Homology, Homotopy and Applications, vol.3, No.7, 2001, pp.139–163

HOMOTOPY RINEHART COHOMOLOGY OF HOMOTOPY LIE-RINEHART PAIRS

LARS KJESETH

(communicated by Larry Lambe)

Abstract We define homotopy Lie-Rinehart pairs and the associated homotopy Rinehart algebra in the context of coalgebras. We define homotopy Lie-Rinehart resolutions and present conditions under which the associated homotopy Rinehart algebra is a cohomological model for the Rinehart algebra of the resolved Lie-Rinehart pair.

1. Introduction

The Rinehart cohomology of homotopy Lie-Rinehart resolutions of Lie-Rinehart pairs arose from 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, for example, [Kim92b], [Kim93], [Sta92], [Kim92a], [FHST89], [Sta88], [Sta96], [KS87] and [HT92]) is a differential graded Poisson algebra which, in certain cases, is a cohomological model for the Rinehart algebra (Alt_{B}(𝔤, B), δ_{R}) of alternating B-multilinear functions from a Lie algebra 𝔤 into the associative algebra B [Rin63]. The Rinehart complex is a subcomplex of the Chevalley-Eilenberg complex ([CE48]) precisely when (B, 𝔤) is a Lie-Rinehart pair satisfying certain Lie-Rinehart relations (see below).

In homological algebra, a traditional strategy is to replace both B and 𝔤 in Alt_{B}(𝔤, B) with free or projective resolutions R_{B} and R_{𝔤} which have the same or similar algebraic structures, at least up to homotopy. The new object Alt_{R_{B}}(R_{𝔤}, R_{B}) is a cohomological model for Alt_{B}(𝔤, B) if Alt_{R_{B}}(R_{𝔤}, R_{B}) is a differential graded commutative algebra (a dcga) with the same cohomology as Alt_{B}(𝔤, B). In contrast, the BFV construction of the classical BRST algebra (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 to construct a differential. Attempting to construct the dcga Alt_{R_{B}}(R_{𝔤}, R_{B}) fails if we resolve B 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

Received January 8, 2001, revised May 3, 2001; published on June 21, 2001. 2000 Mathematics Subject Classification: primary 17B55, 17B81; secondary 17B56, 17B99. Key words and phrases: strongly homotopy Lie algebras, Lie-Rinehart pairs, Rinehart cohomology, BRST cohomology. \begin{matrix}c \\ ◯ \end{matrix} 2001, Lars Kjeseth. Permission to copy for private use granted.

be Rinehart-like. But to resolve a Lie-Rinehart pair with a new pair (R_{B}, R_{𝔤}) which is Lie-Rinehart, we must give up a strictly Lie structure on R_{𝔤} in favor of a strongly homotopy Lie structure.

In its earliest incarnation (see [Sul77] and [SS]) a strongly homotopy Lie (shLie) algebra was defined on the tensor coalgebra T^{c}L of a graded module L. In [LM95] and [LS93], the shLie structure was shifted (almost by brute force) onto the graded symmetric coalgebra on the suspension of L, that is, on ⋀(sL). This transition seemed inelegant and we felt there must be smoother means and more satisfying motivations for making the move. By taking a close look at choices of grading and specific actions of the symmetric group on a graded vector space, we discovered we could identify a subcoalgebra ⋀ L of T^{c}L, which is isomorphic to ⋀(sL) and on which an shLie structure can be defined as it was on T^{c}L in [LM95] and [LS93].

Grading and Sign Conventions In order to understand the language of this paper, we must first establish our grading and sign conventions.

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

Let V = {V_{p}}_{0}^{∞} be a graded vector space over k or, more generally, a graded module over an algebra A over k. There are four distinct gradings on the tensor module ⨂ V = {V^{⊗n}}_{n=0}^{∞}: the external, internal, combined and total gradings. We shall denote v_{1} ⊗ ⋯ ⊗ v_{n} ∈ V^{⊗n} by v_{[1 to n]}. The external degree, ed(v_{[1 to n]}) is given by the number n of tensor components. The internal degree, id(v_{[1 to n]}) comes from the internal grading on V; each tensor component v_{i} is an element of V_{p_{i}} for some p_{i}. The internal degree is the sum ∑_{i=1}^{n} p_{i}. The combined degree ||v_{[1 to n]}|| is the vector ⟨ed(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 : ⨂ V → ⨂ V is of homogeneous degree r with respect to a specific grading if f ((⨂ V)_{p}) ⊂ (⨂ V)_{p+r} for all p. For each of the four gradings, the degree of the homogeneous map f is the difference between the degree of v_{[1 to n]} ∈ ⨂ V and the degree of its non-zero image under f.

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

Since an shLie algebra R_{𝔤} which resolves a Lie algebra 𝔤 is defined on a graded coalgebra, it is convenient to realize the Lie algebra 𝔤 on a graded colagebra as well. To do this, we must decide what degree to assign to each element of the Lie algebra. If each element of 𝔤 is assigned an external degree of one and an internal degree of zero, we have essentially three choices for describing 𝔤 as a coalgebra. In each case, the skew-commutativity of l_{2} appears as the result of applying the Koszul sign convention. If we select the natural bigrading on T^{c}𝔤, the element x has bidegree ⟨1, 0⟩, which we treat as a vector, and

l_{2}(x ⊗ y) = (−1)^{⟨1,0⟩·⟨1,0⟩}l_{2}(y ⊗ x) = −l_{2}(y ⊗ x).

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

l_{2}(x ⊗ y) = (−1)^{1·1}l_{2}(y ⊗ x) = −l_{2}(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 l_{2} again appears skewcommutative, but is actually graded commutative.

When it resolves a Lie algebra 𝔤, an shLie algebra R_{𝔤} must operate as a differential object, i.e., its differential must have degree −1 with respect to a single grading. So the shLie algebra resolution R_{𝔤} of 𝔤 must be graded either with respect to the total degree or with respect to the suspended internal degree. Since the differential graded commutative (dgc) algebra R_{B} which resolves B is graded by internal degree, the suspended internal grading for the shLie algebra resolution is the preferred choice. Therefore, we will also view the Lie algebra 𝔤 in its suspended form s𝔤.

Background For any smooth manifold M, the set of smooth functions, C^{∞}(M), is an associative commutative algebra and the set of smooth vector fields on the manifold, Γ(T M), forms a Lie algebra. Both C^{∞}(M) and Γ(T M) are modules over each other. Moreover, the the module actions satisfy the equations (fX) · g = f(X · g) and [X, fY] = f[X, Y] + (X · f)Y for all f, g ∈ C^{∞}(M) and X, Y ∈ Γ(T M). 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 over each other.

Definition 1.1. [Rin63] We denote the left B-module action μ on 𝔤 by μ(a⊗α) := aα. Let ω : 𝔤 ⊗ B ⟶ B (or, alternatively, ω : 𝔤 → Der(B)) denote the 𝔤-module action on B. The pair (B, 𝔤) is a Lie-Rinehart pair, provided the Lie-Rinehart relations (LRa) and (LRb) are satisfied for all a, b ∈ B and x, y ∈ 𝔤:

LRa: ω(ax ⊗ b) = a · ω(x ⊗ b), where · indicates the multiplication on B.

LRb: [x, ay] = a[x, y] + ω(x ⊗ a)y.

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

We will suppose that 𝔤 is finitely generated over B, that is to say, with respect to a generating set {X_{α}}, every element x of 𝔤 can be expressed as the sum b^{α}X_{α} where b^{α} ∈ B. (We use the Einstein summation convention throughout.) The first Lie-Rinehart relation (LRa) states that ω is B-linear in 𝔤. We can use the second Lie-Rinehart relation (LRb) and its skew-symmetric counterpart to show that the bracket must have the form

[b^{α}X_{α}, b^{β}X_{β}] = b^{α}b^{β}[X_{α}, X_{β}] + ω(b^{α}X_{α} ⊗ b^{β})X_{β} − ω(b^{β}X_{β} ⊗ b^{α})X_{α} (1)

for any element b^{α}X_{α} ⊗ b^{β}X_{β} of 𝔤 ⊗ 𝔤. In the rest of the paper, we will replace the second Lie-Rinehart relation (LRb) in 1 with this equivalent, but more symmetric form.

For any Lie algebra 𝔤 and 𝔤-module B, an n-multilinear function f_{n} : 𝔤^{×n} → B is alternating if

f_{n}(x_{1}, ..., x_{i}, x_{i+1}, ..., x_{n}) = −f_{n}(x_{1}, ..., x_{i+1}, x_{i}, ..., x_{n}).

The Chevalley-Eilenberg complex is the set of all alternating multilinear functions Alt_{k}(𝔤, B), graded by n, and equipped with a degree +1 differential δ_{CE} : Alt_{k}^{n}(𝔤, B) → Alt_{k}^{n+1}(𝔤, B) given by

\begin{matrix}δ_{CE}f_{n}(x_{0}, ..., x_{n}) = ∑_{i=0}^{n}(−1)^{i}ω(x_{i} ⊗ f_{n}(x_{0}, ..., x̂_{i}, ..., x_{n}) \\ −∑_{i<j}(−1)^{i+j−1}f_{n}([x_{i}, x_{j}], x_{0}, ..., x̂_{i}, ..., x̂_{j}, ..., x_{σ(n+1)}), \end{matrix}

(2)

where x̂_{k} indicates that x_{k} should be omitted. The image under δ_{CE} of any element b ∈ B ≈ Alt_{k}^{0}(𝔤, B) is defined by setting δ_{CE}b(x) = ω(x ⊗ b). When B is an algebra, the Chevalley-Eilenberg complex is a differential graded commutative algebra. For f_{n} and g_{m} in Alt_{k}(𝔤, B), the product f_{n} ⌣ g_{m} is given by

(f_{n} ⌣ g_{m})(x_{1}, ..., x_{n+m}) = ∑_{unshuffles^{(n,m)}^{σ}} f_{n}(x_{σ(1)}, ..., x_{σ(n)})g_{m}(x_{σ(n+1)}, ..., x_{σ(n+m)}).

An (n, m)−unshuffle is any permutation σ in the symmetric group Σ_{n+m} such that

σ(1) < ⋯ < σ(n)

first σ hand

and σ(n + 1) < ⋯ < σ(n + m),

second σ hand

where σ(j) is the element of the set {1, ..., n + m} moved to the j^{th} position under σ. The differential δ_{CE} acts as a derivation with respect to this multiplication. The cohomology of this complex with respect to δ_{CE} is the Chevalley-Eilenberg cohomology of 𝔤 with coefficients in B [CE48].

Suppose (B, 𝔤) is a Lie-Rinehart pair and the alternating function f_{n} is Bmultilinear, i.e., f_{n}(a_{1}x_{1}, ..., a_{n}x_{n}) = a_{1} ⋯ a_{n}f_{n}(x_{1}, ..., x_{n}). Because the Lie action map ω maps 𝔤 into the derivations of B and as a result of the Lie-Rinehart relations, the image of f_{n} under the Chevalley-Eilenberg differential δ_{CE} is again B-multilinear, despite the fact that the bracket is not B-multilinear. The Rinehart algebra R = Alt_{B}(𝔤, B) with differential δ_{R} = δ_{CE} is a subcomplex of the Chevalley-Eilenberg algebra and the cohomology with respect to δ_{R} is the Rinehart cohomology of 𝔤 with coefficients in B [Rin63].

Summary

First, we define the Lie algebras of subordinate derivation sources, resting coderivations and shared Lie modules (§2). Following the lead of [LM95] and [LS93], we define both homotopy and non-homotopy Lie algebras, Lie algebra modules, Lie-Rinehart pairs and Rinehart cohomology, all in the coalgebra setting (§§3 and 4). We piece together homotopy Lie-Rinehart resolutions for Lie-Rinehart pairs and present conditions under which the homotopy Rinehart algebra for a homotopy Lie-Rinehart resolution is a model for the Rinehart cohomology complex for a Lie-Rinehart pair (§5).

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

2. Subordinate derivation sources, resting coderivations and shared Lie modules

Motivation The following brief and not unduly precise reexamination of the Lie-Rinehart relations and the Rinehart differential provides motivation for why coalgebras are an appropriate setting for both the homotopy and the non-homotopy versions of Lie algebras, Lie modules, Chevalley-Eilenberg cohomology and Rinehart cohomology. In particular, the discussion below highlights why we define subordinate and resting coderivations, as well as shared Lie modules.

The underlying module structure for the tensor coalgebra T^{c}𝔤 on 𝔤 is ⨂ 𝔤, which is the collection {𝔤^{⊗n}}_{n=0}^{∞}.

With malice aforethought, we rename the bracket as l_{2} and extend l_{2} to a map from 𝔤^{⊗n} → 𝔤^{⊗n−1} for any n > 2 by setting (up to sign)

l_{2}(x_{[1 to n]}) =∑_{i=1}^{n−1} ± (1^{⊗i−1} ⊗ l_{2} ⊗ 1^{⊗n−(i+1)})(x_{[1 to n]}),  (3)

where (1^{⊗i−1} ⊗ l_{2} ⊗ 1^{⊗n−(i+1)})(x_{[1 to n]}) =

±x_{[1 to i−1]} ⊗ l_{2}(x_{i} ⊗ x_{i+1}) ⊗ x_{[i+2 to n]}.

The bracket l_{2} is now a coderivation on T^{c}𝔤 ([Lad99] and [GL89]), although it has lost its skew-commutativity. We regain the symmetry of the bracket by passing to an appropriate subcoalgebra ⋀ 𝔤, which we define below. We will denote elements

of this coalgebra by x_{1} ∧ ⋯ ∧x_{n} or x ∧_{[1 to n]}. In this context, the Jacobi identity is the fact that l_{2} ∘ l_{2} = 0. Or, since the set of coderivations on a coalgebra forms a Lie algebra, the Jacobi identity is equivalent to [l_{2}, l_{2}] = 0.

Similarly, we rename the Lie action map ω as m_{2} and extend m_{2} as a coderivation

on ⋀ 𝔤 ⊗ B which is subordinate to l_{2} by setting m_{2}(x ∧_{[1 to n]} ⊗ b) =

l_{2}(x ∧_{[1 to n]}) ⊗ b ± ∑_{(n−1,1)^{σ}}x ∧_{σ[1 to n−1]} ⊗ m_{2}(x_{σ(n)} ⊗ b)  (4)

for any n > 1. The symbol x ∧_{σ[1 to n]} is shorthand for x_{σ(1)} ∧ ⋯ ∧x_{σ(n)}, where σ(i) is the element of the ordered set {1, ..., k} which moves to the i^{th} position under the (n − 1, 1)-unshuffle σ. The condition any Lie action must satisfy becomes m_{2} ∘ m_{2} = 0 or [m_{2}, m_{2}] = 0.

For the Lie-Rinehart pair (B, 𝔤), the Lie algebra is a module over B, but the bracket l_{2} is not B-linear. Rather, the second Lie-Rinehart relation states that l_{2}(ax ∧by) consists of a B-linear piece, abl_{2}(x ∧y), and two other terms, m_{2}(ax ⊗ b)y − m_{2}(by ⊗ a)x (compare with equation 1). Any coderivation which shares this property with a specific subordinate coderivation is called a resting coderivation. In particular, l_{2} rests on m_{2}.

Finally, any alternating function f : 𝔤^{⊗n} → B is a linear function on 𝔤 ^{∧n}. We

extend f_{n} to a function from ⋀ 𝔤 into ⋀ 𝔤 ⊗ B by setting f_{n}(x ∧_{[1 to k+n]}) =

∑_{(k,n)^{σ}} ±x ∧_{σ[1 to k]} ⊗ f_{n}(x ∧_{σ[k+1 to k+n]})  (5)

on 𝔤 ^{∧k+n} and setting f_{n} = 0 on 𝔤 ^{∧i} for i < n. Using equations (3), (4) and (5), the Chevalley-Eilenberg differential (and hence, the Rinehart differential) becomes

δ_{CE}f_{n} = m_{2} ∘ f_{n} ± f_{n} ∘ l_{2},

which looks remarkably like some sort of commutator bracket action, whence the definiton of a shared Lie module given below.

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

1. The graded Lie algebra Coder_{W}^{W} (⋀(sV)) is the set of all coderivations on the free graded commutative algebra ⋀(sV), each of which rests on its own specific subordinate coderivation in the Lie algebra Coder_{W}^{W} (⋀(sV) ⊗ W) (see 2).

2. For a graded commutative algebra W, the Lie algebra Coder_{W}^{W} (⋀(sV) ⊗ W) is the set of all coderivations, each of which is subordinate to a specific resting coderivation and is also a W -derivation source. A W -derivation source m is the extension as a coderivation onto ⋀(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 Hom_{W} (⋀(sV), ⋀(sV) ⊗ W) is the set of all W -linear maps on ⋀(sV) with coefficients in ⋀(sV)⊗W and it admits a shared Lie module structure over both Coder_{W}^{W} (⋀(sV) ⊗ W) and Coder_{W}^{W} (⋀(sV)).

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

Coalgebras and Subcoalgebras

A graded coassociative coalgebra is a pair (C, Δ), where C is a graded module over k together with a 0-degree coassociative comultiplication Δ : C → C ⊗ C. A function f with degree |f| = r is a coderivation on C if Δf = (f ⊗ 1_{C} + 1_{C} ⊗ f)Δ.

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)^{|f||g|}gf for all f and g ∈ Coder(C), where |f| and |g| are the degrees of f and g. For a graded module V, the tensor module ⨂ V = {V^{⊗n}} is a graded coassociative coalgebra. We will denote an element of V^{⊗n} by v_{[1 to n]}. The coalgebra T^{c}V is equipped with the standard comultiplication Δ : T^{c}V → T^{c}V ⊗ T^{c}V given by

Δ(v_{[1 to n]}) = ∑_{j=0}^{n}(v_{[1 to j]}) ⊗ (v_{[j+1 to n]}),

where V^{⊗0} ≈ k. The terms with j = 0 and j = n are 1 ⊗ v_{[1 to n]} and v_{[1 to n]} ⊗ 1, respectively.

Actions of the symmetric group Σ_{∞} on T^{c}V An action ρ_{n} of the symmetric group Σ_{n} on V^{⊗n} is any homomorphism

ρ_{n} : Σ_{n} → End(V^{⊗n}).

An action ρ = {ρ_{n}}_{n=1}^{∞} of {Σ_{n}}_{n=1}^{∞} on T^{c}V is defined in the obvious way. The ρ-invariant subspace of T^{c}V forms a subcoalgebra. The two actions and associated subcoalgebras which play a role in this paper are listed below. The first one is the familiar graded commutative coalgebra ⋀ V. The other one helps place this paper in its historical context.

1. The internal graded action ρ^{∧} recognizes only the internal grading on V^{⊗n} and is given by σ·(v_{[1 to n]}) = �_{id}(σ)v_{σ[1 to n]} for all n, σ ∈ Σ_{n} and v_{[1 to n]} ∈ V^{⊗n}. Since Σ_{n} acts transitively on itself, the ρ^{∧}-invariant subspace ⋀ V in T^{c}V is the internal graded symmetric coalgebra on V and is generated by elements of the form

∑_{σ∈Σ_{n}} �_{id}(σ)v_{σ[1 to n]},

which we will denote by v_{[1 to n]}^{∧}. For all v_{[1 to n]}^{∧} ∈ V^{∧n} and σ ∈ Σ_{n}, we have the commutation relation v_{[1 to n]}^{∧} = �_{id}(σ)v_{σ[1 to n]}^{∧}. The coassociative comultiplication on ⋀ V is given by

Δ(v_{[1 to n]}^{∧}) = ∑_{j=0}^{n} ∑_{(j,n−j)−unshuffle^{ρ}} �_{id}(ρ) (v_{ρ[1 to j]}^{∧}) ⊗ (v_{ρ[j+1 to n]}^{∧}) .

2. The combined graded action ρ ∧ is given by σ · (v_{[1 to n]}) = �_{|| ||}(σ)v_{σ[1 to n]}.

The ρ ∧-invariant subcoalgebra is the combined graded symmetric coalgebra ⋀ V, generated by elements of the form

∑_{σ∈Σ_{n}} �_{|| ||}(σ)v_{σ[1 to n]}.

The commutation relation is v ∧_{[1 to n]} = �_{|| ||}(σ)v ∧_{σ[1 to n]}, and the comultiplication is given by

Δ(v ∧_{[1 to n]}) = ∑_{j=0}^{n} ∑_{(j,n−j)^{ρ}} �_{|| ||}(ρ) (v ∧_{ρ[1 to j]}) ⊗ (v ∧_{ρ[j+1 to n]}) ,

where it is understood that the second sum is over all (j, n − j)-unshuffles. Let sV be the suspension of V, that is to say, (sV)_{p} = V_{p−1} for all p. The

subcoalgebra ⋀ V of the tensor coalgebra T^{c}V is isomorphic to the subcoalgebra ⋀(sV) of T^{c}(sV). This isomorphism replaces the bigrading on ⋀ V with its total grading, allowing it to function more easily as a differential object. The coalgebra ⋀(sV) is graded by internal degree, which will be the preferred setting when looking at graded maps from ⋀(sV) into a differential graded commutative algebra, which are traditionally graded by internal degree.

The coalgebra isomorphism S : ⋀ V → ⋀(sV) is not of homogeneous degree;

each map S^{n} : V ^{∧n} → (sV)^{∧n} increases the internal degree of an element by n. Exchanging a map f with S^{n} will produce a sign of (−1)^{⟨0,n⟩·⟨ed(f),id(f)⟩} = (−1)^{n·id(f)}. The inverse isomorphism of S^{n} shall be denoted by S^{−n}. The map S respects the appropriate actions of Σ_{∞}.

Although we work with maps on T^{c}(sV), the definition of action invariant maps applies to maps on T^{c}V as well. Let W be a graded module.

Definition 2.1. A map of graded modules l̂_{n} : (sV)^{⊗n} → W is ρ_{n}-invariant if, for all sv_{[1 to n]} and σ ∈ Σ_{n}, l̂_{n}(σ · (sv_{[1 to n]})) = l̂_{n}(sv_{[1 to n]}). Similarly, a map l̂ : T^{c}(sV) → W is ρ-invariant if l̂(σ · (sv_{[1 to n]})) = l̂(sv_{[1 to n]}) for all n and σ ∈ Σ_{n}.

If l̂: T^{c}(sV) → W is ρ-invariant, then l̂ restricted to the ρ-invariant subcoalgebra is well-defined because Σ_{n} acts transitively on itself. The maps of greatest interest to us in this paper are either suspended internal graded symmetric (ρ^{∧}-invariant)

or combined graded symmetric (ρ ∧-invariant). The map l̂_{n} is suspended internal graded symmetric if ˆ �_{id}(σ)l̂_{n}(sv_{σ[1 to n]}) = l̂_{n}(sv_{[1 to n]}) and l_{n} is combined graded symmetric if �_{|| ||}(σ)l_{n}(v_{σ[1 to n]}) = l_{n}(v_{[1 to n]}) for all v_{[1 to n]} and σ ∈ Σ_{n}. We do

not lose any information if we restrict the map l̂_{n} to (sV)^{∧n} (or restrict l_{n} to V ^{∧n}) because

l̂_{n}(sv_{[1 to n]}^{∧}) = l̂_{n} ( ∑_{σ∈Σ_{n}} ˆ �_{id}(σ)sv_{σ[1 to n]})
= ∑_{σ∈Σ_{n}} ˆ �_{id}(σ)l̂_{n}(sv_{σ[1 to n]})
= ∑_{σ∈Σ_{n}} l̂_{n}(sv_{[1 to n]})
= n!l̂_{n}(sv_{[1 to n]}).

So whenever we consider collections of maps on T^{c}V with the same invariance, we pass to that invariant subcoalgebra. The isomorphism S implies that any map l_{n} :

V ^{∧n} → V induces a map l̂_{n} : (sV)^{∧n} → (sV) by setting l̂_{n} = (−1)^{n·id(l_{n})}S^{1}l_{n}S^{−n}.

This process is invertible, so the set of all maps l_{n} : V ^{∧n} → V is isomorphic to the set of all maps l̂_{n} : (sV)^{∧n} → sV. Therefore, whatever we define and prove in the suspended internal graded setting has its analog in the combined graded setting, e.g., all Lie algebras and graded commutative algebra defined in the remainder of this section.

The graded Lie algebra Coder(⋀(sV )) Following [Lad99] and [GL89], one can extend any map l̂_{n} : (sV)^{⊗n} → (sV) to a coderivation l̂_{n} : T^{c}(sV) → T^{c}(sV) by setting l̂_{n} = 0 on (sV)^{⊗k} for k < n and, for k ⩾ n, setting

l̂_{n}(sv_{[1 to k]}) = ∑_{i=1}^{k−n+1}(1^{⊗i−1} ⊗ l̂_{n} ⊗ 1^{⊗k−n+1−i})(sv_{[1 to k]}).

Here, (1^{⊗i−1} ⊗ l̂_{n} ⊗ 1^{⊗k−n+1−i})(v_{[1 to k]}) =

ˆ �_{id}(l̂_{n}; sv_{[1 to i−1]})sv_{[1 to i−1]} ⊗ l̂_{n}(sv_{[i to i+n−1]}) ⊗ sv_{[i+n to k]},

where ˆ �_{id}(l̂_{n}; sv_{[1 to i−1]}) is the sign introduced by exchanging l̂_{n} with sv_{[1 to i−1]}. Extending any ρ^{∧}-invariant map l̂_{n} as a coderivation on T^{c}(sV) does not preserve its invariance. However, we can extend (rather elegantly) the restriction of l̂_{n} on (sV)^{∧n} to a coderivation on the ρ^{∧}-invariance subcoalgebra ⋀(sV) by setting

l̂_{n}(sv_{[1 to k]}^{∧}) = ∑_{(n,k−n)^{ρ}} ˆ �_{id}(ρ)l̂_{n}(sv_{ρ[1 to n]}^{∧}) ∧ sv_{ρ[n+1 to k]}^{∧}.  (6)

The reader is left to verify that extending a map l̂_{n} as in equation (6) produces a well-defined map on ⋀(sV). The restriction of the coderivation l̂_{n} on T^{c}(sV) to the ρ^{∧}-invariant subcoalgebra ⋀(sV) is equal to the coderivation in equation 6. The set of all coderivations l̂_{n} on ⋀(sV) generate the Lie algebra Coder(⋀(sV)).

Next, we examine maps m̂_{n} : (sV)^{∧n−1}⊗W → W (the Lie action map m̂_{2} is such a map) and f̂_{n} : (sV)^{∧n} → W (elements of Alt_{k}(𝔤, B) are of this form), where W is a graded commutative algebra. We will denote the degree of any element w ∈ W by |w|. Elements of W have no external degree, at least not with respect to ⋀(sV).

Subordinate coderivations and Coder(⋀(sV ) ⊗ W) We begin with maps m̂_{n} : (sV)^{∧n−1} ⊗ W → W. Let l̂_{n} : (sV)^{∧n} → sV be any map of degree id(l̂_{n}) and let l̂ be any coderivation on ⋀(sV) of degree id(l̂).

Definition 2.2. A map m̂_{n} : (sV)^{∧n−1} ⊗ W → W is an l̂_{n}-subordinate map if the degree of m̂_{n} and l̂_{n} agree. A map m̂ : ⋀(sV)⊗W → ⋀(sV)⊗W is an l̂-subordinate coderivation if id(m̂) = id(l̂) and (Δ ⊗ 1_{W})m̂ = (l̂⊗ 1_{⋀(sV)⊗W} + 1_{⋀(sV)} ⊗ m̂)(Δ ⊗ 1_{W}).

Just as the map l̂_{n} extends to a coderivation on ⋀(sV), so too, the l̂_{n}-subordinate map m̂_{n} may be extended to an l̂_{n}-subordinate coderivation on ⋀(sV) ⊗ W. On (sV)^{∧k} ⊗ W, we set m̂_{n} = 0 for k < n − 1 and we set m̂_{n} = l̂_{n} ⊗ 1_{W} + 1^{∧k−n+1} ⊗ m̂_{n} for all k > n − 1.

The set of all subordinate coderivations Coder(⋀(sV) ⊗ W) forms a graded Lie algebra under the graded commutator bracket. The bracket respects subordination, that is to say, if m̂_{i} and m̂_{j} are l̂_{i} and l̂_{j}-subordinate coderivations respectively, then [m̂_{i}, m̂_{j}] is a coderivation subordinate to [l̂_{i}, l̂_{j}]. (Of course, the suspension map S respects subordinate maps and coderivations.)

The graded commutative algebra Hom(⋀(sV ), ⋀(sV ) ⊗ W) We can extend any map f̂_{n} : (sV)^{∧n} → W to a function from ⋀(sV) into ⋀(sV)⊗W by setting f̂_{n} = 0 for k < n and, for k ⩾ n, setting f̂_{n}(sv_{[1 to k]}^{∧}) =

∑_{(k−n,n)^{σ}} ˆ �_{id}(σ)ˆ �_{id}(f̂_{n}; sv_{σ[1 to k−n]}^{∧})sv_{σ[1 to k−n]}^{∧} ⊗ f̂_{n}(sv_{σ[k−n+1 to k]}^{∧}). (7)

The set of all such maps is Hom(⋀(sV), ⋀(sV) ⊗ W).

Suppose m̂_{p} ∈ Coder(⋀(sV) ⊗ W) is subordinate to l̂_{p} ∈ Coder(⋀(sV)). The map

⟨m̂_{p}, ⟩ : Hom(⋀(sV), ⋀(sV) ⊗ W) → Hom(⋀(sV), ⋀(sV) ⊗ W),

given by setting

⟨m̂_{p}, f̂_{n}⟩ = m̂_{p}f̂_{n} − ˆ �_{id}(m̂_{p}; f̂_{n})f̂_{n}l̂_{p}

on (sV)^{∧n+p−1} and extending ⟨m̂_{p}, f̂_{n}⟩ to all of ⋀(sV) as in equation (7) is welldefined. This map exposes an unusual type of graded Lie module structure on Hom(⋀(sV), ⋀(sV)⊗W). It is, in a sense, a Lie-module over both Coder(⋀(sV)⊗W) and Coder(⋀(sV)) simultaneously in the sense that

⟨[m̂_{i}, m̂_{j}], ⟩ = ⟨m̂_{i}, ⟨m̂_{j}, ⟩⟩ − ˆ �_{id}(m̂_{i}; m̂_{j}) ⟨m̂_{j}, ⟨m̂_{i}, ⟩⟩ .

We will call such a Lie module shared.

When (B, s𝔤) is a Lie-Rinehart pair, the Lie module map m̂_{2} : 𝔤⊗B → B should be a map from s𝔤 into Der(B). With this in mind, we introduce the following notion.

Definition 2.3. An l̂_{n}-subordinate map m̂_{n} is a W -derivation source if for every sv_{[1 to n−1]}^{∧} in (sV)^{∧n−1}, the map m̂_{n}(sv_{[1 to n−1]}^{∧} ⊗ ( )) : W → W is a graded derivation on W.

In other words, a W -derivation source m̂_{n} is a map from (sV)^{∧n−1} into Der(W). In practice, we will work with m̂_{n} as a l̂_{n}-subordinate coderivation on ⋀(sV) ⊗ W, where the characterization of m̂_{n} as a map into Der(W) is somewhat obscured. The subset of all W -derivation sources in Coder(⋀(sV) ⊗ W) forms a graded Lie subalgebra denoted by Coder^{W} (⋀(sV) ⊗ W).

The subset of all coderivations l̂_{n} ∈ Coder(⋀(sV)) which admit a subordinate W -derivation source m̂_{n} forms a graded Lie subalgebra which we will denote by Coder^{W} (⋀(sV)).

The algebra structure on W extends to a graded commutative algebra structure on ⋀(sV) ⊗ W, which provides Hom(⋀(sV), ⋀(sV) ⊗ W) with a cup product. If f̂_{n} and ĝ_{s} are maps in Hom(⋀(sV), ⋀(sV) ⊗ W). Then f̂_{n} ⌣ ĝ_{s} is given by setting (f̂_{n} ⌣ ĝ_{s})(sv_{[1 to n+s]}^{∧}) =

∑_{(n,s)^{ρ}} ˆ �_{id}(ρ)ˆ �_{id}(ĝ_{s}; sv_{ρ[1 to n]}^{∧})f̂_{n}(sv_{ρ[1 to n]}^{∧}) · ĝ_{s}(sv_{ρ[n+1 to n+s]}^{∧})

on (sV)^{∧n+s}, where · represents the the multiplication on W (henceforth, the · will be supressed). The map f̂_{n} ⌣ ĝ_{s} is then extended to all of ⋀(sV) as in equation (7). The map ⟨m̂_{p}, ⟩ acts as a derivation with respect to the cup product, so long as m̂_{p} is a W -derivation source.

W -linear maps and resting coderivations For a Lie-Rinehart pair (B, s𝔤), the Rinehart complex of B-linear alternating maps is a subcomplex of the Chevalley-Eilenberg complex without modifying the differential. This is due to the B-linearity of any f̂_{n} in Alt_{B}(s𝔤, B), the Lie-Rinehart relations, and the fact that the Lie module map m̂_{2} is a B-derivation source.

If the graded module V is a module over the graded commutative algebra W, then the tensor coalgebras T^{c}V and T^{c}(sV) are modules over TW. We will follow our general convention and denote w_{1}v_{1} ⊗⋯⊗w_{n}v_{n} by wv_{[1 to n]} and, in the suspended setting, denote w_{1}sv_{1} ⊗ ⋯ ⊗ w_{n}sv_{n} by wsv_{[1 to n]}. The unraveling map

Û : T^{c}(sV) → W ⊗ T^{c}(sV)
wsv_{[1 to n]} ↦ û([1 to n])w_{[1 to n]}^{·} ⊗ sv_{[1 to n]},

where û([1 to n]) is the sign produced when moving the w_{i}’s past the v_{j}’s. The map Û respects the coalgebra structure of T^{c}(sV).

Definition 2.4. A map f̂_{n} : (sV)^{∧n} → W is W -linear if

f̂_{n}(wsv_{[1 to n]}^{∧}) = û([1 to n])ˆ �_{id}(f̂_{n}; w_{[1 to n]}^{·})w_{[1 to n]}^{·}f̂_{n}(sv_{[1 to n]}^{∧}).

Likewise, a map m̂_{n} : (sV)^{∧n−1} ⊗ W → W is W -linear if m̂_{n}(wsv_{[1 to n−1]}^{∧} ⊗ w) =

û([1 to n − 1])ˆ �_{id}(m̂_{n}; w_{[1 to n−1]}^{·})w_{[1 to n−1]}^{·}m̂_{n}(sv_{[1 to n−1]}^{∧} ⊗ w).

For a map m̂_{n} : (sV)^{∧n−1} ⊗ W → W, we define the sV -extension map m̂_{n}^{e} : (sV)^{∧n} → sV by setting

m̂_{n}^{e}(wsv_{[1 to n]}^{∧}) = ∑_{(n−1,1)^{ρ}} ˆ �_{id}(ρ)m̂_{n}(wsv_{ρ[1 to n−1]} ⊗ w_{ρ(n)})sv_{ρ(n)}.

Definition 2.5. Given m̂_{n} ∈ Coder^{W} (⋀(sV) ⊗ W), an W -linear, W -derivation source which is subordinate to l̂_{n} ∈ Coder^{W} (⋀(sV)), the map l̂_{n} is said to rest on m̂_{n} if l̂_{n}(wsv_{[1 to n]}^{∧}) =

û([1 to n])ˆ �_{id}(l̂_{n}; w_{[1 to n]}^{·})w_{[1 to n]}^{·}l̂_{n}(sv_{[1 to n]}^{∧}) + m̂_{n}^{e}(wsv_{[1 to n]}^{∧}).

The subset of Coder^{W} (⋀(sV)) of all coderivations l̂_{p} which rest on W -linear W - derivation sources m̂_{p} forms a graded Lie subalgebra. This subalgebra is denoted by Coder_{W}^{W} (⋀(sV)). Similarly, the subset of Coder^{W} (⋀(sV) ⊗ W) consisting of all W -linear W -derivation sources m̂_{p} subordinate to maps l̂_{p}, which in turn rest on m̂_{p}, forms a graded Lie subalgebra, denoted by Coder_{W}^{W} (⋀(sV) ⊗ W).

Proposition 2.6. Let f̂_{n} : (sV)^{∧n} → W be a W -linear map and suppose the map m̂_{p} : (sV)^{∧p−1} ⊗ W → W is a W -linear, W -derivation source subordinate to the map l̂_{p} : (sV)^{∧p} → V, which in turn rests on m̂_{p}. Then ⟨m̂_{p}, f̂_{n}⟩ = m̂_{p}f̂_{n} −

ˆ �_{id}(m̂_{p}; f̂_{n})f̂_{n}l̂_{p} is W -linear.

Proof of Proposition 2.6. In this proof and in the proofs to follow, we include only those sign arguments which are instructive.

First we compute m̂_{p}f̂_{n}(wsv_{[1 to p+n−1]}). We break up the result into two sums. The first one, (Lin m̂_{p}f̂_{n}), consists of those terms in which all the w_{i}’s can and have been factored out. The remaining terms form the second sum (Nonlin m̂_{p}f̂_{n}). The first sum is over all (p − 1, n)-unshuffles σ:

(Lin m̂_{p}f̂_{n}) = ∑_{σ} ±w_{[1 to p+n−1]}^{·}m̂_{p}(sv_{σ[1 to p−1]}^{∧} ⊗ f̂_{n}(sv_{σ[p to n+p−1]}^{∧}).

The second sum is actually a double sum over all (p − 1, n)-unshuffles σ and all (1, n − 1)-unshuffles γ acting on the second σ-hand:

(Nonlin m̂_{p}f̂_{n}) = ∑_{σ} ∑_{γ}

±m̂_{p}(sv_{σ[1 to p−1]}^{∧} ⊗ w_{σ[1 to p−1]}^{·} · w_{γσ[p to p+n−1]}^{·}) · f̂_{n}(sv_{γσ[p to n+p−1]}^{∧}).

We compute f̂_{n}l̂_{p}(wsv_{[1 to p+n−1]}^{∧}) by again breaking it up into two sums: (Lin f̂_{n}l̂_{p}), is a sum over all (p, n − 1)-unshuffles ρ, i.e.,

(Lin f̂_{n}l̂_{p}) = ∑_{ρ} ±w_{[1 to p+n−1]}^{·}f̂_{n}(l̂_{p}(sv_{σ[1 to p]}^{∧}) ∧ sv_{σ[p+1 to n+p−1]}^{∧})

and (Nonlin f̂_{n}l̂_{p}) is again a double sum over all (p, n − 1)-unshuffles ρ and all (n − 1, 1)-unshuffles τ acting on the first ρ-hand. In other words,

(Nonlin f̂_{n}l̂_{p}) = ∑_{ρ} ∑_{τ}

±m̂_{p}(sv_{τρ[1 to p−1]}^{∧} ⊗ w_{τρ[1 to p]}^{·} · w_{ρ[p+1 to p+n−1]}^{·})f̂_{n}(sv_{τρ(p)} ∧ sv_{ρ[p+1 to n+p−1]}^{∧}).

Recognizing that ||m̂_{p}f̂_{n}|| = ||f̂_{n}l̂_{p}||, it follows that

[(Lin m̂_{p}f̂_{n}) − ˆ�(m̂_{p}; f̂_{n})(Lin f̂_{n}l̂_{p})] = ±w_{[1 to p+n−1]}^{·} ⟨m̂_{p}, f̂_{n}⟩ (sv_{[1 to p+n−1]}^{∧}).

Therefore, once we show that [(Nonlin m̂_{p}f̂_{n}) − ˆ�(m̂_{p}; f̂_{n})(Nonlin f̂_{n}l̂_{p})] = 0, the proof will be complete.

For every pair consisting of a (p − 1, n)-unshuffle σ and a (1, n − 1)-unshuffle γ acting on the second σ hand, there is a unique pair consisting of a (p, n−1)-unshuffle ρ and a (p − 1, 1)-unshuffle τ acting on the first ρ hand which produces the exact same sequence of components, that is to say,

σ[1 to p − 1] = τρ[1 to p − 1],
γσ(p) = τρ(p) and
γσ[p + 1 to p + n − 1] = ρ[p + 1 to p + n − 1].

Why should this be so? A simple counting argument provides the answer. Either combination produces an (p − 1, 1, n − 1)-unshuffle. There are \binom{p+n−2}{p−1} distinct (p − 1, 1, n − 1)-unshuffles which send j ∈ {1, …, p + n − 1} to the p^{th} position because once the p^{th} position is determined, the remaining p+n−2 terms must be unshuffled into a hand of length p − 1 and a hand of length n − 1. How many distinct pairs (σ, γ) send j to the p^{th} position? This question is equivalent to asking how many distinct (p − 1, n)-unshuffles σ send j to the second hand since, once j is in the second hand, there is a unique (1, n − 1) unshuffle γ which will send j to the p^{th} position. The number of (p − 1, n)-unshuffles is \binom{p+n−1}{n} and \binom{p+n−2}{n−1} of them have j in the second hand. Since \binom{p+n−2}{n−1} = \binom{p+n−2}{p−1}, we know there is a one-to-one correspondence between (p − 1, 1, n − 1)-unshuffles and pair (σ, γ) which send j to the p^{th} position. A similar argument shows that the number of (p − 1, 1, n − 1)- unshuffles which send j to the p^{th} position equals the number of (ρ, τ) combinations which do the same. We can show that the (σ, γ) term of (Nonlinm̂_{p}f̂_{n}) and the corresponding (ρ, τ) term of (Nonlinf̂_{n}l̂_{p}) have opposite signs, so they cancel. □

Naturally, the graded commutative algebra Hom_{W} (⋀(sV), ⋀(sV) ⊗ W) is a shared Lie module over the Lie algebras Coder_{W}^{W} (⋀(sV), ⋀(sV) ⊗ W) and Coder_{W}^{W} (⋀(sV)).

3. Lie algebras, their modules and cohomology in the coalgebra setting

Lie algebras and their modules The most familiar definition of a Lie algebra 𝔤 is as an ungraded k-vector space (or, more generally, a module over a k-algebra A) equipped with a skew-symmetric bracket l_{2} = [ , ] that maps 𝔤 ⊗ 𝔤 into 𝔤 and that satisfies the Jacobi identity, i.e., the equality

l_{2}(x, l_{2}(y, z)) = l_{2}(l_{2}(x, y), z) + l_{2}(y, l_{2}(x, z))

holds for all x, y, z ∈ 𝔤. In the context of graded coassociative coalgebras, we can consider 𝔤 a graded vector space where all elements of 𝔤 have combined degree ⟨1, 0⟩. The bracket l_{2} becomes a degree ⟨−1, 0⟩ coderivation on ⋀ 𝔤. No information is lost in the process because

l ∧_{2}(x ∧y) = l_{2} (x ⊗ y + (−1)^{⟨0,1⟩·⟨0,1⟩}y ⊗ x)
= l_{2}(x ⊗ y) − l_{2}(y ⊗ x)
= l_{2}(x ⊗ y) + l_{2}(x ⊗ y)
= 2l_{2}(x ⊗ y).

But we can also define the Lie algebra on ⋀(s𝔤), where the bracket l̂_{2} becomes an internal graded symmetric coderivation of degree -1 on ⋀(s𝔤) with respect to the internal grading. Again, no essential information is lost. We can therefore realize a Lie algebra in three equivalent ways:

1. A Lie algebra is a vector space 𝔤 over k with skew symmetric bracket l_{2} which satisfies the Jacobi identity (the original definition).

2. A Lie algebra is the subcoalgebra ⋀ 𝔤, together with a combined degree ⟨−1, 0⟩ coderivation l_{2} satisfying l_{2} ∘ l_{2} = \frac{1}{2}[l_{2}, l_{2}] = 0, which is equivalent to the Jacobi identity. Since the internal degree is zero, it can be ignored, so l_{2} can be considered a degree −1 coderivation, i.e., a differential on ⋀ 𝔤.

3. A Lie algebra is a subcoalgebra ⋀(s𝔤) together with a suspended internal degree -1 differential l̂_{2}. This is our preferred setting.

The Jacobi identity is a result of the more general fact, true for all coderivations l̂ with odd degree, that [l̂, l̂] = 2l̂∘l̂ and [l̂, l̂] = −[l̂, l̂] = 0. In other words, any degree −1 coderivation l̂ is a differential on the coalgebra ⋀(s𝔤) and if the differential l̂ has external degree −1, then it is the extension of a map l̂ : s𝔤 ∧ s𝔤 → s𝔤 as a coderivation, i.e., l̂ is a Lie bracket.

Suppose B is an ungraded Lie module over 𝔤, i.e., there is a map m̂_{2} : s𝔤⊗B → B such that m̂_{2}(l̂_{2}(sx ∧ sy) ⊗ b) − m̂_{2}(sx ⊗ m̂_{2}(sy ⊗ b)) + m̂_{2}(sy ⊗ m̂_{2}(sx ⊗ b)) = 0, for all x, y ∈ 𝔤 and b ∈ B. The map m̂_{2} is l̂_{2}-subordinate, and we can extend m̂_{2} to an l̂_{2}-subordinate coderivation on ⋀(s𝔤) ⊗ B. The equation above can rewritten as m̂_{2} ∘ m̂_{2} = \frac{1}{2}[m̂_{2}, m̂_{2}] = 0 and since the degree of m̂_{2} is -1, it is a differential on ⋀(s𝔤) ⊗ B. In the coalgebra setting, then, a 𝔤-module structure is any coderivation m̂_{2} on ⋀(s𝔤) ⊗ B which is a differential and which is subordinate to the bracket l̂_{2}.

Chevalley-Eilenberg Cohomology Definition 3.1. A Chevalley-Eilenberg pair (B, 𝔤) is a Lie algebra 𝔤 and Lie module B over 𝔤.

For any Chevalley-Eilenberg pair (B, 𝔤), an n-multilinear function f_{n} : 𝔤^{×n} → B is alternating if

f_{n}(x_{σ(1)}, ..., x_{σ(n)}) = (−1)^{σ}f_{n}(x_{1}, ..., x_{n})

for all σ ∈ Σ_{n}, where (−1)^{σ} is the sign of the permutation. The algebra of all alternating multilinear functions Alt_{k}(𝔤, B) is graded by n and admits a degree +1 differential δ_{CE} : Alt_{k}^{n}(𝔤, B) → Alt_{k}^{n+1}(𝔤, B) given by equation (2). The cohomology of this complex with respect to δ_{CE} is the Chevalley-Eilenberg cohomology of 𝔤 with coefficients in B.

In the coalgebra setting, any alternating n-multilinear function f_{n} can be realized uniquely as a linear function f̂_{n} : (s𝔤)^{∧n} → B, which then can be extended as a coderivation f̂_{n} : ⋀(s𝔤) → ⋀(s𝔤) ⊗ B. We extend b ∈ B as a coderivation b : ⋀(s𝔤) → ⋀(s𝔤) ⊗ B by setting b(sx_{[1 to k]}^{∧}) = sx_{[1 to k]}^{∧} ⊗ b. Any function F̂ ∈

Hom(⋀(s𝔤), ⋀(s𝔤) ⊗ B) is the sum ∑_{n=0}^{∞} f̂_{n}, where f̂_{n} : (s𝔤)^{∧n} → B has degree n. Hence, Hom(⋀(s𝔤), ⋀(s𝔤) ⊗ B) is graded by ℕ and is isomorphic to Alt_{k}(𝔤, B). The differential δ_{CE} on Hom(⋀(s𝔤), ⋀(s𝔤) ⊗ B) has the elegant form ⟨m̂_{2}, ⟩. This formula works even for b ∈ B, since δ_{CE}b = ⟨m̂_{2}, b⟩ = m̂_{2}b + bl̂_{2} and l̂_{2} = 0 on s𝔤. Since [m̂_{2}, m̂_{2}] = −[m̂_{2}, m̂_{2}] = 0, it is now simple to show that

δ_{CE} ∘ δ_{CE} = ⟨m̂_{2} ⟨m̂_{2}, ⟩⟩ = ⟨\frac{1}{2}[m̂_{2}, m̂_{2}], ⟩ = 0.

If B is an algebra, then Hom(⋀(s𝔤), ⋀(s𝔤) ⊗ B) is also an algebra. The differential δ_{CE} acts as 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 definition of a Lie-Rinehart pair below has been modified from definition 1.1 to reflect the coalgebra setting.

Definition 3.2. [Rin63] Let μ : B ⊗ (s𝔤) → (s𝔤) be a left B-module action on s𝔤 denoted by μ(a⊗α) := aα. Let m̂_{2} : (s𝔤)⊗B → B be an l̂_{2}-subordinate B-derivation source, i.e., an s𝔤-module action on B. The pair (B, s𝔤) is a Lie-Rinehart pair, provided the Lie-Rinehart relations (LRa) and (LRb) are satisfied. LRa: m̂_{2} is B-linear. LRb: l̂_{2} rests on m̂_{2} (definition 2.5).

Suppose f̂_{n} is in Hom_{B}(⋀(s𝔤), ⋀(s𝔤) ⊗ B). ¿From proposition 2.6, we know the image of f̂_{n} under the Chevalley-Eilenberg differential δ_{CE} = ⟨m̂_{2}, ⟩ is again Blinear. The Rinehart algebra R = Hom_{B}(⋀(s𝔤), ⋀(s𝔤) ⊗ B) with differential δ_{R} = ⟨m̂_{2}, ⟩ is a subcomplex of the Chevalley-Eilenberg complex and the cohomology of R with respect to δ_{R} is the Rinehart cohomology of 𝔤 with coefficients in B.

4. Strongly homotopy Lie algebras, their modules and cohomology

Strongly homotopy Lie algebras (shLie algebras) first appeared implicitly in [Sul77] and explicitly in [SS] in the context of deformation theory. A concise introduction to shLie algebras is found in [LM95]. The initial definitions below are lifted directly from [LM95] and then modified to fit the language introduced in §2.

Definition 4.1. [LM95] and [LS93] An L(m)-structure on a graded module L is a system of linear maps {l_{k} : 1 ⩽ k ⩽ m ⩽ ∞, k ≠ ∞}, where l_{k} : ⨂^{k}L → L has combined degree ⟨1 − k, k − 2⟩ and each map is combined graded symmetric in the sense that

l_{k}(v_{σ[1 to k]}) = �_{|| ||}(σ)l_{k}(v_{[1 to k]})

for all σ ∈ Σ_{k}. Moreover, the following generalized form of the Jacobi identity, the n^{th} Jacobi identity map, is satisfied for n ⩽ m:

J_{ID}_{n}(v_{[1 to n]}) = ∑_{i,j⩾1^{i+j=n+1}} ∑_{σ} �_{|| ||}(σ)(−1)^{i(j−1)}l_{j}(l_{i}(v_{σ[1 to i]}) ⊗ v_{σ[i+1 to n]}) = 0,

where the second summation runs over all (i, j − 1)-unshuffles. The graded vector space L is a strongly homotopy Lie algebra (or shLie algebra) if L admits an L(∞)- structure.

We can rewrite this definition in terms of maps on ⋀ L because the maps l_{k} are combined graded symmetric. The Jacobi identity maps J_{ID}_{n} should equal 0 on ⋀ L, but it is not immediately evident that the maps J_{ID}_{n} can be extended as coderivations on ⋀ L. Fortunately, we can rewrite the Jacobi identity maps using the bracket on Coder(⋀ L). Consider the bracket of l_{j} with l_{i} with respect to the combined grading on ⋀ L:

[l_{j}, l_{i}] = l_{j}l_{i} − (−1)^{⟨1−j,j−2⟩·⟨1−i,i−2⟩}l_{i}l_{j} = l_{j}l_{i} + (−1)^{j+i}l_{i}l_{j}.

Multiplying both sides by the desired sign for l_{j}l_{i} in the Jacobi identity map J_{ID}_{n}, we find that

(−1)^{i(j−1)}[l_{j}, l_{i}] = (−1)^{i(j−1)}l_{j}l_{i} + (−1)^{i(j−1)+j+i}l_{i}l_{j}
= (−1)^{i(j−1)}l_{j}l_{i} + (−1)^{j(i−1)}l_{i}l_{j}.

We therefore reformulate the definition of an L(m)-algebra and an shLie algebra:

Definition 4.2. An L(m)-structure on a graded module L consists of a system of coderivations {l_{k} : ⋀ L → ⋀ L : 1 ⩽ k ⩽ m ⩽ ∞, k ≠ ∞}, where each l_{k} has combined degree ⟨1 − k, k − 2⟩. Moreover, the following generalized form of the Jacobi identity is satisfied for n ⩽ m:

J_{ID}_{n} = \frac{1}{2} ∑_{i,j⩾1^{i+j=n+1}}(−1)^{i(j−1)}[l_{j}, l_{i}] = 0  (8)

on L ^{∧n}. Again, if L admits an L(∞)-structure, then L is a strongly homotopy Lie algebra (an shLie algebra).

Since the Jacobi identity maps are sums of coderivations, it follows that J_{ID}_{n}

can be extended to a coderivation on ⋀ L. Furthermore, since J_{ID}_{n} = 0 on L ^{∧n}, we have proven the following proposition.

Proposition 4.3. J_{ID}_{n} = 0 on ⋀ L.

In both [LS93] and [LM95], this result was not achieved without first passing to the suspended internal graded symmetric setting. Lada, Markl and Stasheff defined strongly homotopy Lie algebras on the tensor coalgebra T^{c}L rather than on ⋀ L, whose existence had not been recognized. The maps J_{ID}_{n} cannot be extended as coderivations on the tensor coalgebra while preserving the desired symmetry. So instead, they suspended the maps l_{k}, changing the symmetry of the maps from combined graded symmetric to suspended internal graded symmetric. They were then in familiar territory, where the maps \begin{matrix} \\ J_{ID} & n \end{matrix} could be extended as coderivations on ⋀(sL), even though they did not make explicit use of the bracket on Coder(⋀(sL)).

Modules over a strongly homotopy Lie algebra Definition 4.4. [LM95] Let L = (L, l_{i}) be an L(p)-algebra (0 < p < ∞) and let M be a differential graded module with differential m_{1}. Then a left L(k)-module structure over L on M (for k ⩽ p) is a collection {m_{n} : 1 ⩽ n ⩽ k, n ≠ ∞} of l_{n}-subordinate coderivations m_{n} on ⋀ L ⊗ M such that the n^{th} action identity map

ACT_{ID}_{n} = ∑_{i+j=n+1}(−1)^{i(j−1)}m_{j}m_{i}

= \frac{1}{2} ∑_{i,j⩾1^{i+j=n+1}}(−1)^{i(j−1)}[m_{j}, m_{i}]

= 0

on ⋀ L ⊗ M. The differential graded module M is a strongly homotopy Lie module over L (or an L-shLie module) if L admits an L(∞)-structure and M is a module with respect to that L(∞)-structure.

Definition 4.4 implies that the differential m_{1} on M must be l_{1}-subordinate. It is simple to verify that m_{1} is a differential on ⋀ L ⊗ M.

When M is a differential graded commutative algebra, we require that the maps

m_{i} be M-derivation sources, that is to say, for every v ∧_{[1 to i−1]} ∈ L ^{∧i−1}, the map m_{i}(v_{[1 to i−1]} ⊗ ( )) : M → M is in Der(M).

Since Coder(⋀ L) ≈ Coder(⋀(sL)), it follows that if (L, l_{i}) is a strongly homotopy Lie algebra, then (sL, l̂_{i}), where l̂_{i} is the coderivation on ⋀(sL) induced by the isomorphism S : ⋀ L → ⋀(sL), is also an shLie algebra. The suspended internal degree of each map l̂_{i} is −1, so the sum of the L(∞)-structure maps l̂_{i} is a differential on ⋀(sL), first shown in [LS93]. This is not the case for (L, l_{i}) because each l_{i} is bigraded. In order for the sum of the l_{i}’s to be a differential, one would have to make sense of l_{i} as a degree −1 map. Passing to the suspended internal graded symmetric setting is a convenient way to accomplish this because degree of l̂_{i} is ed(l_{i}) + id(l_{i}) = −1, which is the total degree of l_{i}. For (sL, l̂_{i}), the n^{th} Jacobi identity map is

Ĵ_{ID}_{n} = \frac{1}{2} ∑_{i,j⩾1^{i+j=n+1}}[l̂_{j}, l̂_{i}],  (9)

which is 0 on ⋀(sL).

Proposition 4.5. [LS93] If (sL, l̂_{i}) is an shLie algebra, the map D_{sL} = ∑_{i=1}^{∞} l̂_{i} is a differential on ⋀(sL), i.e. it is a map of degree -1 such that D_{sL} ∘ D_{sL} = 0.

Proof of Proposition 4.5. A proof can be found in [LS93]. Here the proof is essentially the same but takes advantage of the bracket of coderivations:

D_{sL} ∘ D_{sL} = (∑_{j=1}^{∞} l̂_{j}) ∘ (∑_{i=1}^{∞} l̂_{i}) = ∑_{j=1}^{∞} ∑_{i=1}^{∞} l̂_{j}l̂_{i} = ∑_{n=1}^{∞} ∑_{i,j⩾1^{i+j=n+1}} l̂_{j}l̂_{i}.

Using equation (9) in definition 4.2, we see that

∑_{n=1}^{∞} ∑_{i,j⩾1^{i+j=n+1}} l̂_{j}l̂_{i} = \frac{1}{2} ∑_{n=1}^{∞} ∑_{i,j⩾1^{i+j=n+1}}[l̂_{j}, l̂_{i}] = \frac{1}{2} ∑_{n=1}^{∞} Ĵ_{ID}_{n} = 0.

Note also that D_{sL} is a degree −1 coderivation, so D_{sL} ∘ D_{sL} = \frac{1}{2}[D_{sL}, D_{sL}] and the graded skew-commutativity of the bracket ensures that [D_{sL}, D_{sL}] = 0. □

Suppose (M, m_{i}) is an shLie module over (L, l_{i}). In the suspended internal graded symmetric setting, the maps m̂_{i} : ⋀(sL) ⊗ M → ⋀(sL) ⊗ M form an shLie module structure on M over (sL, l̂_{i}). Again, the degree of m̂_{i} is −1, the total degree of m_{i}, and the action identity map has the form

\begin{matrix} \\ ACT & ID \end{matrix}_{n} = \frac{1}{2} ∑_{i,j⩾1^{i+j=n+1}}[m̂_{j}, m̂_{i}],  (10)

which is zero on ⋀(sL) ⊗ M.

Proposition 4.6. The map D_{M} = ∑_{i=1}^{∞} m̂_{i} is a differential on ⋀(sL) ⊗ M, i.e., D_{M} ∘ D_{M} = 0.

Proof of Proposition 4.6. Isomorphic to the proof of proposition 4.5. Notice that D_{M} is D_{sL}-subordinate and that D_{M} ∘ D_{M} = \frac{1}{2}[D_{M}, D_{M}] = 0. □

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

Homotopy Chevalley-Eilenberg cohomology Definition 4.7. A homotopy Chevalley-Eilenberg pair (M, L) consists of an shLie algebra L and an L-shLie module M.

A multi-linear alternating function F_{n} : L^{×n} → M can be seen as a linear function F̂_{n} : (sL)^{∧n} → M. So the homotopy Chevalley-Eilenberg algebra Alt_{k}(L, M) is isomorphic to

Hom_{k}(⋀(sL), ⋀(sL) ⊗ M).

Any homogeneous map F̂ splits into a sum ∑_{n=0}^{∞} F̂_{n}, where F̂_{n} : (sL)^{∧n} → M. The map F̂_{n} is extended as a coderivation on ⋀(sL)⊗M. Any function F̂_{0} is an element a ∈ M. The degree of a : (sL)^{∧0} → M as a map is the negative of its degree as an element of M. The map a is extended as a coderivation on ⋀(sL) ⊗ M by setting

a(sv_{[1 to k]}^{∧}) = ˆ �_{id}(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 complex Hom_{k}(⋀(sL), ⋀(sL) ⊗ M) is ℤ-graded by suspended internal degree. For any F̂ ∈ Hom_{k}(⋀(sL), ⋀(sL) ⊗ M) of degree îd(F̂), we define

δ_{hCE}F̂ = D_{M} F̂ − ˆ�_{id}(F̂)F̂D_{sL} = ⟨D_{M}, F̂⟩ ,

which has degree îd(F̂) + 1.

Proposition 4.8. δ_{hCE} ∘ δ_{hCE} = 0.

Proof of Proposition 4.8.

δ_{hCE} ∘ δ_{hCE} = ⟨D_{M}, ⟨D_{M}, ⟩⟩
= ⟨\frac{1}{2}[D_{M}, D_{M}], ⟩
= 0. □

The cohomology with respect to δ_{hCE} is the homotopy Chevalley-Eilenberg cohomology of sL with coefficients in M.

When M is a differential graded commutative algebra, the homotopy Chevalley-Eilenberg complex Hom_{k}(⋀(sL), ⋀(sL) ⊗ M) is a differential graded commutative algebra. Suppose F̂ = ∑_{i=0}^{∞} F̂_{i} and Ĝ = ∑_{j=0}^{∞} Ĝ_{j} are both elements of the complex, where F̂_{i} : ⋀^{i}(sL) → M and Ĝ_{j} : ⋀^{j}(sL) → M. The product F̂ ⌣ Ĝ = ∑_{n=0}^{∞}(F̂ ⌣ Ĝ)_{n} is given by (F̂ ⌣ Ĝ)_{n} = ∑_{i+j=n} F̂_{i} ⌣ Ĝ_{j}. The differential δ_{hCE} acts as a derivation with respect to this multiplication.

Homotopy Lie-Rinehart pairs and homotopy Rinehart cohomology The homotopy Rinehart complex is a straightforward generalization of the Rinehart complex in the ungraded setting. Since the Rinehart complex is defined only for Lie-Rinehart pairs (B, 𝔤), we must define what constitutes a homotopy Lie-Rinehart pair (M, sL). The subset Alt_{M}(sL, M) of Alt_{k}(sL, M) consisting of all M-linear alternating functions is isomorphic to Hom_{M}(⋀(sL), ⋀(sL) ⊗ M).

Definition 4.9. A homotopy Lie-Rinehart pair (M, sL) consists of a differential graded commutative algebra (M, m̂_{i}) which is an shLie module over the shLie algebra (sL, l̂_{i}), which in turn is an M-module. Moreover, the following two homotopy Lie-Rinehart relations must be satisfied for all i ⩾ 1:

(hLRa_{i}): The shLie module structure map m̂_{i} is M-linear.

(hLRb_{i}): The shLie structure map l̂_{i} rests on m̂_{i} (definition 2.5).

An L(p)-Lie-Rinehart pair ((M, m̂_{i}), (sL, l̂_{i})) has maps m̂_{i} and l̂_{i} which satisfy (hLRa_{i}) and (hLRb_{i}) for 1 ⩽ i ⩽ p.

Proposition 4.10. If F̂ is M-linear, then so is δ_{hCE}F̂.

Proof of Proposition 4.10. The image of F̂ under δ_{hCE} is

⟨D_{M}, F̂⟩ = ∑_{i=1}^{∞} ⟨m̂_{i}, F̂⟩ .

The proof follows as a consequence of proposition 2.6. □

We conclude that the subset of all M-linear functions in Hom_{k}(⋀(sL), ⋀(sL) ⊗ M) forms a subcomplex R = Hom_{M}(⋀(sL), ⋀(sL) ⊗ M) with differential δ_{R} = ⟨D_{M}, ⟩. The differential, together with the cup product, provides the homotopy Rinehart complex R with the structure of a differential graded commutative algebra. The cohomology of R with respect to δ_{R} is the homotopy Rinehart cohomology of sL with coefficients 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 those component in order to construct a model for a complex with the same basic algebraic structure and precisely the same (co)homology as the original. (It is not always necessary to replace all of the components, see [Sta92], for example.) When building a model for the Rinehart cohomology for a Lie-Rinehart pair (B, s𝔤) over a k-algebra A, we replace the pair with a pair of 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 resolutions Let (B, s𝔤) be a Lie-Rinehart pair over a k-algebra A with module structure maps μ̂ : B⊗s𝔤 → s𝔤 and m̂_{2} : s𝔤⊗B → B. Let l̂_{2} denote the bracket on s𝔤 and π_{B} denote the multiplication on B. (Henceforth, only maps associated with the Lie-Rinehart pair will have hats.) The basic ingredients of a homotopy Lie-Rinehart resolution for a Lie-Rinehart pair (B, s𝔤) are (sL, l_{i}) and (M, m_{i}, π_{M}) where H_{l_{1}}(sL) = s𝔤 and H_{m_{1}}(M) = B.

Definition 5.1. A homotopy Lie-Rinehart resolution of a Lie-Rinehart pair (B, s𝔤) over an algebra A is a homotopy Lie-Rinehart pair (M, sL) over A, such that

(a) the shLie algebra (sL, l_{i}), seen as the coalgebra ⋀(sL), resolves ⋀(s𝔤), i.e.,

H_{l_{1}}(⋀(sL)) = ⋀(s𝔤),  (11)

where (following the physicists’ notation) H_{l_{1}} denotes the homology with respect to the differential l_{1},

(b) the differential graded commutative algebra (M, m_{i}, π_{M}) satisfies

H_{m_{1}}(⋀(sL) ⊗ M) = ⋀(s𝔤) ⊗ B,  (12)

(c) the dgca (M, m_{i}, π_{M}) also satisfies

H_{m_{1}}(M ⊗ M) = B ⊗ B.  (13)

Furthermore, the following conditions hold:

i. H_{l_{1}}(l_{2}) = l̂_{2} on ⋀(s𝔤).

ii. H_{m_{1}}(m_{2}) = m̂_{2} on ⋀(s𝔤) ⊗ B. iii. H_{m_{1}}(π_{M}) = π_{B} on B ⊗ B.

The conditions above are quite reasonable, but satisfying them is far from automatic. Consider any homotopy Lie-Rinehart pair (M, sL). Since m_{1} is a derivation on M, the multiplication on M is a chain map which induces a well-defined map H_{m_{1}}(π_{M}) from H_{m_{1}}(M ⊗ M) into H_{m_{1}}(M). Both ⋀(sL) and ⋀(sL) ⊗ M are complexes with differentials l_{1} and m_{1}, respectively. Since the maps J_{ID}_{2} and ACT_{ID}_{2} are both zero, it follows that both l_{2} and m_{2} are graded morphisms of complexes and as such, induce well-defined maps H_{l_{1}}(l_{2}) and H_{m_{1}}(m_{2}) on their respective homologies.

In the proposition below, we state conditions on the homotopy Lie-Rinehart pair which guarantees it satisfies the conditions in definition 5.1.

Proposition 5.2. Let (M, sL) be a homotopy Lie-Rinehart pair. Suppose the differential graded algebra (M, m_{i}, π_{M}) is a projective resolution of B over A which respects the algebra structure on B, i.e., condition iii of definition 5.1 is satisfied. Suppose (sL, l_{i}) is a projective resolution s𝔤. Furthermore,

(a) H_{l_{1}}(l_{2}) = l̂_{2} on s𝔤 ∧ s𝔤 and

(b) H_{m_{1}}(m_{2}) = m̂_{2} on s𝔤 ⊗ B. Then (M, sL) is a homotopy Lie-Rinehart resolution of (B, s𝔤).

SDR-Data For the proof of proposition 5.2, we will use strong deformation retract data (SDRdata). In general, SDR-data for two differential graded modules (M, d_{M}) and (N, d_{N}) over a commutative ring R with unity consist of two degree 0 chain maps λ : M → N and p : N → M and a homotopy h : N → N such that

pλ = 1_{M}  and
λp = 1_{N} + d_{N}h + hd_{N},

which guarantee an isomorphism of homology (or cohomology), that is to say, H_{d_{M}} (M) ≈ H_{d_{N}} (N). SDR-data is succinctly described by

((M, d_{M}) [figure: λ; p] (N, d_{N}), h) .

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

Proof of Proposition 5.2. For the deleted projective resolution M of B, the nondeleted projective resolution M^{+} : M →^{p_{M}} B → 0 is acyclic, so there exists a contracting homotopy s_{i} : (M^{+})_{i} → (M^{+})_{i+1} where s_{−1} : B → M_{0} such that 1_{M^{+}} = m_{1}^{+}s + sm_{1}^{+} [Wei94]. Setting λ_{M} = s_{−1} and −(h_{M})_{i} = s_{i} for i ⩾ 0, we produce SDR-data

((B, 0) [figure: λ_{M}; p_{M}] (M, m_{1}), h_{M}) .

By construction, p_{M}λ_{M} = 1_{B}. On M_{i} for i > 0, the map λ_{M}p_{M} = 0 and 1_{M_{i}} = −m_{1}h_{M} −h_{M}m_{1}. For i = 0, the identity map 1_{M_{0}} = −h_{M}m_{1} +λ_{M}p_{M} and m_{1}h_{M} = 0. Therefore λ_{M}p_{M} = 1_{M} + m_{1}h_{M} + h_{M}m_{1}. Likewise, we can construct SDR-data

((s𝔤, 0) [figure: λ_{sL}; p_{sL}] (sL, l_{1}), h_{sL})

for s𝔤 and sL.

We will now repeatedly “tensor” SDR-data to produce new SDR-data, using the Gugenheim tensor trick ([GLS91]) for defining the homotopy h. We illustrate this technique by showing that

((B ⊗ B, 0) [figure: λ_{M} ⊗ λ_{M}; p_{M} ⊗ p_{M}] (M ⊗ M, m_{1}), h_{M⊗M}) ,

is SDR-data, where setting h_{M⊗M} = 1_{M} ⊗ h_{M} + h_{M} ⊗ λ_{M}p_{M} was first proposed by Gugenheim and Lambe in [GL89]. Clearly, (p_{M} ⊗p_{M})(λ_{M} ⊗λ_{M}) = 1_{B⊗B} and it is a simple exercise to show that 1_{M⊗M} +m_{1}h_{M⊗M} +h_{M⊗M}m_{1} = (λ_{M} ⊗λ_{M})(p_{M} ⊗p_{M}). We have verified that equation (13) and condition (iii) in definition 5.1 hold.

The tensor trick can be extended to show that

((T^{c}(s𝔤), 0) [figure: Tλ_{sL}; Tp_{sL}] (T^{c}(sL), l_{1}), Th_{sl})

is SDR-data, where Tλ_{sL} = λ_{sL}^{⊗i} on (s𝔤)^{⊗i}, where Tp_{sL} = p_{M}^{⊗i} on (sL)^{⊗i} and where Th_{sL} = ∑_{i=1}^{∞} T_{i}h_{sL} with T_{i}h_{sL} = ∑_{j=1}^{i} 1^{⊗j−1} ⊗ h_{sL} ⊗ (λ_{sL}p_{sL})^{⊗i−j} on (sL)^{⊗i} (see [GLS91]). It is clear that Tp_{sL}Tλ_{sL} = 1_{T^{c}(s𝔤)}. We can show that Tλ_{sL}Tp_{sL} = 1_{T^{c}(sL)}+l_{1}Th_{sL}+Th_{sL}l_{1} by induction on n. As a result, H_{l_{1}}(T^{c}(sL)) = T^{c}(s𝔤). Similarly, we can define SDR-data

((T^{c}(s𝔤) ⊗ B, 0) [figure: Tλ_{M}; Tp_{M}] (T^{c}(sL) ⊗ M, m_{1}), Th_{M})

where Th_{M} = ∑_{i=1}^{∞} T_{i}h_{M} with

T_{i}h_{M} = (∑_{j=1}^{i−1} 1^{⊗j−1} ⊗ h_{sL} ⊗ (λ_{sL}p_{sL})^{⊗i−j+1} ⊗ λ_{M}p_{M} ) + 1^{⊗i−1} ⊗ h_{M}
= T_{i−1}h_{sL} ⊗ λ_{M}p_{M} + 1^{⊗i−1} ⊗ h_{M}

on (sL)^{⊗i−1} ⊗ M.

Besides the inclusion map i : ⋀(sL) → T^{c}(sL), there is also a splitting map j : T^{c}(sL) → ⋀(sL) which sends sv_{[1 to n]} to \frac{1}{n!}sv_{[1 to n]}^{∧}. Both i and j are chain maps, so the composition ji = 1_{⋀(sL)} induces an isomorphism between H_{l_{1}}(⋀(sL)) and ⋀(H_{l_{1}}(sL)), which equals ⋀(s𝔤). Similarly, using (i ⊗ 1_{M}) and (j ⊗ 1_{M}), we can show that H_{m_{1}}(⋀(sL) ⊗ M) = ⋀(s𝔤) ⊗ B. Having verified that equations (11) and (12) in definition 5.1 hold, it is now straightforward to show that conditions (a) and (b) in proposition 5.2 guarantee that conditions (i) and (ii) are satisfied. □

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 the Lie-Rinehart pair (B, s𝔤), then (R, ⟨D_{M}, ⟩) is a cohomological model for (R, ⟨m̂_{2}, ⟩).

Proof of Proposition 5.3. We have the following SDR-data:

((⋀(s𝔤), 0) [figure: λ_{1}; p_{1}] (⋀(sL), l_{1}), h_{1})

and

((⋀(s𝔤) ⊗ B, 0) [figure: λ_{2}; p_{2}] (⋀(sL) ⊗ M, m_{1}), h_{2}) .

The E_{0} term of the spectral sequence is the algebra R, bigraded by the external degree of a map and the difference between the internal degree and the external degree, e.g, if the map F_{n} has internal degree |F_{n}|, then it has bidegree (n, |F_{n}|−n). The differential ⟨D_{M}, ⟩ splits up into the sum ∑_{i=1}^{∞} ⟨m_{i}, ⟩, each of which has bidegree (i − 1, 2 − i). The map ⟨m_{1}, ⟩ is the differential on E_{0}. If F_{n} is a cocycle, then F_{n} is a chain map between the complexes ⋀(sL) and ⋀(sL) ⊗ M. So the class of F_{n} induces a map [F_{n}] : ⋀(s𝔤) → ⋀(s𝔤) ⊗ B. Therefore, the E_{1} term is contained in R and all elements [F_{n}] of E_{1}, have bidegree (n, 0). It follows that the cohomology of R is the cohomology of the E_{1} term with respect to the differential [⟨m_{2}, ⟩] given by sending [F_{n}] to [⟨m_{2}, F_{n}⟩]. The square of [⟨m_{2}, ⟩] is zero precisely because [⟨m_{2}, ⟨m_{2}, F_{n}⟩⟩] equals [− ⟨m_{3}, ⟨m_{1}, F_{n}⟩⟩ − ⟨m_{1}, ⟨m_{3}, F_{n}⟩⟩]. The first term ⟨m_{3} ⟨m_{1}, F_{n}⟩⟩ is zero because ⟨m_{1}, F_{n}⟩ = 0. The second term is the coboundary of ⟨m_{3}, F_{n}⟩, so its class with respect to ⟨m_{1}, ⟩ is zero.

There is an algebra splitting from R to E_{1} given by sending f_{n} to [λ_{2}f_{n}p_{1}], so E_{1} ≈ R as graded commutative algebras. Furthermore, since H_{m_{1}}(m_{2}) = m̂_{2} and H_{l_{1}}(l_{2}) = l̂_{2}, it follows that [⟨m_{2}, ⟩] = ⟨m̂_{2}, ⟩. Therefore, E_{1} and R are isomorphic as differential graded commutative algebras and have the same cohomology. □

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-Rinehart pair (A/I, I/I^{2}) which arises in the BFV formulation of classical BRST cohomology.

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Lars Kjeseth lkjeseth@elcamino.cc.ca.us

Mathematical Sciences Division El Camino College U.S.A. Torrance 16007 Crenshaw Blvd, Torrance, CA 90506