Homology, Homotopy and Applications, vol.????, No.????, 2001, pp.1{22 ISSN 1512{0139

c°2000

HOMOTOPY RINEHART COHOMOLOGY OF HOMOTOPYLIE-RINEHART PAIRS

LARS KJESETH

(communicated by Larry Lambe)

AbstractWe de¯ne homotopy Lie-Rinehart pairs and the associated homo-

topy Rinehart algebra in the context of coalgebras. We de¯ne homotopyLie-Rinehart resolutions and present conditions under which the asso-ciated homotopy Rinehart algebra is a cohomological model for theRinehart algebra of the resolved Lie-Rinehart pair.

1. Introduction

The Rinehart cohomology of homotopy Lie-Rinehart resolutions of Lie-Rinehart pairs arosefrom the study of the BFV (Batalin, Fradkin and Vilkovisky, [FV75], [BV77], [BF83],[BV83] and [BV85]) formulation of classical BRST cohomology (Becchi, Rouet and Stora[BRS75] and, independently, Tyutin [Tyu75]). The classical BRST algebra (A;D) (see, forexample, [Kim92b], [Kim93], [Sta92], [Kim92a], [FHST89], [Sta88], [Sta96], [KS87] and[HT92]) is a di®erential graded Poisson algebra which, in certain cases, is a cohomologicalmodel for the Rinehart algebra (AltB(g;B); ±R) of alternating B-multilinear functions froma Lie algebra g into the associative algebra B [Rin63]. The Rinehart complex is a subcom-plex of the Chevalley-Eilenberg complex ([CE48]) precisely when (B; g) is a Lie-Rinehart pairsatisfying certain Lie-Rinehart relations (see below).

In homological algebra, a traditional strategy is to replace both B and g in AltB(g;B)with free or projective resolutions RB and Rg which have the same or similar algebraic struc-tures, at least up to homotopy. The new object AltRB

(Rg;RB) is a cohomological modelfor AltB(g;B) if AltRB

(Rg;RB) is a di®erential graded commutative algebra (a dcga) withthe same cohomology as AltB(g;B). In contrast, the BFV construction of the classical BRSTalgebra (see, for example, [Kim93]) begins by replacing B with the Koszul-Tate resolution[Tat57], but side-steps replacing g with a suitable resolution, opting instead to adjoin formal(ghost) variables to the Koszul-Tate resolution. They then exploit a graded Poisson bracket toconstruct a di®erential. Attempting to construct the dcga AltRB

(Rg;RB) fails if we resolveB and g without preserving (as far as possible) the Lie-Rinehart structure of the pair (B; g).Our view in this paper is that a model for a Rinehart complex should be Rinehart-like. Butto resolve a Lie-Rinehart pair with a new pair (RB;Rg) which is Lie-Rinehart, we must giveup a strictly Lie structure on Rg in favor of a strongly homotopy Lie structure.

In its earliest incarnation, [Sul77] and [SS], a strongly homotopy Lie (shLie) algebras wasde¯ned on the tensor coalgebra TcL of a graded module L. In [LM95] and [LS], the shLiestructure was shifted (almost by brute force) onto the graded symmetric coalgebra on thesuspension of L, that is, on

V(sL). Parts of this transition were quite rough and we felt there

must be smoother means and more satisfying motivations for making the move. By taking aclose look at choices of grading and speci¯c actions of the symmetric group on a graded vector

Received 8 January 2001, revised ????; published on ????.2000 Mathematics Subject Classi¯cation: primary 17B55, 17B81; secondary 17B56, 17B99Key words and phrases: strongly homotopy Lie algebras, Lie-Rinehart pairs, Rinehart cohomology, BRSTcohomologyc° 2001, Lars Kjeseth. Permission to copy for private use granted.

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 2

space, we discovered we could identify a subcoalgebra jVL of TcL, which is isomorphic toV(sL) and on which an shLie structure can be de¯ned as it was on TcL in [LM95] and [LS]

(See x2).

Grading and Sign Conventions

In order to understand the language of this paper, we must establish our grading and signconventions ¯rst.

All vector spaces, algebras and coalgebras in this paper are over a ¯eld k of characteristiczero. All tensor products are over k. All maps are at least k-linear or k-multilinear.

Let V = fVpg10 be a graded vector space over k or, more generally, a graded module overan algebra A over k. There are four distinct gradings on the tensor module

NV = fV ­ng1n=0:

the external, internal, combined and total gradings. We shall denote v1 ­ ¢ ¢ ¢ ­ vn 2 V ­n byv[1 to n]. The external degree, ed(v[1 to n]) is given by the number n of tensor components. Theinternal degree, id(v[1 to n]) comes from the internal grading on V ; each tensor component viis an element of Vpi for some pi. The internal degree is the sum

Pni=1 pi. The combined degree

jjv[1 to n]jj is the vector­ed(v[1 to n]); id(v[1 to n])

®. Finally, for any bigraded object, the total

degree td(v[1 to n]) is the sum ed(v[1 to n]) + id(v[1 to n]).

A function f :NV !N

V is of homogeneous degree r with respect to a speci¯c grading iff ((NV )p) ½ (

NV )p+r for all p. For each of the four gradings, the degree of the homogeneous

map f is the di®erence between the degree of v[1 to n] 2NV and the degree of its non-zero

image under f .

The Koszul sign convention states that exchanging two objects of homogeneous degrees pand q (whether elements or maps) introduces a factor of (¡1)pq. In this paper, we apply theKoszul sign convention consistently, no matter which grading. (In the case of the combinedgrading, the product of p and q will be the dot product of the degree vectors.)

Resolving a Lie algebra with an shLie algebra resolution is much simpler if we view theLie algebra as a graded object. After all, since the shLie algebra resolution is de¯ned on agraded coalgebra, it makes sense that the original Lie algebra should be realized on a gradedcoalgebra as well, where, at the very least, every element has an external degree. We must naildown the relation between the symmetry of the Lie bracket and the degree we assign to theelements of the Lie algebra. If each element of the Lie algebra g is assigned an external degreeof one and an internal degree of zero, we have essentially three choices for describing g as acoalgebra. In each case, the skew-commutativity of l2 appears as the result of applying theKoszul sign convention. If we select the natural bigrading on Tcg, the element x has bidegreeh1; 0i, which we treat as a vector, and

l2(x­ y) = (¡1)h1;0i¢h1;0il2(y ­ x) = ¡l2(y ­ x):

We might instead choose the total degree, in which case the element x has degree 0 + 1 = 1.Here again,

l2(x­ y) = (¡1)1¢1l2(y ­ x) = ¡l2(y ­ x):

Finally, if we consider the tensor coalgebra on the suspension of g under the internal grading,then the element x again has degree 1 and l2 again appears skew-commutative, but is actuallygraded commutative.

When it resolves a Lie algebra, an shLie algebra must operate as a di®erential object.The di®erential must have degree ¡1 with respect to a single grading. So the shLie algebraresolution Rg of g must be graded either with respect to the total degree or with respect tothe suspended internal degree. Since the di®erential graded commutative (dgc) algebra RB

which resolves B is graded by internal degree, the suspended internal grading for the shLiealgebra resolution is the preferred choice. Therefore, we will also view the Lie algebra g in itssuspended form sg.

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BackgroundFor any smooth manifoldM , the set of smooth functions C1(M) is an associative commutativealgebra and the set of smooth vector ¯elds on the manifold, ¡(TM), forms a Lie algebra. BothC1(M) and ¡(TM) are modules over each other. Moreover, the the module actions satisfythe equations (fX) ¢ g = f(X ¢ g) and [X;fY ] = f [X;Y ] + (X ¢ f)Y for all f; g 2 C1(M) andX; Y 2 ¡(TM). A Lie-Rinehart pair is a couple (B; g) which admits the analogous structure,where g is a Lie algebra (we will suspend it later), B is an algebra, and both are modules overeach other.

De¯nition 1.1. : [Rin63] We denote the left B-module action ¹ on g by ¹(a­®) := a®. Let! : g ­ B ¡! B (or, alternatively, ! : g ! Der(B)) denote the g-module action on B. Thepair (B; g) is a Lie-Rinehart pair, provided the Lie-Rinehart relations (LRa) and (LRb) aresatis¯ed for all a; b 2 B and x; y 2 g:

LRa: !(ax­ b) = a ¢ !(x­ b), where ¢ indicates the multiplication on B.LRb: [x; ay] = a[x; y] + !(x­ a)y.

The term Lie-Rinehart pair is not widely used. More often, g has been called a (B; k)-Liealgebra, [Rin63] [Pal61] and [Her72]. More recently, Lie-Rinehart pairs have appeared asLie algebroids (see [dSW]).

The ¯rst Lie-Rinehart relation (LRa) states that ! is B-linear in g.Suppose g is ¯nitely generated over B, that is to say, with respect to a generating set fX®g,

every element x of g can be expressed as the sum b®X® where b® 2 B. (We use the Einsteinsummation convention throughout.) The second Lie-Rinehart relation (LRb) and its skew-symmetric counterpart,[ax; y] = a[x; y]¡!(y­ a)x; imply that for any element b®X® ­ b¯X¯

of g­ g the bracket must have the form

[b®X®; b¯X¯] = b®b¯[X®;X¯ ] + !(b®X® ­ b¯)X¯ ¡ !(b¯X¯ ­ b®)X®: (1)

The second Lie-Rinehart relation (LRb) is equivalent to equation (1).For any Lie algebra g and g-module B, an n-multilinear function fn : g£n ! B is alternating

if

fn(x1; :::; xi; xi+1; :::; xn) = ¡fn(x1; :::; xi+1; xi; :::; xn):

The Chevalley-Eilenberg complex is the set of all alternating multilinear functions Altk(g;B),graded by n, and equipped with a degree +1 di®erential ±CE : Altnk(g;B) ! Altn+1

k (g;B)given by

±CEfn(x0; :::; xn) =nX

i=0

(¡1)i!(xi ­ fn(x0; :::; bxi; :::; xn)

¡X

i<j

(¡1)i+j¡1fn([xi; xj]; x0; :::; bxi; :::; bxj; :::; x¾(n+1));

(2)

where cxk indicates that xk should be omitted. Any element b 2 B is considered a 0-cochain.The image of b under ±CE is de¯ned by setting ±CEb(x) = !(x ­ b): When B is an algebra,the Chevalley-Eilenberg complex is a di®erential graded commutative algebra. For fn and gmin Altk(g;B), the product fn ^ gm is given by

(fn ^ gm)(x1; :::; xn+m) =X

¾(n;m)

unshu²es

fn(x¾(1); :::; x¾(n))gm(x¾(n+1); :::; x¾(n+m)):

An (n;m)¡unshu²e is any permutation ¾ in the symmetric group §n+m such that

¾(1) < ¢ ¢ ¢ < ¾(n)| {z }¯rst ¾ hand

and ¾(n+ 1) < ¢ ¢ ¢ < ¾(n+m);| {z }second ¾ hand

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where ¾(j) is the element of the set f1; :::; n + mg moved to the jth position under ¾. Thedi®erential ±CE acts as a derivation with respect to this multiplication. The cohomology ofthis complex with respect to ±CE is the Chevalley-Eilenberg cohomology of g with coe±cientsin B [CE48].

Suppose (B;g) is a Lie-Rinehart pair and the alternating function fn is B-multilinear, i.e.,fn(a1x1; :::; anxn) = a1 ¢ ¢ ¢anfn(x1; :::; xn): Because the Lie action map ! maps g into thederivations of B and as a result of the Lie-Rinehart relations, the image of fn under theChevalley-Eilenberg di®erential ±CE is again B-multilinear, despite the fact that the bracketis not B-multilinear. The Rinehart algebra R = AltB(g;B) with di®erential ±R = ±CE is asubcomplex of the Chevalley-Eilenberg algebra and the cohomology with respect to ±R is theRinehart cohomology of g with coe±cients in B [Rin63].

SummaryFirst, we de¯ne the Lie algebras of subordinate derivation sources, resting coderivations and

shared Lie modules (x2). Following the lead of [LM95] and [LS], we de¯ne both homotopy andnon-homotopy Lie algebras, Lie algebra modules, Lie-Rinehart pairs and Rinehart cohomology,all in the coalgebra setting (xx3 and 4). We piece together homotopy Lie-Rinehart resolutionsfor Lie-Rinehart pairs and present conditions under which the homotopy Rinehart algebra fora homotopy Lie-Rinehart resolution is a model for the Rinehart cohomology complex for aLie-Rinehart pair (x5).

We have omitted all sign arguments from the proofs in this paper, as they are not generallyinstructive.

2. Subordinate derivation sources, resting coderivations and sharedLie modules

MotivationThe following brief and not unduly precise reexamination of the Lie-Rinehart relations andthe Rinehart di®erential provides motivation for why coalgebras are an appropriate setting forboth the homotopy and the non-homotopy versions of Lie algebras, Lie modules, Chevalley-Eilenberg cohomology and Rinehart cohomology. In particular, the discussion below highlightswhy we de¯ne subordinate and resting coderivations, as well as shared Lie modules.

The underlying module structure for the tensor coalgebra Tcg on g isN

g, which is thecollection fg­ng1n=0.

We rename the bracket as l2 and extend l2 to a map from g­n ! g­n¡1 for any n > 2 bysetting (up to sign)

l2(x1 ­ ¢ ¢ ¢ ­ xn) =X

¾

(2;n¡2)

§ l2(x¾(1) ­ x¾(2))­ x¾(3) ­ ¢ ¢ ¢ ­ x¾(n); (3)

where the sum runs over all (2; n ¡ 2)-unshu²es ¾. The bracket l2 is now a coderivation onTcg, although it has lost its skew-commutativity. We regain the symmetry by passing to anappropriate subcoalgebra. In this context, we express the Jacobi identity by l2±l2 = 0. Or, sincethe set of coderivations on a coalgebra forms a Lie algebra, the Jacobi identity is equivalentto [l2; l2] = 0.

Similarly, we rename the Lie action map ! as m2 and extend m2 as a coderivation onTcg­B which is subordinate to l2 by settingm2(x1 ­ ¢ ¢ ¢ ­ xn ­ b) =

l2(x1 ­ ¢ ¢ ¢ ­ xn)­ b §X

½

(n¡1;1)

§ x½(1) ­ ¢ ¢ ¢ ­ x½(n¡1) ­m2(x½(n) ­ b) (4)

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 5

for any n > 1, where l2 has been extended as in equation (3). Passing again to the appropriatesubcoalgebra, we regain the desired symmetry. The condition any Lie action must satisfybecomes m2 ±m2 = 0 or [m2;m2] = 0.

For the Lie-Rinehart pair (B;g), the Lie algebra is a module over B, but the bracket l2is not B-linear. Rather, the second Lie-Rinehart relation states that l2(ax ­ by) consists of aB-linear piece

abl2(x­ y);

and two other terms

m2(ax­ b)y ¡m2(by ­ a)x

(compare with equation 1). Any coderivation which shares this property with a speci¯c sub-ordinate coderivation is called a resting coderivation, e.g., l2 rests on m2.

Finally, we extend an alternating function fn : g­n ! B to a new alternating function fnfrom Tcg into Tcg­B by setting fn(x1 ­ ¢ ¢ ¢ ­ xn+k) =

X

¾

(k;n)

§x¾(1) ­ ¢ ¢ ¢ ­ x¾(k) ­ fn(x¾(k+1) ­ ¢ ¢ ¢ ­ x¾(k+n)) (5)

on g­k+n and then passing to the appropriate subcoalgebra. Using equations (3), (4) and (5),the Chevalley-Eilenberg di®erential (and hence, the Rinehart di®erential) becomes

±CEfn = m2 ± fn § fn ± l2;which looks remarkably like some sort of commutator bracket action.

In our more rigorous discussion of these ideas, we focus on the following algebraic objects.

1. The graded Lie algebra CoderWW (V

(sV )) is the set of all coderivations on the free gradedcommutative algebra

V(sV ), each of which rests on its own speci¯c subordinate coderiva-

tion in the Lie algebra CoderWW (V

(sV )­W ) (see 2).

2. For a graded commutative algebra W , the Lie algebra CoderWW (V

(sV ) ­W ) is the setof all coderivations, each of which is subordinate to a speci¯c resting coderivation and isalso a W -derivation source. A W -derivation source m is the extension as a coderivationonto

V(sV ) of a map m on (sV )^n ­W which acts like a Lie module structure map,

i.e., m : (sV )^n ! Der(W ).

3. The graded commutative algebra HomW (V

(sV );V

(sV )­W ) is the set of all W -linearmaps on

V(sV ) with coe±cients in

V(sV ) ­ W and it admits a shared Lie module

structure over both CoderWW (V

(sV )­W ) and CoderWW (V

(sV )).

We begin our more formal treatment with a return to the basics.

Coalgebras and SubcoalgebrasA graded coassociative coalgebra is a pair (C;¢), where C is a graded module over k together

with a 0-degree coassociative comultiplication ¢ : C ! C­C. The comultiplication breaksup into pieces ¢p;q : Cp+q ! Cp­Cq, based on the bigrading of the range. A function f withdegree jfj = r is a coderivation on C if, for all p and q, the diagram

Cp+r ­Cq + Cp ­Cq+r f­1 + 1­f¡! Cp ­Cqx??¢p+r;q + ¢p;q+r

x??¢p;q

Cp+q+r f¡! Cp+q

commutes, where again (f ­ 1)(cp ­ cq) = f(cp)­ cq and (1­ f)(cp ­ cq) = (¡1)rpcp ­ f(cq).The set of all coderivations on a coalgebra C, denoted Coder(C), is a graded Lie algebra

under the graded commutator bracket, that is to say, [f; g] = fg ¡ (¡1)jf jjgjgf for all f andg 2 Coder(C), where jf j and jgj are the degrees of f and g. For a graded module V , the

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 6

tensor moduleNV = fV ­ng is a graded coassociative coalgebra TcV with the standard

comultiplication ¢ : TcV ! TcV ­TcV given by

¢(v[1 to n]) =nX

j=0

(v[1 to j])­ (v[j+1 to n]);

where V ­0 ¼ k and the terms with j = 0 and j = n are 1­v[1 to n] and v[1 to n]­1 respectively.

Actions of the symmetric group §1 on TcVAn action ½n of the symmetric group §n on V ­n is any homomorphism

½n : §n ! End(V ­n):

An action ½ = f½ng1n=1 of f§ng1n=1 on TcV is de¯ned in the obvious way. The ½-invariantsubspace of TcV forms a subcoalgebra. The two actions and associated subcoalgebras whichplay a role in this paper are listed below. The ¯rst one is the familiar graded commutativecoalgebra

VV . The other one helps place this paper in its historical context.

1. The internal graded action ½^ recognizes only the internal grading on V ­n and is givenby ¾ ¢ (v[1 to n]) = kid(¾)v¾[1 to n] for all n, ¾ 2 §n and v[1 to n] 2 V ­n. The symbolv¾[1 to n] is shorthand for v¾(1)­ ¢ ¢ ¢ ­ v¾(n), where ¾(i) is the element of the ordered set

f1; :::; kg which moves to the ith position under ¾. Since §n acts transitively on itself,the ½^-invariant subspace

VV in TcV is the internal graded symmetric coalgebra on V

and is generated by elements of the formX

¾2§n

kid(¾)v¾[1 to n];

which we will denote by v^[1 to n]. For all v^[1 to n] 2 V ^n and ¾ 2 §n, we have the

commutation relation v^[1 to n] = kid(¾)v^¾[1 to n]: The coassociative comultiplication onVV is given by

¢(v^[1 to n]) =nX

j=0

X

½

(j;n¡j)¡unshu²e

kid(½)³v^½[1 to j]

´­³v^½[j+1 to n]

´:

2. The combined graded action ½ eis given by ¾ ¢ (v[1 to n]) = kjj jj(¾)v¾[1 to n]: The ½ e-

invariant subcoalgebra is the combined graded symmetric coalgebra jVV , generated byelements of the form X

¾2§n

kjj jj(¾)v¾[1 to n]:

The commutation relation is v e[1 to n] = kjj jj(¾)v e

¾[1 to n]; and the comultiplication isgiven by

¢(ve

[1 to n]) =nX

j=0

X

½

(j;n¡j)

kjj jj(½)

µv

e½[1 to j]

¶­µv

e½[j+1 to n]

¶;

where it is understood that the second sum is over all (j; n¡ j)-unshu²es.

Let sV be the suspension of V , that is to say, (sV )p = Vp¡1 for all p. The subcoalgebrajVV of the tensor coalgebra TcV is isomorphic to the subcoalgebraV

(sV ) of Tc(sV ). This

isomorphism replaces the bigrading on jVV with its total grading, allowing it to function moreeasily as a di®erential object. The coalgebra

V(sV ) is graded by internal degree, which will

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 7

be the preferred setting when looking at graded maps fromV

(sV ) into a di®erential gradedcommutative algebra, which are traditionally graded by internal degree.

The coalgebra isomorphism S : jVV ! V(sV ) is not of homogeneous degree; each map

Sn : V en ! (sV )^n increases the internal degree of an element by n. Exchanging a map fwith Sn will produce a sign of (¡1)h0;ni¢hed(f);id(f)i = (¡1)n¢id(f): The inverse isomorphism ofSn shall be denoted by S¡n. The map S respects the appropriate actions of §1.

Although we de¯ne action invariant maps on Tc(sV ), the de¯nition applies to TcV as well.Let W be a graded module.

De¯nition 2.1. : A map of graded modules bln : (sV )­n ! W is ½n-invariant if, for all

sv[1 to n] and ¾ 2 §n, bln(¾ ¢ (sv[1 to n])) = bln(sv[1 to n]). Similarly, a map bl : Tc(sV )! W is

½-invariant if bl(¾ ¢ (sv[1 to n])) = bl(sv[1 to n]) for all n and ¾ 2 §n.

If bl : Tc(sV ) ! W is ½-invariant, then bl restricted to the ½-invariant subcoalgebra is well-de¯ned because §n acts transitively on itself. The maps of greatest interest to us in this paperare either suspended internal graded symmetric (½^-invariant) or combined graded symmetric

(½ e-invariant). The map bln is suspended internal graded symmetric if dkid(¾)bln(sv¾[1 to n]) =bln(sv[1 to n]) and ln is combined graded symmetric if kjj jj(¾)ln(v¾[1 to n]) = ln(v[1 to n]) for

all v[1 to n] and ¾ 2 §n. We do not lose any information if we restrict the map bln to (sV )^n

(or restrict ln to V en) because

bln(sv^[1 to n]) = blnÃX

¾2§n

dkid(¾)sv¾[1 to n]

!

=X

¾2§n

dkid(¾)bln(sv¾[1 to n])

=X

¾2§n

bln(sv[1 to n])

= n!bln(sv[1 to n]):

So whenever we consider collections of maps on TcV with the same invariance, we pass to that

invariant subcoalgebra. The isomorphism S implies that any map ln : V en ! V induces amap bln : (sV )^n ! (sV ) by setting bln = (¡1)n¢id(ln)S1lnS¡n: This process is invertible, so the

set of all maps ln : V en ! V is isomorphic to the set of all maps bln : (sV )^n ! sV . Therefore,whatever we de¯ne and prove in the suspended internal graded setting has its analog in thecombined graded setting, e.g., all Lie algebras and graded commutative algebra de¯ned in theremainder of this section.

The graded Lie algebra Coder(V

(sV ))

Following [Lad], one can extend any map bln : (sV )­n ! (sV ) to a coderivation bln : Tc(sV )!Tc(sV ) by setting bln = 0 on (sV )­k for k < n and, for k > n, setting

bln(sv[1 to k]) =k¡n+1X

i=1

(1­i¡1 ­ bln ­ 1­k¡n+1¡i)(sv[1 to k]):

Here, (1­i¡1 ­ bln ­ 1­k¡n+1¡i)(v[1 to k]) =

dkid(bln; sv[1 to i¡1])sv[1 to i¡1] ­ bln(sv[i to i+n¡1])­ sv[i+n to k];

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 8

where dkid(bln; sv[1 to i¡1]) is the sign introduced by exchanging bln with sv[1 to i¡1]. However,

extending bln as a coderivation on Tc(sV ) does not guarantee that its invariance is preserved.

Exploiting the commutation relations, we extend (rather elegantly) any map bln on (sV )^n

to a coderivation on the coalgebraV

(sV ) by setting

bln(sv^[1 to k]) =X

½

(n;k¡n)

dkid(½)bln(sv^½[1 to n]) ^ sv^½[n+1 to k]: (6)

The reader is left to verify that extending a map bln as in equation (6) produces a well-de¯nedmap on

V(sV ).

Although for any action ½n, extending an ½n-invariant map bln as a coderivation on Tc(sV )does not result in a ½-invariant map on the tensor coalgebra, the restriction of the coderivationbln on Tc(sV ) to the ½-invariant subcoalgebra is equal to the coderivation one produces by

extending bln restricted to the ½n-invariant subspace of (sV )­n as a coderivation on the ½-

invariant subcoalgebra of Tc(sV ). The set of all coderivations bln generate the Lie algebraCoder(

V(sV )). Next, we examine maps cmn : (sV )­n¡1 ­W ! W (the Lie action map cm2

is such a map) and cfn : (sV )­n ! W (elements of Altk(g;B) are of this form), where Wis a graded commutative algebra. We will denote the degree of any element w 2 W by jwj.Elements of W have no external degree, at least not with respect to Tc(sV ) or its subcoalgebraV

(sV ).

Subordinate coderivations and Coder(V

(sV )­W )

We begin with maps cmn : (sV )­n¡1 ­W ! W . Let bln : (sV )­n ! sV be any map of degree

id(bln) and let bl be any coderivation on Tc(sV ) of degree id(bl).

De¯nition 2.2. A map cmn : (sV )­n¡1 ­W ! W is an bln-subordinate map if the degree of

cmn and bln agree. A map bm : Tc(sV )­W ! Tc(sV )­W is an bl-subordinate coderivation if

id(bm) = id(bl) and, for all p and q,

(¢p;q ­ 1W ) bm = (bl ­ 1)(¢p+n¡1;q ­ 1W ) + (1­ bm)(¢p;q+n¡1 ­ 1W ):

Just as the map bln extends to a coderivation on Tc(sV ), so too, the map cmn may be

extended as an bln-subordinate coderivation. On (sV )­k ­W , we set cmn = 0 for k < n¡ 1

and we set cmn = bln ­ 1W + 1­k¡n+1 ­ cmn for all k > n¡ 1.

De¯nition 2.3. The map cmn is ½n¡1-invariant if, for all ¾ 2 §n¡1,

cmn(¾ ¢ (sv[1 to n¡1])­w) = cmn(sv[1 to n¡1] ­ w):

If both bln and a subordinate coderivation cmn are invariant with respect to the same action½, then we may pass to the ½-invariant subcoalgebra, where cmn remains subordinate to bln. Weextend the map cmn on (sV )^n¡1 ­W as a coderivation on

V(sV )­W by setting

cmn(sv^[1 to k] ­w) = bln(sv^½[1 to k])­w +

X

¾

(k¡n+1;n¡1)

dkid(¾)dkid(cmn; sv^¾[1 to k¡n+1]) ¢

sv^¾[1 to k¡n+1] ­ cmn(sv^¾[k¡n+2 to k] ­ w):

(7)

The set of all subordinate coderivations Coder(V

(sV ) ­W ) forms a graded Lie algebraunder the graded commutator bracket. The bracket respects subordination, that is to say, ifcmi and cmj are bli and blj -subordinate coderivations respectively, then [cmi; cmj] is a coderiva-

tion subordinate to [bli; blj]. (Of course, the suspension map S respects subordinate maps andcoderivations.)

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 9

The graded commutative algebra Hom(V

(sV );V

(sV )­W )

Suppose the map cfn : (sV )^n ! W is suspended internal graded symmetric. We can extendcfn to an internal graded symmetric function from

V(sV ) into

V(sV )­W by setting cfn = 0

for k < n and, for k > n, setting cfn(sv^[1 to k]) =

X

¾

(k¡n;n)

dkid(¾)dkid(cfn; sv^¾[1 to k¡n])sv^¾[1 to k¡n] ­cfn(sv^¾[k¡n+1 to k]): (8)

The set of all such maps is Hom(V

(sV );V

(sV )­W ).

Suppose cmp 2 Coder(V

(sV )­W ) is subordinate to blp 2 Coder(V

(sV )). The map

hcmp; i : Hom(^

(sV );^

(sV )­W )! Hom(^

(sV );^

(sV )­W );

given by setting Dcmp;cfn

E= cmp

cfn ¡dkid(cmp;cfn)cfnblp

on (sV )^n+p¡1 and extendingDcmp;cfn

Eto all of

V(sV ) as in equation (8) is well-de¯ned. This

map exposes an unusual type of graded Lie module structure on Hom(V

(sV );V

(sV )­W ). Itis, in a sense, a Lie-module over both Coder(

V(sV )­W ) and Coder(

V(sV )) simultaneously

in the sense that

h[cmi; cmj]; i = hcmi; hcmj; ii ¡dkid(cmi; cmj) hcmj; hcmi; ii :We will call such a Lie module shared.

When (B; sg) is a Lie-Rinehart pair, the Lie module map cm2 : g­B! B should be a mapfrom sg into Der(B). With this in mind, we introduce the following notion.

De¯nition 2.4. : An bln-subordinate map cmn is a W -derivation source if for every sv^[1 to n¡1]

in (sV )^n¡1, the map cmn(sv^[1 to n¡1] ­ ( )) : W !W is a graded derivation on W .

In other words, a W -derivation source cmn is a map from (sV )^n¡1 into Der(W ). In prac-tice, we will work with cmn as a coderivation on

V(sV ) ­W , where the characterization of

cmn as a map into Der(W) is somewhat obscured. The subset of all W -derivation sources inCoder(

V(sV )­W ) forms a graded Lie subalgebra denoted by CoderW (

V(sV )­W ).

The subset of all coderivations bln 2 Coder(V

(sV )) which admit a subordinate W -derivationsource cmn forms a graded Lie subalgebra which we will denote by CoderW (

V(sV )).

The algebra structure onW extends to a graded commutative algebra structure onV

(sV )­W , which provides Hom(

V(sV );

V(sV ) ­W ) with a cup product. If cfn and bgs are maps in

Hom(V

(sV );V

(sV )­W ). Then cfn ^ bgs is given by setting (cfn ^ bgs)(sv^[1 to n+s]) =

X

½

(n;s)

dkid(½)dkid( bgs; sv^½[1 to n])cfn(sv^½[1 to n]) ¢ bgs(sv^½[n+1 to n+s])

on (sV )^n+s, where ¢ represents the the multiplication on W (henceforth, the ¢ will be su-

pressed). The map cfn ^ bgs is then extended to all ofV

(sV ) as in equation (8). The maphcmp; i acts as a derivation with respect to the cup product, so long as cmp is a W -derivationsource.

W -linear maps and resting coderivationsFor a Lie-Rinehart pair (B; sg), the Rinehart complex of B-linear alternating maps is a sub-complex of the Chevalley-Eilenberg complex without modifying the di®erential. This is due to

the B-linearity of any cfn in AltB(sg;B), the Lie-Rinehart relations and the fact that the Liemodule map cm2 is a B-derivation source.

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 10

If the graded module V is a module over the graded commutative algebra W , then the tensorcoalgebras TcV and Tc(sV ) are modules over TW . We will follow our general convention anddenote w1v1­¢ ¢ ¢­wnvn by wv[1 to n] and, in the suspended setting, denote w1sv1­¢ ¢ ¢­wnsvnby wsv[1 to n]. The unraveling map

bU : Tc(sV )!W ­ Tc(sV )

wsv[1 to n] 7! bu([1 to n])w¢[1 to n] ­ sv[1 to n];

where bu([1 to n]) is the sign produced when moving the wi's past the vj's. The map bU respectsthe coalgebra structure of Tc(sV ), that is to say, for all p and q, the diagram

(sV )­p ­ (sV )­q(M­1)T2;3(bU­bU)¡! W ­ (sV )­p ­ (sV )­qx??¢p;q

x??1­¢p;q

(sV )­p+qbU¡! W ­ (sV )­p+q

commutes.

De¯nition 2.5. : A suspended internal graded symmetric map cfn : (sV )^n !W is W -linearif

cfn(wsv^[1 to n]) = bu([1 to n])dkid(cfn;w¢[1 to n])w¢[1 to n]

cfn(sv^[1 to n]):

Likewise, a suspended internal graded symmetric map cmn : (sV )^n¡1 ­W ! W is W -linearifcmn(wsv^[1 to n¡1] ­ w) =

bu([1 to n¡ 1])dkid(cmn;w¢[1 to n¡1])w¢[1 to n¡1]cmn(sv^[1 to n¡1] ­w):

For a map cmn : (sV )^n¡1 ­W ! W , we de¯ne the V -extension map cmne : (sV )^n ! V

by setting

cmne(wsv^[1 to n]) =

X

½

(n¡1;1)

dkid(½)cmn(wsv½[1 to n¡1] ­ w½(n))sv½(n):

De¯nition 2.6. : Given cmn 2 CoderW (V

(sV )­W ), an W -linear, W -derivation source which

is subordinate to bln 2 CoderW (V

(sV )), the map bln is said to rest on cmn if bln(wsv^[1 to n]) =

bu([1 to n])dkid(bln;w¢[1 to n])w¢[1 to n]

bln(sv^[1 to n]) + cmne(wsv^[1 to n]):

The subset of CoderW (V

(sV )) of all coderivations blp which rest on W -linear W -derivation

sources cmp forms a graded Lie subalgebra. This subalgebra is denoted by CoderWW (V

(sV )).

Similarly, the subset of CoderW (V

(sV )­W ) consisting of all W -linear W -derivation sources

cmp subordinate to maps blp, which in turn rest on cmp, forms a graded Lie subalgebra, denoted

by CoderWW (V

(sV ) ­ W ). Proposition 2.7 is crucial and stated below; its proof is in theappendix.

Proposition 2.7. : Let cfn : (sV )^n ! W be a W -linear map and suppose the map cmp :

(sV )^p¡1­W !W is a W -linear, W -derivation source subordinate to the map blp : (sV )^p !V , which in turn rests on cmp. Then

Dcmp;cfn

E= cmp

cfn ¡dkid(cmp;cfn)cfnblp is W-linear.

Proof of Proposition 2.7. : First we compute cmpcfn(wsv[1 to p+n¡1]). We break up the result

into two sums. The ¯rst one, (Lin cmpcfn), consists of those terms in which all the wi's can and

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 11

have been factored out. The remaining terms form the second sum (Nonlin cmpcfn). The ¯rst

sum is over all (p¡ 1; n)-unshu²es ¾:

(Lin cmpcfn) =

X

¾

§w¢[1 to p+n¡1]cmp(sv^¾[1 to p¡1] ­cfn(sv^¾[p to n+p¡1]):

The second sum is actually a double sum over all (p ¡ 1; n)-unshu²es ¾ and all (1; n ¡ 1)-unshu²es ° acting on the second ¾-hand:

(Nonlin cmpcfn) =

X

¾

X

°

§cmp(sv^¾[1 to p¡1] ­ w¢¾[1 to p¡1] ¢ w¢°¾[p to p+n¡1]) ¢cfn(sv^°¾[p to n+p¡1]):

We compute cfnblp(wsv^[1 to p+n¡1]) by again breaking it up into two sums: (Lin cfnblp), is a

sum over all (p; n¡ 1)-unshu²es ½, i.e.,

(Lin cfnblp) =X

½

§w¢[1 to p+n¡1]cfn(blp(sv^¾[1 to p]) ^ sv^¾[p+1 to n+p¡1])

and (Nonlin cfnblp) is again a double sum over all (p; n ¡ 1)-unshu²es ½ and all (n ¡ 1; 1)-unshu²es ¿ acting on the ¯rst ½-hand. In other words,

(Nonlin cfnblp) =X

½

X

¿

§cmp(sv^¿½[1 to p¡1] ­w¢¿½[1 to p] ¢w¢½[p+1 to p+n¡1])

cfn(sv¿½(p) ^ sv^½[p+1 to n+p¡1]):

Recognizing that jjcmpcfnjj = jjcfnblpjj, it follows that

h(Lin cmp

cfn)¡ bk(cmp;cfn)(Lin cfnblp)i

= §w¢[1 to p+n¡1]

Dcmp;cfn

E(sv^[1 to p+n¡1]):

Therefore, once we show thath(Nonlin cmp

cfn)¡ bk(cmp;cfn)(Nonlin cfnblp)i

= 0; the proof will

be complete.For every pair consisting of a (p ¡ 1; n)-unshu²e ¾ and a (1; n¡ 1)-unshu²e ° acting on

the second ¾ hand, there is a unique pair consisting of a (p; n¡ 1)-unshu²e ½ and a (p¡ 1; 1)-unshu²e ¿ acting on the ¯rst ½ hand which produces the exact same sequence of components,that is to say,

¾[1 to p ¡ 1] = ¿½[1 to p¡ 1];

°¾(p) = ¿½(p) and

°¾[p+ 1 to p+ n¡ 1] = ½[p+ 1 to p+ n¡ 1]:

Why should this be so? A simple counting argument provides the answer. Either combinationproduces an (p ¡ 1; 1; n¡ 1)-unshu²e. There are

¡p+n¡2p¡1

¢distinct (p ¡ 1; 1; n¡ 1)-unshu²es

which send j 2 f1; : : : ; p+n¡1g to the pth position because once the pth position is determined,the remaining p + n ¡ 2 terms must be unshu²ed into a hand of length p ¡ 1 and a handof length n ¡ 1. How many distinct pairs (¾; °) send j to the pth position? This question isequivalent to asking how many distinct (p¡1; n)-unshu²es ¾ send j to the second hand since,once j is in the second hand, there is a unique (1; n¡ 1) unshu²e ° which will send j to thepth position. The number of (p¡1; n)-unshu²es is

¡p+n¡1n

¢and

¡p+n¡2n¡1

¢of them have j in the

second hand. Since¡p+n¡2n¡1

¢=¡p+n¡2p¡1

¢, we know there is a one-to-one correspondence between

(p¡ 1; 1; n¡ 1)-unshu²es and pair (¾; °) which send j to the pth position. A similar argumentshows that the number of (p ¡ 1; 1; n¡ 1)-unshu²es which send j to the pth position equalsthe number of (½; ¿) combinations which do the same. We can show that the (¾; °) term of

(Nonlincmpcfn) and the corresponding (½; ¿) term of (Nonlincfnblp) have opposite signs, so they

cancel. 2

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 12

Naturally, the graded commutative algebra HomW (V

(sV );V

(sV ) ­ W ) is a shared Liemodule over the Lie algebras CoderWW (

V(sV );

V(sV )­W ) and CoderWW (

V(sV )).

3. Lie algebras, their modules and cohomology in the coalgebra set-ting

Lie algebras and their modulesThe most familiar de¯nition of a Lie algebra g is as an ungraded k-vector space (or, moregenerally, a module over a k-algebra A) equipped with a skew-symmetric linear map l2 =[ ; ] : g­ g! g which satis¯es the Jacobi identity, that is to say the equality

l2(x; l2(y; z)) = l2(l2(x; y); z) + l2(y; l2(x; z))

holds for all x; y; z 2 g. In the context of graded coassociative coalgebras, we can considerg a graded vector space where all elements of g have combined degree h1; 0i. The bracket l2

becomes a degree h¡1; 0i coderivation on jV g. No information is lost in the process because

le

2 (x ey) = l2

³x­ y + (¡1)h0;1i¢h0;1iy ­ x

´

= l2(x­ y)¡ l2(y ­ x)

= l2(x­ y) + l2(x­ y)

= 2l2(x­ y):

But we can also de¯ne the Lie algebra onV

(sg), where the bracket bl2 becomes an internalgraded symmetric coderivation of degree -1 on

V(sg) with respect to the internal grading.

Again, no essential information is lost. We can therefore realize a Lie algebra in three equivalentways:

1. A Lie algebra is a vector space g over k with skew symmetric bracket l2 which satis¯esthe Jacobi identity (the original de¯nition).

2. A Lie algebra is the subcoalgebra jVg, together with a combined degree h¡1; 0i coderiva-tion l2 satisfying l2±l2 = 1

2 [l2; l2] = 0, which is equivalent to the Jacobi identity. Since theinternal degree is zero, it can be ignored, so l2 can be considered a degree¡1 coderivation,

i.e., a di®erential on jVg.

3. A Lie algebra is a subcoalgebraV

(sg) together with a suspended internal degree -1

di®erential bl2. This is our preferred setting.

The Jacobi identity is a result of the more general fact, true for all coderivations bl with odddegree, that [bl;bl] = 2bl ±bl and [bl;bl] = ¡[bl;bl] = 0. In other words, any degree ¡1 coderivation blis a di®erential on the coalgebra

V(sg) and if the di®erential bl has external degree ¡1, then it

is the extension of a map bl : sg ^ sg! sg as a coderivation, i.e., bl is a Lie bracket.Suppose B is an ungraded Lie module over g, i.e., there is a map cm2 : sg ­B ! B such

that cm2(bl2(sx^ sy)­ b)¡ cm2(sx­ cm2(sy­ b)) + cm2(sy­ cm2(sx­ b)) = 0; for all x; y 2 g and

b 2 B. The map cm2 is bl2-subordinate, and we can extend cm2 to an bl2-subordinate coderivationonV

(sg)­B. The equation above can rewritten as cm2 ± cm2 = 12[cm2; cm2] = 0 and since the

degree of cm2 is -1, it is a di®erential onV

(sg)­B. In the coalgebra setting, then, an g-modulestructure is any coderivation cm2 on

V(sg)­B which is a di®erential and which is subordinate

to the bracket bl2.

(Cartan-)Chevalley-Eilenberg CohomologyDe¯nition 3.1. : A Chevalley-Eilenberg pair (B;g) is a Lie algebra g and Lie module B overg.

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 13

For any Chevalley-Eilenberg pair (B; g), an n-multilinear function fn : g£n ! B is alter-nating if

fn(x¾(1); :::; x¾(n)) = (¡1)¾fn(x1; :::; xn)

for all ¾ 2 §n, where (¡1)¾ is the sign of the permutation. The algebra of all alternatingmultilinear functions Altk(g;B) is graded by n and admits a degree +1 di®erential ±CE :Altnk (g;B)! Altn+1

k (g;B) given by equation (2). The cohomology of this complex with respectto ±CE is the Chevalley-Eilenberg cohomology of g with coe±cients in B.

In the coalgebra setting, any alternating n-multilinear function fn can be realized uniquely

as a linear function cfn : (sg)^n ! B, which then can be extended as a coderivation cfn :V(sg) ! V

(sg) ­B. We extend b 2 B as a coderivation b :V

(sg) ! V(sg) ­ B by setting

b(sx^[1 to k]) = sx^[1 to k] ­ b: Any function bF 2 Hom(V

(sg);V

(sg)­ B) is the sumP1

n=0cfn,

where cfn : (sg)^n ! B has degree n. Hence, Hom(V

(sg);V

(sg) ­ B) is graded by N andis isomorphic to Altk(g;B). The di®erential ±CE on Hom(

V(sg);

V(sg)­B) has the elegant

form hcm2; i. This formula works even for b 2 B, since ±CEb = hcm2; bi = cm2b+ bbl2 and bl2 = 0on sg. Since [cm2; cm2] = ¡[cm2; cm2] = 0, it is now simple to show that

±CE ± ±CE = hcm2 hcm2; ii =

¿1

2[cm2; cm2];

À= 0:

If B is an algebra, then Hom(V

(sg);V

(sg)­B) is also an algebra. The di®erential ±CE actsas a derivation with respect to the multiplication.

Lie-Rinehart pairs and Rinehart cohomology

Suppose B is an algebra over k which is also a g-module. The de¯nition of a Lie-Rinehart pairbelow has been modi¯ed from de¯nition 1.1 to re°ect the coalgebra setting.

De¯nition 3.2. : [Rin63] Let ¹ : B­ (sg) ! (sg) be a left B-module action on sg denoted

by ¹(a ­ ®) := a®. Let cm2 : (sg) ­ B ! B be an bl2-subordinate B-derivation source, i.e.,an sg-module action on B. The pair (B; sg) is a Lie-Rinehart pair, provided the Lie-Rinehartrelations (LRa) and (LRb) are satis¯ed.

LRa: cm2 is B-linear.

LRb: bl2 rests on cm2 (de¯nition 2.6).

Suppose cfn is in HomB(V

(sg);V

(sg) ­ B). From proposition 2.7, we know the image ofcfn under the Chevalley-Eilenberg di®erential ±CE = hcm2; i is again B-linear. The Rinehartalgebra R = HomB(

V(sg);

V(sg)­B) with di®erential ±R = hcm2; i is a subcomplex of the

Chevalley-Eilenberg complex and the cohomology of R with respect to ±R is the Rinehartcohomology of g with coe±cients in B.

4. Strongly homotopy Lie algebras, their modules and cohomology

Strongly homotopy Lie algebras (shLie algebras) ¯rst appeared implicitly in [Sul77] andexplicitly in [SS] in the context of deformation theory. A concise introduction to shLie algebrasis found in [LM95]. The initial de¯nitions below are lifted directly from [LM95] and thenmodi¯ed to ¯t the language introduced in x2.

De¯nition 4.1. : [LM95] and [LS] An L(m)-structure on a graded module L is a system

of linear maps flk : 1 6 k 6 m 6 1; k 6= 1g, where lk :NkL ! L has combined degree

h1¡ k; k ¡ 2i and each map is combined graded symmetric in the sense that

lk(v¾[1 to k]) = kjj jj(¾)lk(v[1 to k])

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 14

for all ¾ 2 §k. Moreover, the following generalized form of the Jacobi identity, the nth Jacobiidentity map, is satis¯ed for n 6 m:

JIDn(v[1 to n]) =X

i+j=n+1

i;j>1

X

¾

kjj jj(¾)(¡1)i(j¡1)lj(li(v¾(1 to i))­ v¾(i+1 to n)) = 0;

where the second summation runs over all (i; j ¡ 1)-unshu²es. The graded vector space L isa strongly homotopy Lie algebra (or shLie algebra) if L admits an L(1)-structure.

We can rewrite this de¯nition in terms of maps on jVL because the maps lk are combined

graded symmetric. The Jacobi identity maps JIDn should equal 0 on jVL, but it is not imme-

diately evident that the maps JIDn can be extended as coderivations on jVL. Fortunately, we

can rewrite the Jacobi identity maps using the bracket on Coder( jVL). Consider the bracket

of lj with li with respect to the combined grading on jVL:

[lj ; li] = ljli ¡ (¡1)h1¡j;j¡2i¢h1¡i;i¡2ililj = lj li + (¡1)j+ililj :

Multiplying both sides by the desired sign for ljli in the Jacobi identity map JIDn, we ¯ndthat

(¡1)i(j¡1)[lj ; li] = (¡1)i(j¡1)ljli + (¡1)i(j¡1)+j+ililj

= (¡1)i(j¡1)ljli + (¡1)j(i¡1)lilj:

We therefore reformulate the de¯nition of an L(m)-algebra and an shLie algebra:

De¯nition 4.2. : An L(m)-structure on a graded module L consists of a system of coderiva-

tions flk : jVL ! jVL : 1 6 k 6 m 6 1; k 6= 1g, where each lk has combined degreeh1¡ k; k ¡ 2i. Moreover, the following generalized form of the Jacobi identity is satis¯ed forn 6 m:

JIDn =1

2

X

i+j=n+1

i;j>1

(¡1)i(j¡1)[lj ; li] = 0 (9)

on Len. Again, if L admits an L(1)-structure, then L is a strongly homotopy Lie algebra

(an shLie algebra).

Since the Jacobi identity maps are sums of coderivations, it follows that JIDn can be

extended to a coderivation on jVL. Furthermore, since JIDn = 0 on Len, we have proven

the following proposition.

Proposition 4.3. : JIDn = 0 on jVL.

In both [LS] and [LM95], this result was not achieved without ¯rst passing to the suspendedinternal graded symmetric setting. Lada, Markl and Stashe® de¯ned strongly homotopy Lie

algebras on the tensor coalgebra TcL rather than on jVL, whose existence had not beenrecognized. The maps JIDn cannot be extended as coderivations on the tensor coalgebrawhile preserving the desired symmetry. So instead, they suspended the maps lk, changingthe symmetry of the maps from combined graded symmetric to suspended internal graded

symmetric. They were then in familiar territory, where the maps [JIDn could be extendedas coderivations on

V(sL), even though they did not make explicit use of the bracket on

Coder(V

(sL)).

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 15

Modules over a strongly homotopy Lie algebraDe¯nition 4.4. : [LM95] Let L = (L; li) be an L(p)-algebra (0 < p < 1) and let M be adi®erential graded module with di®erential m1. Then a left L(k)-module structure over L onM (for k 6 p) is a collection fmn : 1 6 n 6 k; n6=1g of ln-subordinate coderivations mn

on jVL­M such that the nth action identity map

ACT IDn =X

i+j=n+1

(¡1)i(j¡1)mjmi

=1

2

X

i+j=n+1

i;j>1

(¡1)i(j¡1)[mj ;mi]

= 0

on jVL­M . The di®erential graded module M is a strongly homotopy Lie module over L (oran L-shLie module) if L admits an L(1)-structure and M is a module with respect to thatL(1)-structure.

De¯nition 4.4 implies that the di®erential m1 on M must be l1-subordinate. It is simple to

verify that m1 is a di®erential on jVL­M .When M is a di®erential graded commutative algebra, we require that the maps mi be M-

derivation sources, that is to say, for every v e[1 to i¡1] 2 L

ei¡1, the map mi(v[1 to i¡1]­ ( )) :

M !M is in Der(M).

Since Coder( jVL) ¼ Coder(V

(sL)), it follows that if (L; li) is a strongly homotopy Lie

algebra, then (sL; bli), where bli is the coderivation onV

(sL) induced by the isomorphism

S : jVL ! V(sL), is also an shLie algebra. The suspended internal degree of each map bli is

¡1, so the sum of the L(1)-structure maps bli is a di®erential onV

(sL), ¯rst shown in [LS].This is not the case for (L; li) because each li is bigraded. In order for the sum of the li'sto be a di®erential, one would have to make sense of li as a degree ¡1 map. Passing to thesuspended internal graded symmetric setting is a convenient way to accomplish this becausedegree of bli is ed(li) + id(li) = ¡1, which is the total degree of li. For (sL; bli), the nth Jacobiidentity map is

dJIDn =1

2

X

i+j=n+1

i;j>1

[blj; bli]; (10)

which is 0 onV

(sL).

Proposition 4.5. : [LS] If (sL; bli) is an shLie algebra, the map DsL =P1

i=1bli is a di®erential

onV

(sL), i.e. it is a map of degree -1 such that DsL ±DsL = 0.

Proof of Proposition 4.13. : A proof can be found in [LS]. Here the proof is essentially thesame but takes advantage of the bracket of coderivations:

DsL ±DsL =

0@1X

j=1

blj

1A ±

à 1X

i=1

bli!

=1X

j=1

1X

i=1

bljbli =1X

n=1

X

i+j=n+1

i;j>1

bljbli:

Using equation (10) in de¯nition 4.2, we see that

1X

n=1

X

i+j=n+1

i;j>1

bljbli =1

2

1X

n=1

X

i+j=n+1

i;j>1

[blj; bli] =1

2

1X

n=1

dJIDn = 0:

Note also that DsL is a degree ¡1 coderivation, so DsL ±DsL = 12 [DsL;DsL] and the graded

skew-commutativity of the bracket ensures that [DsL;DsL] = 0. 2

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 16

Suppose (M;mi) is an shLie module over (L; li). In the suspended internal graded symmetricsetting, the maps cmi :

V(sL)­M ! V

(sL)­M form an shLie module structure on M over

(sL; bli). Again, the degree of cmi is ¡1, the total degree of mi, and the action identity map hasthe form

\ACT IDn =1

2

X

i+j=n+1

i;j>1

[cmj;cmi]; (11)

which is zero onV

(sL)­M .

Proposition 4.6. : The map DM =P1

i=1 cmi is a di®erential onV

(sL)­M, i.e., DM ±DM =0.

Proof of Proposition 4.14. : Isomorphic to the proof of proposition 4.5. Notice that DM isDsL-subordinate and that DM ±DM = 1

2[DM ;DM ] = 0. 2

It is only for historic reasons that we present shLie structures and shLie module structures¯rst in the combined graded symmetric setting ¯rst and then in the suspended internal gradedsymmetric one. We will now use the suspended internal graded symmetric setting exclusively.

Homotopy Chevalley-Eilenberg cohomologyDe¯nition 4.7. : A homotopy Chevalley-Eilenberg pair (M;L) consists of an shLie algebraL and an L-shLie module M .

A multi-linear alternating function Fn : L£n ! M can be seen as a linear function cFn :(sL)^n !M . So the homotopy Chevalley-Eilenberg algebra Altk(L;M) is isomorphic to

Homk(^

(sL);^

(sL)­M):

Any homogeneous map bF splits into a sumP1

n=0bFn, where bFn : (sL)^n !M. The map bFn is

extended as a coderivation onV

(sL)­M. Any function bF0 is an element a 2M . The degreeof a : (sL)^0 ! M as a map is the negative of its degree as an element of M . The map a isextended as a coderivation on

V(sL)­M by setting

a(sv^[1 to k]) = dkid(a; sv^[1 to k])sv^[1 to k] ­ a:

Instead of N-graded, as was the case in the ungraded setting, the Chevalley-Eilenberg com-plex Homk(

V(sL);

V(sL) ­ M) is Z-graded by suspended internal degree. For any bF 2

Homk(V

(sL);V

(sL)­M) of degree bid( bF ), we de¯ne

±hCE bF = DMbF ¡ bkid( bF ) bFDsL =

DDM ; bF

E;

which has degree bid( bF ) + 1.

Proposition 4.8. : ±hCE ± ±hCE = 0:

Proof of Proposition 4.17. :

±hCE ± ±hCE = hDM ; hDM ; ii

=

¿1

2[DM ;DM ];

À

= 0: 2

The cohomology with respect to ±hCE is the homotopy Chevalley-Eilenberg cohomology ofsL with coe±cients in M .

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 17

When M is a di®erential graded commutative algebra, the homotopy Chevalley-Eilenbergcomplex Homk(

V(sL);

V(sL) ­ M) is a di®erential graded commutative algebra. Suppose

bF =P1

i=0bFi and bG =

P1j=0

cGj are both elements of the complex, where bFi :Vi(sL) ! M

and cGj :Vj(sL) ! M . The product bF ^ bG =

P1n=0( bF ^ bG)n is given by ( bF ^ bG)n =P

i+j=nbFi ^ cGj. The di®erential ±hCE acts as a derivation with respect to this multiplication.

Homotopy Lie-Rinehart pairs and homotopy Rinehart cohomologyThe homotopy Rinehart complex is a straightforward generalization of the Rinehart complexin the ungraded setting. Since the Rinehart complex is de¯ned only for Lie-Rinehart pairs(B;g), we must de¯ne what constitutes a homotopy Lie-Rinehart pair (M; sL). The subsetAltM (sL;M) of Altk(sL;M) consisting of all M -linear alternating functions is isomorphic toHomM (

V(sL);

V(sL)­M).

De¯nition 4.9. : A homotopy Lie-Rinehart pair (M; sL) consists of a di®erential graded

commutative algebra (M;cmi) which is an shLie module over the shLie algebra (sL; bli), whichin turn is an M-module. Moreover, the following two homotopy Lie-Rinehart relations mustbe satis¯ed for all i > 1:

(hLRai): The shLie module structure map cmi is M -linear.

(hLRbi): The shLie structure map bli rests on cmi (de¯nition 2.6).

An L(p)-Lie-Rinehart pair ((M;cmi); (sL; bli)) has maps cmi and bli which satisfy (hLRai) and(hLRbi) for 1 6 i 6 p.

Proposition 4.10. : If bF is M-linear, then so is ±hCE bF .

Proof of Proposition 4.18. : The image of bF under ±hCE is

DDM ; bF

E=1X

i=1

Dcmi; bF

E:

The proof follows as a consequence of proposition 2.7. 2

We conclude that the subset of all M -linear functions in Homk(V

(sL);V

(sL)­M) forms asubcomplex R = HomM (

V(sL);

V(sL)­M) with di®erential ±R = hDM ; i. The di®erential,

together with the cup product, provides the homotopy Rinehart complex R with the structureof a di®erential graded commutative algebra. The cohomology of R with respect to ±R is thehomotopy Rinehart cohomology of sL with coe±cients in M.

5. Homotopy Lie-Rinehart resolutions of Lie-Rinehart pairs

It is often necessary to replace the components of a complex with resolutions of thosecomponent in order to construct a model for a complex with the same basic algebraic structureand precisely the same (co)homology as the original. (It is not always necessary to replaceall of the components, see [Sta92], for example.) When building a model for the Rinehartcohomology for a Lie-Rinehart pair (B; sg) over a k-algebra A, we replace the pair with a pairof resolutions (M;sL), which retain as much of the algebraic structure of (B; sg) as possible.It makes sense, then, that (M;L) should form a homotopy Lie-Rinehart pair.

Piecing together Lie-Rinehart resolutionsLet (B; sg) be a Lie-Rinehart pair over a k-algebra A with module structure maps b¹ : B­sg!sg and cm2 : sg­B! B. Let bl2 denote the bracket on sg and ¼B denote the multiplication onB. (Henceforth, only maps associated with the Lie-Rinehart pair will have hats.) The basicingredients of a homotopy Lie-Rinehart resolution for a Lie-Rinehart pair (B; sg) are (sL; li)and (M;mi; ¼M ) where Hl1(sL) = sg and Hm1(M) = B.

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 18

De¯nition 5.1. : A homotopy Lie-Rinehart resolution of a Lie-Rinehart pair (B; sg) over analgebra A is a homotopy Lie-Rinehart pair (M; sL) over A, such that

(a) the shLie algebra (sL; li), seen as the coalgebraV

(sL), resolvesV

(sg), i.e.,

Hl1(^

(sL)) =^

(sg); (12)

where (following the physicists' notation) Hl1 denotes the homology with respect to thedi®erential l1,

(b) the di®erential graded commutative algebra (M;mi; ¼M ) satis¯es

Hm1(^

(sL)­M) =^

(sg)­B; (13)

(c) the dgca (M;mi; ¼M ) also satis¯es

Hm1(M ­M) = B­B: (14)

Furthermore, the following conditions hold:

i. Hl1(l2) = bl2 onV

(sg).

ii. Hm1(m2) = cm2 onV

(sg)­B.

iii. Hm1(¼M ) = ¼B on B­B.

The conditions above are quite reasonable, but satisfying them is far from automatic. Con-sider any homotopy Lie-Rinehart pair (M; sL). Since m1 is a derivation on M , the multiplica-tion on M is a chain map which induces a well-de¯ned map Hm1

(¼M ) from Hm1(M ­M) into

Hm1(M). BothV

(sL) andV

(sL)­M are complexes with di®erentials l1 and m1, respectively.Since the maps JID2 and ACT ID2 are both zero, it follows that both l2 and m2 are gradedmorphisms of complexes and as such, induce well-de¯ned maps Hl1(l2) and Hm1(m2) on theirrespective homologies.

In the proposition below, we state conditions on the homotopy Lie-Rinehart pair whichguarantees it satis¯es the conditions in de¯nition 5.1.

Proposition 5.2. : Let (M;sL) be a homotopy Lie-Rinehart pair. Suppose the di®erentialgraded algebra (M;mi; ¼M ) is a projective resolution of B over A which respects the algebrastructure on B, i.e., condition iii of de¯nition 5.1 is satis¯ed. Suppose (sL; li) is a projectiveresolution sg. Furthermore,

(a) Hl1(l2) = bl2 on sg ^ sg and

(b) Hm1(m2) = cm2 on sg­B.

Then (M;sL) is a homotopy Lie-Rinehart resolution of (B; sg).

SDR-DataFor the proof of proposition 5.2, we will use strong deformation retract data (SDR-data). Ingeneral, SDR-data for two di®erential graded modules (M; dM ) and (N; dN ) over a commuta-tive ring R with unity consist of two degree 0 chain maps ¸ : M ! N and p : N !M and ahomotopy h : N ! N such that

p¸ = 1M and

¸p = 1N + dNh+ hdN ;

which guarantee an isomorphism of homology (or cohomology), that is to say, HdM (M) ¼HdN (N). SDR-data is succinctly described by

³(M; dM )

¸ -¾p

(N;dN); h´:

When the inclusion map ¸ and the projection map p also respect additional algebraic structureon M and N , then that structure is carried over to homology (or cohomology).

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 19

Proof of Proposition 5.2. : For the deleted projective resolution M of B, the non-deleted pro-

jective resolution M+ : MpM! B ! 0 is acyclic, so there exists a contracting homotopy

si : (M+)i ! (M+)i+1 where s¡1 : B !M0 such that 1M+ = m+1 s+ sm+

1 [Wei94]. Setting¸M = s¡1 and ¡(hM )i = si for i > 0, we produce SDR-data

³(B; 0)

¸M -¾pM

(M;m1); hM´:

By construction, pM¸M = 1B. On Mi for i > 0, the map ¸MpM = 0 and 1Mi= ¡m1hM ¡

hMm1. For i = 0, the identity map 1M0 = ¡hMm1 + ¸MpM and m1hM = 0. Therefore¸MpM = 1M +m1hM + hMm1. Likewise, we can construct SDR-data

³(sg; 0)

¸sL -¾psL

(sL; l1); hsL

´

for sg and sL.We will now repeatedly \tensor" SDR-data to produce new SDR-data, using the Gugenheim

tensor trick ([GLS91]) for de¯ning the homotopy h. We illustrate this technique by showingthat

³(B­B; 0)

¸M ­ ¸M-¾pM ­ pM

(M ­M;m1); hM­M´;

is SDR-data, where setting hM­M = 1M­hM +hM­¸MpM was ¯rst proposed by Gugenheimand Lambe in [GL89]. Clearly, (pM ­ pM)(¸M ­ ¸M ) = 1B­B and it is a simple exercise toshow that 1M­M + m1hM­M + hM­Mm1 = (¸M ­ ¸M )(pM ­ pM ). We have veri¯ed thatequation (14) and condition (iii) in de¯nition 5.1 hold.

The tensor trick can be extended to show that

³(Tc(sg); 0)

T¸sL-¾TpsL

(Tc(sL); l1); Thsl´

is SDR-data, where T¸sL = ¸sL­i on (sg)­i, where TpsL = pM

­i on (sL)­i and where

ThsL =P1

i=1 TihsL with TihsL =Pi

j=1 1­j¡1­hsL­(¸sLpsL)­i¡j on (sL)­i (see [GLS91]).It is clear that TpsLT¸sL = 1Tc(sg). We can show that T¸sLTpsL = 1Tc(sL) + l1ThsL+ThsLl1by induction on n. As a result, Hl1(Tc(sL)) = Tc(sg). Similarly, we can de¯ne SDR-data

³(Tc(sg)­B; 0)

T¸M-¾TpM

(Tc(sL)­M;m1); ThM

´

where ThM =P1

i=1 TihM with

TihM =

0@i¡1X

j=1

1­j¡1 ­ hsL ­ (¸sLpsL)­i¡j+1 ­ ¸MpM

1A+ 1­i¡1 ­ hM

= Ti¡1hsL ­ ¸MpM + 1­i¡1 ­ hMon (sL)­i¡1 ­M .

Besides the inclusion map i :V

(sL) ! Tc(sL), there is also a splitting map j : Tc(sL) !V(sL) which sends sv[1 to n] to 1

n!sv^[1 to n]. Both i and j are chain maps, so the composition

ji = 1V(sL) induces an isomorphism between Hl1(V

(sL)) andV

(Hl1(sL)), which equalsV

(sg). Similarly, using (i­ 1M) and (j ­ 1M ), we can show that Hm1

(V

(sL)­M) =V

(sg)­ B:Having veri¯ed that equations (12) and (13) in de¯nition 5.1 hold, it is now straightforwardto show that conditions (a) and (b) in proposition 5.2 guarantee that conditions (i) and (ii)are satis¯ed. 2

Homology, Homotopy and Applications, vol. ????, No. ????, 2001 20

We can use a spectral sequence argument to prove the following result.

Theorem 5.3. : If the pair (M;sL) is a projective homotopy Lie-Rinehart resolution of theLie-Rinehart pair (B; sg), then (R; hDM ; i) is a cohomological model for (R; hcm2; i).Proof of Proposition 5.3. : We have the following SDR-data:

³(^

(sg); 0)¸1 -¾p1

(^

(sL); l1); h1

´

and ³(^

(sg)­B; 0)¸2 -¾p2

(^

(sL)­M;m1); h2

´:

The E0 term of the spectral sequence is the algebra R, bigraded by the external degree ofa map and the di®erence between the internal degree and the external degree, e.g, if the mapFn has internal degree jFnj, then it has bidegree (n; jFnj ¡ n). The di®erential hDM ; i splitsup into the sum

P1i=1 hmi; i, each of which has bidegree (i ¡ 1; 2 ¡ i). The map hm1; i is

the di®erential on E0. If Fn is a cocycle, then Fn is a chain map between the complexesV

(sL)and

V(sL) ­M . So the class of Fn induces a map [Fn] :

V(sg) ! V

(sg) ­ B. Therefore,the E1 term is contained in R and all elements [Fn] of E1, have bidegree (n; 0). It followsthat the cohomology of R is the cohomology of the E1 term with respect to the di®erential[hm2; i] given by sending [Fn] to [hm2; Fni]. The square of [hm2; i] is zero precisely because[hm2; hm2; Fnii] equals [¡hm3; hm1; Fnii ¡ hm1; hm3; Fnii]. The ¯rst term hm3 hm1; Fnii iszero because hm1; Fni = 0. The second term is the coboundary of hm3; Fni, so its class withrespect to hm1; i is zero.

There is an algebra splitting from R to E1 given by sending fn to [¸2fnp1], so E1 ¼ R

as graded commutative algebras. Furthermore, since Hm1(m2) = cm2 and Hl1(l2) = bl2, it

follows that [hm2; i] = hcm2; i. Therefore, E1 and R are isomorphic as di®erential gradedcommutative algebras and have the same cohomology. 2

Constructing homotopy Lie-Rinehart resolutions for a Lie-Rinehart pair is no easy task.In an upcoming paper, we construct a homotopy Lie-Rinehart resolution for the Lie-Rinehartpair (A

±I; I±I2) which arises in the BFV formulation of classical BRST cohomology.

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Lars Kjeseth lkjeseth@elcamino.cc.ca.us

Mathematical Sciences DivisionEl Camino CollegeU.S.A.Torrance16007 Crenshaw Blvd, Torrance, CA 90506