, such as cosk` 1.sk` 1.X //: it is a more economic realization of this homotopy type, and has aC more geometricC character. 280 EZRAGETZLER
We now recall Duskin’s notion of higher groupoid: he calls these `-dimensional hypergroupoids, but we simply call them weak `-groupoids.
Definition 2.3. A Kan complex X is a weak `-groupoid if …`.X / X n D or, equivalently, if the maps i are bijective for n > `; it is a weak `-group if in addition it is reduced (has a single 0-simplex).
The 0-simplices of an `-groupoid are interpreted as its objects and the 1- simplices as its morphisms. The composition gh of a pair of 1-morphisms with @1g @0h equals @1z, where z X2 is a filler of the horn D 2 2 .g; ; h/ sSet.ƒ1;X /: 2 If ` > 1, this composition is not canonical — it depends on the choice of the filler z X2 — but it is associative up to a homotopy, by the existence of fillers in 2 dimension 3. A weak 0-groupoid is a discrete set, while a weak 1-groupoid is the nerve of a groupoid, by Proposition 2.2. Duskin [Dus01] identifies weak 2-groupoids with the nerves of bigroupoids. A bigroupoid G is a bicategory whose 2-morphisms are invertible and whose 1-morphisms are equivalences; the nerve N G of G is a simplicial set whose 0-simplices are the objects of G, whose 1-simplices are the morphism of G, and whose 2-simplices x are the 2-morphisms with source @2x @0x and target @1x. ı The singular complex of a topological space is the simplicial set
n Sn.X/ Map. ; X/: D To see that this is a Kan complex, we observe that there is a continuous retraction from n n to ƒn . The fundamental `-groupoid of a topological space X is D j j j i j the weak `-groupoid …`.S .X//. For ` 0, this equals 0.X/, while for ` 1, it D D is the nerve of the fundamental groupoid of X. Often, weak `-groupoids come with explicit choices for fillers of horns: tenta- tively, we refer to such weak `-groupoids as `-groupoids. (Often, this term is used for what we call strict `-groupoids, but the latter are of little interest for ` > 2.) We may axiomatize `-groupoids by a weakened form of the axioms for simplicial T -complexes, studied by Dakin [Dak83] and Ashley [Ash88].
Definition 2.4. An `-groupoid is a simplicial set X together with a set of thin elements Tn Xn for each n > 0, satisfying the following conditions: (i) every degenerate simplex is thin; (ii) every horn has a unique thin filler; (iii) every n-simplex is thin if n > `. LIE THEORY FOR NILPOTENT L -ALGEBRAS 281 1
If g is an `-groupoid and n > `, we denote by x0; : : : ; xi 1; ; xi 1; : : : ; xn h C i the unique thin filler of the horn n .x0; : : : ; xi 1; ; xi 1; : : : ; xn/ sSet.ƒi ;X /: C 2 Definition 2.5. An -groupoid is a simplicial set X together with a set of 1 thin elements Tn Xn for each n > 0, satisfying the following conditions: (i) every degenerate simplex is thin; (ii) every horn has a unique thin filler. It is clear that every `-groupoid is a weak `-groupoid and that every - 1 groupoid is a Kan complex. Not every weak `-groupoid underlies an `-groupoid. However, if X is a weak `-groupoid, then any minimal simplicial subcomplex Z of X underlies an `-groupoid; it suffices to take the set of thin n-simplices n n Tn Zn to be a section of the map 0 Zn sSet.ƒ0;Z /, taking care to select W ! n the (necessarily unique) degenerate simplex in each fiber of 0 when there is one. Note also that while the nerve of a bigroupoid is a weak 2-groupoid in our sense, it is not in general a 2-groupoid unless the identity and inverse 1-morphisms are strict. The Dold-Kan simplicial set K .V / is an `-groupoid if and only if Vi vanishes for i > `; it is minimal if and only if V has vanishing differential. In Section 5, we will find analogues of these observations for L -algebras. 1 3. The simplicial de Rham theorem
Let n be the free graded commutative algebra over K with generators ti of degree 0 and dti of degree 1, and relations Tn 0 and dTn 0, where Tn D D D t0 tn 1: C C n KŒt0; : : : ; tn; dt0; : : : ; dtn=.Tn; dTn/: D There is a unique differential on n such that d.ti / dti and d.dti / 0. D D The dg commutative algebras n are the components of a simplicial dg commutative algebra : the simplicial map f Œk Œn acts by the formula W ! X f ti tj for 0 i n: D Ä Ä f .j / i D Using the simplicial dg commutative algebra , we can define the dg commutative algebra of piecewise polynomial differential forms .X / on a simplicial set X [Sul77], [BG76], [Dup76], [Dup78]. Definition 3.1. The complex of differential forms .X / on a simplicial set X is the space .X / sSet.X ; / of simplicial maps from X to . D 282 EZRAGETZLER
When K ޒ is the field of real numbers, .X / may be identified with the D complex of differential forms on the realization X that are polynomial on each j j geometric simplex of X . j j The following lemma may be found in [BG76]; we learned this short proof from a referee. LEMMA 3.2. For each k 0, the simplicial abelian group k is contractible. Proof. The homotopy groups of the simplicial set k equal the homology groups of the complex C k with differential D n X i @ . 1/ @i Cn Cn 1: D W ! i 0 D Thus, to prove the lemma, it suffices to construct a contracting chain homotopy for the complex C . n 1 n For 0 i n, let i be the affine map Ä Ä W C ! i .t0; : : : ; tn 1/ .t0; : : : ; ti 1; ti tn 1; ti 1; : : : ; tn/: C D C C C n 1 Pn Define a chain homotopy Cn Cn 1 by ! . 1/ C i 0 ti i!. For W ! C D ! k, we see that D 2 n @i ! if 0 i n; @i ! Ä Ä D . 1/n 1 ! if i n 1: C D C It follows that .@ @/! !. C D Given a sequence .i0; : : : ; i / of elements of the set 0; : : : ; n , let k f g Ii :::i n K 0 k W ! be the integral over the k-chain on the n-simplex spanned by the sequence of vertices .ei0 ; : : : ; eik /; this is defined by the explicit formula
a1Š a Š I t a1 t ak dt dt k : i0:::ik i1 i i1 ik k D .a1 a k/Š C C k C Specializing K to the field of real numbers, this becomes the usual Riemann integral. The space Cn of elementary forms is spanned by the differential forms k X j !i :::i kŠ . 1/ ti dti dtb i dti : 0 k D j 0 j k j 0 D (The coefficient kŠ normalizes the form so that Ii :::i .!i :::i / 1.) The spaces 0 k 0 k D Cn are closed under the action of the exterior differential, that is, n X d!i :::i !ii :::i ; 0 k D 0 k i 0 D LIE THEORY FOR NILPOTENT L -ALGEBRAS 283 1 and assemble to a simplicial subcomplex of . The complex Cn is isomorphic to the complex of simplicial chains on n, and this isomorphism is compatible with the simplicial structure. Whitney [Whi57] constructs an explicit projection Pn from n to Cn: n X X (3-6) Pn! !i :::i Ii :::i .!/: D 0 k 0 k k 0 i0<
Proof. To show that P s 0, we must check that Ii0:::ik sn 0 for each D ı D sequence .i0 : : : ik/. By the compatibility of s with simplicial maps, this follows from the formula I s 0; 0:::k ı k D k which is clear, since sk! is a differential form on of degree less than k. The second part of the lemma is a simple calculation. Dupont [Dup76], [Dup78] found an explicit contraction: we now recall his n n formula. Given 0 i n, define the dilation map 'i Œ0; 1 by the Ä Ä W ! 284 EZRAGETZLER formula 'i .u; t/ ut .1 u/ei : D C n 1 n Let .Œ0; 1 / . / be integration along the fibers of the W ! projection Œ0; 1 n n. Define the operator hi 1 by the W ! n W n ! n formula i (3-8) hn! 'i!; D i Let "n n K be evaluation at the vertex ei . Stokes’s theorem implies the W ! i i Poincare´ lemma, that hn is a chain homotopy between the identity and "n: i i i (3-9) dh h d idn " : n C n D n Pn The flow 'i .u/ is generated by the vector field Ei j 0.tj ıij /@j . Let D D i be the contraction .Ei /: we have
(3-10) j 'i .u/ 'i .u/.uj .1 u/i / D C and also k X p 1 (3-11) i !i :::i k . 1/ ıii ! : 0 k D p i0:::{p:::ik p 0 b D i The formula (3-8) for hn may be written more explicitly as Z 1 i 1 hn u 'i .u/i du: D 0 LEMMA 3.5. hi hj hj hi 0. C D n n Proof. Let 'ij Œ0; 1 Œ0; 1 be the map W ! 'ij .u; v; t/ uvt .1 u/ei u.1 v/ej : D k C C i j Then we have h h ! 'ij !. We have 'j i .u; v/ 'ij .v; u/, where u and v D D Q Q Q Q are determined implicitly by the equations .1 u/v 1 u and 1 v .1 v/u: D Q D Q Q Since this change of variables is a diffeomorphism of the interior of the square Œ0; 1 Œ0; 1, the lemma follows. k ik ik 1 i0 LEMMA 3.6. Ii :::i .!/ . 1/ "n hn : : : hn !. 0 k D Proof. For k 0, this holds by definition. We argue by induction on k. We D may assume that ! has positive degree and hence that ! d is exact. By Stokes’s D theorem, k X j 1 Ii :::i .d/ . 1/ I ./: 0 k D i0:::b{j :::ik j 0 D LIE THEORY FOR NILPOTENT L -ALGEBRAS 285 1
On the other hand, by (3-9), we have
k 1 ik ik 1 i0 X j ik ik 1 ij i0 " h h d . 1/ " h Œd; hn h n n n D n n n j 0 D k 1 X j ik ik 1 ij i0 k ik ik 1 ik 2 i0 . 1/ " h bhn h . 1/ " " h h : D n n n C n n n n j 0 D ik ik 1 ik 1 But "n "n "n . D THEOREM 3.7 (Dupont). The operators
n 1 X X ik i0 (3-12) sn !i :::i h h for n 0 D 0 k n n k 0 i0< n k X X j X ik ij i0 . 1/ !i :::i h Œd; h h 0 k k 0j 0 i0< In fact, Dupont’s operator s is a gauge. But by a trick of Lambe and Stasheff [LS87], any contraction gives rise to a gauge. PROPOSITION 3.9. If s is a contraction, then the operator s s ds .id P / Q D is a gauge. If s is a gauge, then s s . Q D Proof. Let s be the contraction s s .id P /. By construction, we have N N D s P 0; hence by Lemma 3.4, Œd; .s /2 0. Then s s ds is a contraction: N D N D Q DN N Œd; s Œd; s ds Œd; s ds s dŒd; s Q D N N D N N CN N .id P /ds s d.id P / D N CN d.id P /s s .id P /d Œd; s id P : D N CN D N D Since d.s /2d .s /2d 2 0, the operator s is a gauge: N D N D Q .s /2 .s ds /.s ds / s d.s /2ds 0: Q D N N N N DN N N D If s happens to be a gauge, then s P 0 by Lemma 3.4. It follows that D s s s .ds .id P / id/ Q D s .ds id/ D s .s d P / .s /2d s P 0: D C D C D We now turn to the proof that Dupont’s operator s is a gauge. Denote by ".˛/ the operation of multiplication by a differential form ˛ on n. LEMMA 3.10. If i i0; : : : ; i , then … f kg i k i i ".!i :::i /h . 1/ h ".!i :::i / ".!i :::i i /h : 0 k D 0 k C 0 k Proof. We have Z 1 k i k 1 . 1/ h ".!i0:::ik / . 1/ w 'i .w/i ".!i0:::ik /dw D 0 Z 1 k ".!i0:::ik / w 'i .w/i dw: D 0 On the other hand, by (3-11), Z 1 Z 1 k i i k 1 . 1/ h ".!i0:::ik i /h . 1/ .uv/ 'i .u/i ".!i0:::ik i /'i .v/i dvdu D 0 0 Z 1 Z 1 1 .k 1/ .uv/ 'i .u/".!i0:::ik /'i .v/i dvdu D C 0 0 Z 1 Z 1 k 1 .k 1/".!i0:::ik / u v 'i .uv/i dvdu: D C 0 0 LIE THEORY FOR NILPOTENT L -ALGEBRAS 287 1 Changing variables from u to w uv, we see that D Z 1 Z 1 Z 1ÂZ 1 à k 1 k 2 k u v 'i .uv/ dvdu v dv w 'i .w/ dw 0 0 D 0 w Z 1 1 1 k .k 1/ .w w /'i .w/ dw; D C 0 establishing the lemma. THEOREM 3.11. The operator s is a gauge. Proof. By induction on k, the above lemma shows that n 1 ik i0 X k` ` X ik i0 j` j0 h h s . 1/ !j :::j h h h h : D C 0 ` ` 0 j0< 1 X k` ` X ik i0 j` j0 (3-14) . 1/ !i :::i !j :::j h h h h : C 0 k 0 ` k;` 0 i0< ik i0 j` j0 !i :::i !j :::j h h h h 0 k 0 ` k` .k 1/.` 1/ j` j0 ik i0 . 1/ !j :::j !i :::i h h h h : D C C C 0 ` 0 k The expression (3-14) changes sign on exchange of .i0; : : : ; ik/ and .j0; : : : ; j`/, and thus vanishes. 4. The Maurer-Cartan set of an L -algebra 1 L -algebras are a generalization of dg Lie algebras in which the Jacobi rule is only1 satisfied up to a hierarchy of higher homotopies. In this section, we start by recalling the definition of L -algebras. Following [Sul77] and [Hin97], we represent the homotopy type of1 an L -algebra g by the simplicial set MC .g/ 1 D MC.g /. We prove that this is a Kan complex, and that under certain additional ˝ hypotheses, it is a homotopy invariant of the L -algebra g. 1 An operation Œx1; : : : ; xk on a graded vector space g is called graded antisym- metric if xi xi 1 Œx1; : : : ; xi ; xi 1; : : : ; xk . 1/j jj C jŒx1; : : : ; xi 1; xi ; : : : ; xk 0 C C C D Vk for all 1 i k 1. Equivalently, Œx1; : : : ; x is a linear map from g to g, Ä Ä k where Vk g is the k-th exterior power of the graded vector space g, that is, the k-th 1 symmetric power of s g. 288 EZRAGETZLER Definition 4.1. An L -algebra is a graded vector space g with a sequence 1 Œx1; : : : ; x for k > 0 of graded antisymmetric operations of degree 2 k, or k equivalently, homogeneous linear maps Vk g g of degree 2, such that for each ! n > 0, the n-Jacobi rule holds: n X k X " . 1/ . 1/ ŒŒxi1 ; : : : ; xik ; xj1 ; : : : ; xjn k 0: D k 1 i1< If g is a nilpotent L -algebra, the curvature 1 X1 1 Ᏺ.˛/ ı˛ Œ˛ ` g2 D C `Š ^ 2 ` 2 D is defined and polynomial in ˛. If g is a dg Lie algebra, the curvature equals Ᏺ.˛/ ı˛ 1 Œ˛; ˛ D C 2 I this expression is familiar from the theory of connections on principal bundles. Definition 4.3. The Maurer-Cartan set MC.g/ of a nilpotent L -algebra g is the set of those ˛ g1 satisfying the Maurer-Cartan equation 1 2 (4-15) Ᏺ.˛/ 0: D An L -algebra is abelian if the bracket Œx1; : : : ; xk vanishes for k > 1. In this case,1 the Maurer-Cartan set is the set of 1-cocycles Z1.g/ of g. Let g be a nilpotent L -algebra. For any element ˛ g1, the formula 1 2 X1 1 ` Œx1; : : : ; x ˛ Œ˛ ; x1; : : : ; x k D `Š ^ k ` 0 D ` defines a new sequence of brackets on g, where Œ˛^ ; x1; : : : ; xk is an abbreviation for Œ˛; : : : ; ˛; x1; : : : ; xk, in which ˛ occurs l times. PROPOSITION 4.4. If ˛ MC.g/, then the brackets Œx1; : : : ; x ˛ make g into 2 k an L -algebra. 1 Proof. Applying the .m n/-Jacobi relation to the sequence C m .˛^ ; x1; : : : ; xn/ and summing over m, we obtain the n-Jacobi relation for the brackets Œx1; : : : ; xk˛. LEMMA 4.5. The curvature satisfies the Bianchi identity X1 1 (4-16) ıᏲ.˛/ Œ˛ `; Ᏺ.˛/ 0: C `Š ^ D ` 1 D n Proof. The n-Jacobi relation for .˛^ / shows that n X 1 Œ˛ `; Œ˛ .n `/ 0: `Š.n `/Š ^ ^ D ` 0 D Summing over n > 0, we obtain the lemma. 290 EZRAGETZLER If g is an L -algebra and is a dg commutative algebra, then the tensor product g is1 an L -algebra, with brackets ˝ 1 x Œx a Œx a . 1/j jx da; ˝ D ˝P C ˝ xi aj Œx1 a1; : : : ; x a . 1/ i If ˛ MCn.g/, we see that 2 ˛ "i ˛ .d ı/hi ˛ hi .d ı/˛ D n C C n C n C X1 1 "i ˛ Ri ˛ hi Œ˛ `; D n C n `Š n ^ ` 2 D i i 0 where R .d ı/h . Introduce the space mcn.g/ .d ı/˛ ˛ .g / : n D C n D f C j 2 ˝ g i i LEMMA 4.6. Let g be a nilpotent L -algebra. The map ˛ ."n˛; Rn˛/ 1 7! induces an isomorphism between MCn.g/ and MC.g/ mcn.g/. Proof. Given MC.g/ and mcn.g/, let ˛0 , and define differential 2 2 D C forms .˛k/k>0 inductively by the formula X1 1 i ` (4-18) ˛k 1 ˛0 hnŒ˛k^ : C D `Š ` 2 D Then for all k, we have "i ˛ and Ri ˛ . The sequence is eventually n k D n k D constant, since by induction, we see that ` X1 1 X i j 1 ` j k 1 ˛k 1 ˛k hn ˛k^ 1 ; ˛k 1 ˛k; ˛k^ F C g n: C D `Š 2 ˝ ` 2 j 1 D D The limit ˛ limk ˛k satisfies D !1 X1 1 i ` ˛ ˛0 h Œ˛ : D `Š n ^ ` 2 D Applying the operator d ı, we see that C X1 1 .d ı/˛ ı .d ı/hi Œ˛ ` C D `Š C n ^ ` 2 D and hence that X1 1 X1 1 Ᏺ.˛/ ı Œ˛ ` .d ı/hi Œ˛ ` D C `Š ^ `Š C n ^ ` 2 ` 2 D D X1 1 Ᏺ./ hi .d ı/Œ˛ ` Ᏺ./ hi .d ı/Ᏺ.˛/: D C `Š n C ^ D C n C ` 2 D The Bianchi identity (4-16) implies that X1 1 X1 1 Ᏺ.˛/ Ᏺ./ hi Œ˛ `; Ᏺ.˛/ hi Œ˛ `; Ᏺ.˛/: D `Š n ^ D `Š n ^ ` 1 ` 1 D D 292 EZRAGETZLER The nilpotence of g implies that Ᏺ.˛/ 0; it follows that ˛ is an element of MCn.g/ i i D with "n˛ and Rn˛ . D D i i i If ˛ and ˇ are a pair of elements of MCn.g/ such that "n˛ "nˇ and Rn˛ i D D Rnˇ, then ` X1 1 X ˛ ˇ hi ˛ j 1; ˛ ˇ; ˇ ` j : D `Š n ^ ^ ` 2 j 1 D D This shows, by induction, that ˛ ˇ F i g for all i > 0 and hence, by the nilpotence 2 of g, that ˛ ˇ. D The following result applies when g is a dg Lie algebra. PROPOSITION 4.7 [Hin97]. If f g h is a surjective morphism of nilpotent W ! L -algebras, the induced morphism MC .f / MC .g/ MC .h/ is a fibration of 1 W ! simplicial sets. n Proof. Let 0 i n. Given a horn ˇ sSet.ƒi ; MC .g// and an n-simplex Ä Ä 2 MCn.h/ such that @j f .@j ˇ/ for j i, we wish to construct an element 2 1 D ¤ ˛ f . / MCn.g/ such that @j ˛ @j ˇ for j i. 2 D ¤ Since f g h is a Kan fibration, there exists an extension W ˝ ! ˝ g n of ˇ of total degree 1 such that f ./ ˛. Let ˛ be the unique element 2 ˝ i i i i D i i of MCn.g/ such that "n˛ "n and Rn˛ Rn. If j i, we have "n@j ˛ "n@j ˇ i i D D ¤ D and R @j ˛ R @j ˇ and hence, by Lemma 4.6, @j ˛ @j ˇ. Thus, ˛ fills the n D n D horn ˇ. Also f ."i ˛/ f ."i / "i and f .Ri ˛/ f .Ri / Ri ; hence n D n D n n D n D n f .˛/ . D The category of nilpotent L -algebras concentrated in degrees . ; 0 is 1 1 a variant of Quillen’s model [Qui69] for rational homotopy of nilpotent spaces. By the following theorem, the functor MC .g/ carries quasi-isomorphisms of such L -algebras to homotopy equivalences of simplicial sets. 1 THEOREM 4.8. If g and h are both L -algebras concentrated in degrees . ; 0 and if f g h is a quasi-isomorphism1 , then 1 W ! MC .f / MC .g/ MC .h/ W ! is a homotopy equivalence. Proof. Filter g by L -algebras F j g, where 1 80 if i j > 0; 80 if i j > 0; 2j i < j C 2j 1 i < j C .F g/ Z .g/ if i j 0; .F C g/ B .g/ if i j 0; D C D D C D :gi if i j < 0; :gi if i j < 0; C C LIE THEORY FOR NILPOTENT L -ALGEBRAS 293 1 and similarly for h. If j > k, there is a morphism of fibrations of simplicial sets MC .F j g/ / MC .F kg/ / MC .F kg=F j g/ MC .F j h/ / MC .F kh/ / MC .F kh=F j h/: We have 2j 2j 1 j j 2j 2j 1 MC .F g=F C g/ MC .H .g// MC .H .h// MC .F h=F C h/: Š Š Š The simplicial sets 2j 1 2j 2 j j 1 MC .F C g=F C g/ B .g/ C and Š ˝ 2j 1 2j 2 j j 1 MC .F C h=F C h/ B .h/ C Š ˝ are contractible by Lemma 3.2. The proposition follows. Let m be a nilpotent commutative ring; that is, m` 1 0 for some `. If g is an C D L -algebra, then g m is nilpotent; this is the setting of formal deformation theory. 1 ˝ In this context too, the functor MC .g; m/ MC .g m/ takes quasi-isomorphisms D ˝ of L -algebras to homotopy equivalences of simplicial sets. 1 PROPOSITION 4.9. If f g h is a quasi-isomorphism of L -algebras and W ! 1 m is a nilpotent commutative ring, then MC .f; m/ MC .g; m/ MC .h; m/ W ! is a homotopy equivalence. Proof. We argue by induction on the nilpotence length ` of m. There is a morphism of fibrations of simplicial sets MC .g; m2/ / MC .g; m/ / MC .g m=m2/ ˝ MC .h; m2/ / MC .h; m/ / MC .h m=m2/: ˝ The abelian L -algebras g m=m2 and h m=m2 are quasi-isomorphic; hence 1 ˝ ˝ the morphism MC .g m=m2/ MC .h m=m2/ is a homotopy equivalence. ˝ ! ˝ The result follows by induction on `. 5. The functor .g/ In this section, we study the functor .g/; we prove that it is homotopy equivalent to MC .g/ and show that it specializes to the Deligne groupoid when g 294 EZRAGETZLER is concentrated in degrees Œ0; /. Fix a gauge s , for example Dupont’s operator 1 (3-12). The simplicial set .g/ associated to a nilpotent L -algebra is the simplicial subset of MC .g/ consisting of those Maurer-Cartan forms1 annihilated by s : (5-19) .g/ ˛ MC .g/ s ˛ 0 : D f 2 j D g For any simplicial set X , the set of simplicial maps sSet.X ; .g// equals the set of Maurer-Cartan elements ˛ MC.g;X / such that s ˛ 0. This is reminiscent 2 D of gauge conditions, such as the Coulomb gauge, in gauge theory. PROPOSITION 5.1. If g is abelian, then there is a natural isomorphism .g/ Š K .gŒ1/. Proof. If ˛ n.g/, then .d ı/˛ sn˛ 0. Hence by (3-7), 2 C D D ˛ Pn˛ sn.d ı/˛ .d ı/sn˛ Pn˛: D C C C C D Thus n.g/ Kn.gŒ1/. Conversely, if ˛ Kn.gŒ1/, then Pn˛ ˛, and hence 2 D sn˛ 0. Thus Kn.gŒ1/ n.g/. D We show that .g/ is an -groupoid and, in particular, a Kan complex: the 1 heart of the proof is an iteration, similar to the iteration (4-18), that solves the n Maurer-Cartan equation on the n-simplex in the gauge sn˛ 0. D Definition 5.2. An n-simplex ˛ n.g/ is thin if I0:::n.˛/ 0. 2 D i i LEMMA 5.3. If g is a nilpotent L -algebra, the map ˛ ."n˛; PnRn˛/ 1 7! induces an isomorphism between n.g/ and MC.g/ PnŒmcn.g/. Proof. Let 0 i n. By (3-7), we see that Ä Ä idn Pn .d ı/sn sn.d ı/ D C C C C i i i " .d ı/.Pnh sn/ .Pnh sn/.d ı/: D n C C n C C n C C It follows that if ˛ n.g/, 2 i i X1 1 i ` (5-20) ˛ " ˛ PnR ˛ .Pnh sn/Œ˛ : D n C n `Š n C ^ ` 2 D Given MC.g/ and PnŒmcn.g/, let ˛0 , and define differential 2 2 D C forms .˛k/k>0 inductively by the formula X1 1 i ` ˛ ˛0 .Pnh sn/Œ˛ : k D `Š n C k^ 1 ` 2 D LIE THEORY FOR NILPOTENT L -ALGEBRAS 295 1 i i Then for all k, we have sn˛ 0, " ˛ and PnR ˛ . The sequence k D n k D n k D .˛k/ is eventually constant since, by induction, we see that ` X1 1 X i j 1 ` j ˛k ˛k 1 .Pnhn sn/ ˛k^ 2 ; ˛k 2 ˛k 1; ˛k^ 1 D `Š C ` 2 j 1 D D k F g n: 2 ˝ The limit ˛ limk ˛k satisfies D !1 X1 1 i ` ˛ ˛0 .P h sn/Œ˛ : D `Š n C ^ ` 2 D By the same argument as in the proof of Lemma 4.6, it follows that X1 1 i ` Ᏺ.˛/ Ᏺ./ .P h sn/Œ˛ ; Ᏺ.˛/ D `Š n C ^ ` 1 D X1 1 i ` .P h sn/Œ˛ ; Ᏺ.˛/: D `Š n C ^ ` 1 D The nilpotence of g implies that Ᏺ.˛/ 0; it follows that ˛ is an element of n.g/ i i D with "n˛ and PRn˛ . D D i i i If ˛ and ˇ are a pair of elements of n.g/ such that "n˛ "nˇ and PnRn˛ i D D PnRnˇ, then ` X1 1 X i j 1 ` j ˛ ˇ .Pnh sn/ ˛ ; ˛ ˇ; ˇ : D `Š n C ^ ^ ` 2 j 1 D D This shows, by induction, that ˛ ˇ F i g for all i > 0, and hence, by the nilpotence 2 of g, that ˛ ˇ. D THEOREM 5.4. If g is a nilpotent L -algebra, .g/ is an -groupoid. If g is 1 1 concentrated in degrees . `; /, respectively . `; 0, then .g/ is an `-groupoid, 1 respectively an `-group. n Proof. Let ˇ sSet.ƒi ; .g// be a horn in .g/. The differential form 2 n 1 i X X ˛0 " ˇ .d ı/ !i :::i Iii :::i .ˇ/ MC.g/ PnŒmcn.g/ D n C C 1 k ˝ 1 k 2 k 1 i1< n constructed in Lemma 5.3 is thin and i .˛/ ˇ. Thus .g/ is an -groupoid. 1 n D 1 1 If g 0, it is clear that every n-simplex ˛ n.g/ is thin, while if g 0, D 2 D then .g/ is reduced. 1 k Given MC.g/ and xi :::i g for 1 i1 < < i n, let 2 1 k 2 Ä k Ä ˛ .xi :::i / n.g/ n 1 k 2 0 be the solution of (5-20) with " ˛n .xi :::i / and n 1 k D n 0 X X R ˛ .xi :::i / !i :::i xi :::i : n n 1 k D 1 k ˝ 1 k k 1 1 i1< 2 n .xi :::i / I1:::n.˛ .xi :::i // g : n 1 k D n 1 k 2 If g is concentrated in degrees . ; 0, then the Maurer-Cartan element 1 equals 0 and may be omitted from the notation for ˛n.xi1:::ik / and n.xi1:::ik /. Since ˛2 .x1; x2; x12/ is a flat connection 1-form on the 2-simplex, its mon- odromy around the boundary must be trivial. (The 2-simplex is simply connected.) In terms of the generalized Campbell-Hausdorff series 2 .x1; x2; x12/, this gives x .x ;x ;x / x the equation e 1 e 2 1 2 12 e 2 in the Lie group associated to the nilpotent D Lie algebra g0. Thus, the simplicial set .g/ (indeed, its 2-skeleton) determines 2 .x1; x2; x12/ as a function of x1, x2 and x12. In the Dupont gauge, modulo terms involving more than two brackets, it equals 1 1 .x1; x2; x12/ x1 x2 Œx1; x2 Œx12 2 D C 2 C 2 1 1 Œx1 x2; Œx1; x2 ŒŒx1 x2; x1; x2 C 12 C C 6 C 1 1 ŒŒx1 x2; x12 Œx1 x2; Œx12 : C 6 C 12 C C Definition 5.6. A nilpotent L -algebra g is minimal if the following two conditions hold: 1 (i) g is concentrated in degrees . ; 0 and 1 (ii) the differential ı of g vanishes. An L -algebra g is minimal if and only if the dg commutative algebra C .g/ is minimal1 in the sense of [Sul77]. The following result was suggested to the author by P. Severa.ˇ PROPOSITION 5.7. If L is minimal, .L/ is a minimal Kan complex. LIE THEORY FOR NILPOTENT L -ALGEBRAS 297 1 1 n Proof. If L is minimal, Ᏺ ˛ !0:::n x is independent of x L . It C ˝ 2 follows that ˛n.xi :::i / ˛n.xi :::i / !0:::n x1:::n 1 k D 1 k k PROPOSITION 5.8. If g is a dg Lie algebra, the composition .x1; x2/ 2 W g0 g0 g0 is associative. ˝ ! Proof. It suffices to show that .x1; x2; x3; xij 0/ 0; in other words, if 3 D D three faces of a thin 3-simplex are thin, then the fourth is. The iteration leading to the solution ˛ of (5-20) with initial conditions ˛0 .d ı/.t1x1 t2x2 t3x3/ D C C C C 0 1 1 0 1 lies in the space g g ; hence I123.˛/ 0 g . ˝ 3 ˚ ˝ 3 D 2 In particular, if g is a dg Lie algebra concentrated in degrees . 2; /, then 1 .g/ is the nerve of a strict 2-groupoid, that is, a groupoid enriched in groupoids; in this way, we see that .g/ generalizes the Deligne 2-groupoid [Del94], [Get02]. Although it is not hard to derive explicit formulas for the generalized Campbell- Hausdorff series up to any order, we do not know any closed formulas for them except when n 1, in which case it is independent of the gauge. We now derive a D closed formula for 1 .x/, which resembles Cayley’s famous formula for the series solution of the ordinary differential equation x .t/ f .x.t//. 0 D To each rooted tree, associate the word obtained by associating to a vertex with i branches the operation Œx; a1; : : : ; ai . Multiply the resulting word by the number of total orders on the vertices of the tree such that each vertex precedes its k parent. Let e.x/ be the sum of these terms over all rooted trees with k vertices. 1 2 For example, e .x/ Œx, e .x/ Œx; Œx, and D D 3 e .x/ Œx; Œx; Œx Œx; Œx; Œx: D C k The coefficient of a tree T in e.x/ equals the number of monotone orderings of its vertices, that is, total orderings such that each vertex is greater than its parent. k The pictures below show the trees contributing to e.X/ for k < 5. 298 EZRAGETZLER e1(X)= e2(X)= e3(X)= + ¡¡AA s s s s s s s s s ¡A @ e4(X)= s + s + 3 ¡ sA + s@ ¡A s ¡ sA s s s s s s s s s s s s s s e5(X)= + + 3 ¡A + @ s s ¡ sA s@ ¡A s ¡ sA s s s s s s s s s s QQ ¡¡AA ¡¡AA @@ ¡¡AA Q + 4 s + 3 s + 6 s + s s s s s s s s s s s s s s s s s PROPOSITION 5.9. The 1-simplex ˛ .x/ 1.g/ that is determined by 1 2 2 MC.g/ and x g0 is given by the formula 2 k X1 t ˛ .x/ ek .x/ x dt: 1 D kŠ C k 1 D Proof. To show that ˛1 .x/ 1.g/, we must show that it satisfies the Maurer- 2 P k k Cartan equation. Let ˛.t/ k1 1.t =kŠ/e.x/: It must be shown that P nD D ˛0.t/ n1 0.1=nŠ/Œ˛.t/^ ; x 0 or, in other words, that C D D 1 n k 1 X . 1/ X kŠ k1 kn eC .x/ Œe .x/; : : : ; e .x/; x˛ D nŠ k1Š knŠ n 0 k1 kn k D CC D 1 X 1 X kŠ k1 kn Œx; e .x/; : : : ; e .x/˛: D nŠ k1Š knŠ n 0 k1 kn k D CC D This is easily proved by induction on k. ½ LIE THEORY FOR NILPOTENT L -ALGEBRAS 299 1 Proposition 5.9 implies a formula for the generalized Campbell-Hausdorff ˛ series 1 .x/: X1 1 .x/ ek .x/: 1 D kŠ k 1 D If g is a dg Lie algebra, only trees with vertices of valence 0 or 1 contribute to k e˛.x/, and we recover the formula (1-3) figuring in the definition of the Deligne groupoid for dg Lie algebras. There is a relative version of Theorem 5.4, analogous to Proposition 4.7: THEOREM 5.10. If f g h is a surjective morphism of nilpotent L - W ! 1 algebras, the induced morphism .f / .g/ .h/ is a fibration of simplicial W ! sets. n Proof. Let 0 i n. Given a horn ˇ sSet.ƒi ; .g// and an n-simplex Ä Ä 2 n.h/ such that f .@j ˇ/ @j for j i, our task is to construct an element 2 1 D ¤ ˛ f . / n.g/ such that @j ˛ @j ˇ if j i. 2 1 n D ¤ 1 n Choose a solution x g of the equation f .x/ I0:::n. / h . Let ˛ 2 i i D 2 be the unique element of n.g/ such that " ˛ " ˇ and n D n  n 1 à i X X i PnR ˛ .d ı/ !i :::i Iii :::i .ˇ/ . 1/ ! x : n D C 1 k ˝ 1 k C 0:::b{:::n ˝ k 1 i1< 1 MCn.g/= n.g/ .d ı/sn.g n/ : Š C ˝ This simplicial abelian group is a retract of the contractible simplicial abelian group g and hence is itself contractible. ˝ 300 EZRAGETZLER Let F i g be the lower central series of g. Given i > 0, we have a morphism of principal fibrations of simplicial sets given by i 1 i i i 1 .F C g/ / .F g/ / .F g=F C g/ i 1 i i i 1 MC .F C g/ / MC .F g/ / MC .F g=F C g/: i i 1 i i 1 i i 1 Since F g=F C g is abelian, we see that .F g=F C g/ MC .F g=F C g/. ' The result follows by induction on `. When g is a nilpotent Lie algebra, the isomorphism 0. .g// 0.MC .g// Š is equivalent to the surjectivity of the exponential map. The above corollary may be viewed as a generalization of this fact. Acknowledgments I am grateful to D. Roytenberg for the suggestion to extend Deligne’s groupoid to L -algebras. 1 References [Ash88] N.ASHLEY, Simplicial T -complexes and crossed complexes: a nonabelian version of a theorem of Dold and Kan, Dissertationes Math. .Rozprawy Mat./ 265 (1988), 61. MR 89k:55028 Zbl 1003.55500 [Bek04] T. BEKE, Higher Cechˇ theory, K-Theory 32 (2004), 293–322. MR 2005i:18007 Zbl 1070. 18008 [BG76]A.K.B OUSFIELD and V. K. A. M. GUGENHEIM, On PL de Rham Theory and Rational Homotopy Type, Mem. Amer. Math. Soc. 8:179, 1976. MR 54 #13906 Zbl 0338.55008 [Dak83] M.K.DAKIN, Kan complexes and multiple groupoid structures, Esquisses Math. 32, Univ. Amiens, 1983. MR 766238 Zbl 0566.55010 [Del94] P. DELIGNE, letter to L. Breen, Feb. 28, 1994. Available at http://math.northwestern.edu/ ~getzler/Papers/deligne.pdf [Dol58] A.DOLD, Homology of symmetric products and other functors of complexes, Ann. of Math. 68 (1958), 54–80. MR 20 #3537 Zbl 0082.37701 [Dup76] J.L.DUPONT, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976), 233–245. MR 54 #1243 Zbl 0331.55012 [Dup78] , Curvature and Characteristic Classes, Lecture Notes in Math. 640, Springer- Verlag, New York, 1978. MR 58 #18477 Zbl 0373.57009 [Dus79] J.DUSKIN, Higher-dimensional torsors and the cohomology of topoi: the abelian theory, in Applications of Sheaves (Univ. Durham, 1977) (M. P. FOURMAN, C.J.MULVEY, and D.S.SCOTT, eds.), Lecture Notes in Math. 753, Springer-Verlag, New York, 1979, pp. 255–279. MR 82c:18006 Zbl 0444.18014 LIE THEORY FOR NILPOTENT L -ALGEBRAS 301 1 [Dus01] J. W. DUSKIN, Simplicial matrices and the nerves of weak n-categories. I: Nerves of bicategories, Theory Appl. Categ. 9 (2001), 198–308. MR 2003f:18005 Zbl 1046.18009 [EML53] S.EILENBERG and S.MAC LANE, On the groups of H.…; n/. I, Ann. of Math. 58 (1953), 55–106. MR 15,54b Zbl 0050.39304 [FL91] Z.FIEDOROWICZ and J.-L.LODAY, Crossed simplicial groups and their associated homol- ogy, Transactions Amer. Math. Soc. 326 (1991), 57–87. MR 91j:18018 Zbl 0755.18005 [Get02] E.GETZLER, A Darboux theorem for Hamiltonian operators in the formal calculus of variations, Duke Math. J. 111 (2002), 535–560. MR 2003e:32026 Zbl 1100.32008 [Gle82] P. G. GLENN, Realization of cohomology classes in arbitrary exact categories, J. Pure Appl. Algebra 25 (1982), 33–105. MR 83j:18016 Zbl 0487.18015 [GM88] W. M. GOLDMAN and J.J.MILLSON, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 43–96. MR 90b:32041 Zbl 0678.53059 [Hin97] V. HINICH, Descent of Deligne groupoids, Internat. Math. Res. Notices 5 (1997), 223–239. MR 98h:22019 Zbl 0948.22016 [Kan58] D.M.KAN, Functors involving c.s.s. complexes, Trans. Amer. Math. Soc. 87 (1958), 330–346. MR 24 #A1720 Zbl 0090.39001 [Kur62]M.K URANISHI, On the locally complete families of complex analytic structures, Ann. of Math. (2) 75 (1962), 536–577. MR 25 #4550 Zbl 0106.15303 [LM95] T. LADA and M.MARKL, Strongly homotopy Lie algebras, Comm. Algebra 23 (1995), 2147–2161. MR 96d:16039 Zbl 0999.17019 [LS87] L.LAMBE and J.STASHEFF, Applications of perturbation theory to iterated fibrations, Manuscripta Math. 58 (1987), 363–376. MR 89d:55041 Zbl 0632.55011 [May92] J. P. MAY, Simplicial Objects in Algebraic Topology, Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 1992. MR 93m:55025 Zbl 0769.55001 [NR66] A.NIJENHUIS and R. W. RICHARDSON,JR., Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc. 72 (1966), 1–29. MR 33 #4190 Zbl 0136.30502 [Pao07] S.PAOLI, Semistrict Tamsamani n-groupoids and connected n-types, preprint, 2007. arXiv math/0701655v2 [Qui69] D.QUILLEN, Rational homotopy theory, Ann. of Math. 90 (1969), 205–295. MR 41 #2678 Zbl 0191.53702 [Sul77] D.SULLIVAN, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331. MR 58 #31119 Zbl 0374.57002 [Whi57]H.W HITNEY, Geometric Integration Theory, Princeton Univ. Press, Princeton, NJ, 1957. MR 19,309c Zbl 0083.28204 (Received November 7, 2004) (Revised May 11, 2006) E-mail address: [email protected] DEPARTMENT OF MATHEMATICS,NORTHWESTERN UNIVERSITY, 2033 SHERIDAN ROAD, EVANSTON, IL 60208-2730, UNITED STATES