Todd Trimble
Notes on the Lie Operad

… abstractly by the formula

(FG)[S]= W,XℱℬF[W]G[X]ℱℬ(S,WX).(F \otimes G)[S] = \int^{W, X \in \mathcal{FB}} F[W] \otimes G[X] \otimes \mathcal{FB}(S, W \oplus X).

Since each finite bijection ϕ:SWX\phi: S \to W \oplus X induces a decomposition of SS as a disjoint union T+UT+U of subsets (setting T=ϕ 1(W)T = \phi^{-1}(W) and U=ϕ 1(X)U = \phi^{-1}(X)), this coend formula may be simplified:

(FG)[S]= S=T+UF[T]G[U].(F \otimes G)[S] = \sum_{S = T+U} F[T] \otimes G[U].

(For the remainder of this paragraph, categorical terms such as “category”, “functor”, “colimit”, etc. refer to their VV-enriched analogues.) The product FGF \otimes G preserves colimits in the separate arguments FF and GG (i.e., G- \otimes G and FF \otimes - are cocontinuous for all FF and GG). Since FF and GG may be canonically presented as colimits of representables, one may define a symmetric monoidal structure on this product, uniquely up to monoidal isomorphism, so that the Yoneda embedding y:ℱℬ opV ℱℬy: \mathcal{FB}^{op} \to V^{\mathcal{FB}} is a symmetric monoidal functor, i.e., so that there is a coherent isomorphism hom(WX,)hom(W,)hom(X,)\hom(W \oplus X, -) \cong \hom(W, -) \otimes \hom(X, -). In conjunction with the universal property of ℱℬ\mathcal{FB}, we may state a universal property of V ℱℬV^{\mathcal{FB}}: let CC be a (VV-)category which is symmetric monoidally cocomplete (meaning its monoidal product is separately cocontinuous), and let AA be an object of CC. Then there exists a cocontinuous symmetric monoidal functor V ℱℬCV^{\mathcal{FB}} \to C sending hom(1,)\hom(\mathbf{1}, -) to AA, and this functor is unique up to monoidal isomorphism.

This universal property may be exploited to yield a second monoidal structure on V ℱℬV^{\mathcal{FB}}. Let [V ℱℬ,C][V^{\mathcal{FB}}, C] denote the category of cocontinuous symmetric monoidal functors V ℱℬCV^{\mathcal{FB}} \to C; then the universal property may be better expressed as saying that the functor [V ℱℬ,C]C[V^{\mathcal{FB}}, C] \to C which evaluates a species F:ℱℬVF: \mathcal{FB} \to V at 1ℱℬ\mathbf{1} \in \mathcal{FB} is an equivalence. In the case C=V ℱℬC = V^{\mathcal{FB}}, the left-hand side of the equivalence carries a monoidal structure given by endofunctor composition. This monoidal structure transports across the equivalence to yield a monoidal product on V ℱℬV^{\mathcal{FB}}, denoted by \circ.

An explicit formula for \circ is given as follows. Under one set of conventions, a VV-species FF may be regarded as a right module over the permutation category P\mathbf{P}, so that the component F[n]F[n] carries an action F[n]S nF[n]F[n] \otimes S_n \to F[n]. The nn-fold Day convolution G nG^{\otimes n} carries, under the same conventions, a left S nS_n-action S nG nG nS_n \otimes G^{\otimes n} \to G^{\otimes n}. Thn the coend formula for FGF \circ G may be written as

(FG)[S]= n0F[n] S nG n[S].(F \circ G)[S] = \sum_{n \geq 0} F[n] \otimes_{S_n} G^{\otimes n}[S].

A special case of this “substitution product” \circ is the analytic functor construction. For each object XX in VV there is a VV-species X^\widehat{X} such that X^[0]=X\widehat{X}[0] = X and X^[n]=0\widehat{X}[n] = 0 otherwise. Letting X nX^n denote the nn-fold tensor product in VV, we have

(FX^)[0]= n0F[n] S nX n(F \circ \widehat{X})[0] = \sum_{n \geq 0} F[n] \otimes_{S_n} X^n

and (FX^)[n]=0(F \circ \widehat{X})[n] = 0 otherwise; we write the right-hand side of the above equation as F(X)F(X). This defines a functor F():VVF(-): V \to V, which we call the analytic functor attached to the species FF. We often commit an abuse of language and write F(X)F(X) for the analytic functor, thinking of the XX as a variable or placeholder for an argument, much as one abuses language by referring to a function f(x)=sin(x)f(x) = \sin(x).

The analytic functor F():VVF(-): V \to V determines, up to isomorphism, its generating species FF; we describe this determination for the category V=Vect kV = Vect_k of vector spaces over a ground field kk. Let F n(X)F_n(X) denote the n thn^{th}-degree component F[n] S nX nF[n] \otimes_{S_n} X^n, and let XX be a vector space freely generated from a set {x 1,,x n}\{x_1, \ldots, x_n\} whose cardinality equals that degree. Then the species value F[n]F[n] can be recovered as the subspace of F n(X)F_n(X) spanned by equivalence classes of those expressions τx i(1)x i(n)\tau \otimes x_{i(1)} \otimes \ldots \otimes x_{i(n)} in which each x ix_i occurs exactly once. We use the notations F[n]F[n] and F(X)[n]F(X)[n] interchangeably for these species values.

If G[0]=0G[0] = 0, we have G n[S]=0G^{\otimes n}[S] = 0 whenever nn exceeds the cardinality S|S|, in which case FGF \circ G makes sense for VV finitely cocomplete. For general nn we have in that case

G n[S]= S=T 1++T nG[T 1]G[T n]G^{\otimes n}[S] = \sum_{S = T_1 + \ldots + T_n} G[T_1] \otimes \ldots \otimes G[T_n]

where the sum is indexed over ordered partitions of SS into nn nonempty subsets T iT_i. The group S nS_n permutes such ordered partitions in such a way that the orbits correspond to unordered partitions, which are tantamount to equivalence relations on SS. Let Eq(S)Eq(S) denote the set of such equivalence relations, and let π:SS/R\pi: S \to S/R denote the canonical projection of SS onto the set of RR-equivalence classes. Then the substitution product may be rewritten as

(FG)[S]= REq(S)F[S/R] xS/RG[π 1(x)](F \circ G)[S] = \sum_{R \in Eq(S)} F[S/R] \otimes \bigotimes_{x \in S/R} G[\pi^{-1}(x)]

whenever G[0]=0G[0] = 0.

Operads

Definition

An operad in VV is a monoid in the monoidal category (V ℱℬ,)(V^{\mathcal{FB}}, \circ).

The unit for the monoidal product \circ will be denoted XX; it is defined by X[n]=0X[n] = 0 if n1n \geq 1, and X[1]=IX[1] = I where II is the monoidal unit of VV. (We repeat that we also use XX as abusive notation for a placeholder or variable with values ranging over objects of VV.)

Clearly an operad MM induces a monad M:V ℱℬV ℱℬM \circ -: V^{\mathcal{FB}} \to V^{\mathcal{FB}}, which in turn restricts to the analytic monad M():VVM(-): V \to V along the embedding ()^:VV ℱℬ\widehat{(-)}: V \to V^{\mathcal{FB}} if VV is cocomplete. Many algebraic structures arising in practice are algebras of analytic monads. For V=Vect kV = Vect_k, we have, e.g.,

  1. The tensor algebra T(X)=1+X+X 2+T(X) = 1 + X + X^{\otimes 2} + \ldots, denoted 11X\frac1{1-X}. The algebras of T()T(-) are associative algebras. The species value T[n]T[n] is the space freely generated from the set of linear orders on an nn-element set, with the evident S nS_n-action.

  2. The symmetric algebra S(X)=1+X+X 2/S 2+S(X) = 1 + X + X^{\otimes 2}/S_2 + \ldots, denoted exp(X)\exp(X). Algebras of S()S(-) are commutative associative algebras. The species value exp[n]\exp[n] is the trivial 1-dimensional representation of S nS_n.

  3. The Lie operad L[]L[-] may be presented as an operad generated by a binary operation [,]L[2][-, -] \in L[2], subject to the Jacobi relation [,[,]]+[,[,]]σ+[,[,]]σ 2=0[-, [-, -]] + [-, [-, -]]\sigma + [-, [-, -]]\sigma^2 = 0 and the alternating relation [,]+[,]τ=0[-, -] + [-, -]\tau = 0, where σ\sigma is a 3-cycle and τ\tau is a 2-cycle. The algebras of the analytic monad L()L(-) are Lie algebras.

By way of contrast, Boolean algebras are not algebras of an analytic monad, since the equation xx=xx \wedge x = x inevitably involves the use of a diagonal map not available in ℱℬ\mathcal{FB}.

Although analytic monads are obviously important, we stress that they are simply restrictions of monads V ℱℬV ℱℬV^{\mathcal{FB}} \to V^{\mathcal{FB}}, and that it is often more flexible to work in the latter setting. For example, if VV is only finitely cocomplete, then analytic monads cannot be defined in general; for example, the free commutative monoid construction does not define an analytic monad on finite-dimensional vector spaces. However, if VV is finitely cocomplete and MM is a VV-operad such that M[0]=0M[0] = 0, then there is a monad MM \circ - acting on the orthogonal complement V V ℱℬV^\perp \hookrightarrow V^{\mathcal{FB}}, i.e., the full subcategory of VV-species GG such that G[0]=0G[0] = 0.

This situation occurs often. For example, consider the operad M(X)=exp(X)1M(X) = \exp(X)-1, whose Vect kVect_k-algebras are commutative monoids without unit. This operad induces a monad MM \circ - on V V^\perp where VV is the category of finite-dimensional vector spaces; the algebras are again commutative (V V^\perp-)monoids without unit. The operad itself can be regarded as the free commutative algebra without unit, MXM \circ X, generated by the monoidal unit XX considered as living in the subcategory V V^\perp. For other reasons, Markl (see references) has also considered algebras over monads MM \circ - and M- \circ M, more general than algebras of analytic monads; he refers to the former as ”MM-modules”.

A look ahead

In the next few sections, we will reprise the beautiful work of Joyal which leads up to a computation of the Lie species LL. Our general methodology is to reinterpret Joyal’s approach via virtual species by appeal to dg-structures on finite-dimensional super vector spaces; ultimately we feel that a proper approach to virtual species should draw on the standard model category structure on this category.

A second point of our approach is to place Joyal’s calculations within the context of a particular bar construction. This will better enable us to compare these calculations with those of Ginzburg-Kapranov, which involve a slightly different bar construction.

Poincaré-Birkhoff-Witt

From this section on, we fix a ground field kk of characteristic 0, and VV henceforth denotes the category of finite-dimensional vector spaces over kk.

To study the Lie operad L[]L[-], Joyal and others (see reference to Hinich and Schechtman) take as starting point the Poincaré-Birkhoff-Witt (PBW) theorem. If LL is a Lie algebra, then its universal enveloping algebra U(L)U(L) carries a canonical filtration, inherited as a quotient of the tensor algebra T(L)T(L) equipped with the degree filtration. Embedded in T(L)T(L) as a filtered subspace is the symmetric algebra S(L)S(L), whose homogeneous components S n(L)S_n(L) may be realized as images of symmetrizing operators acting on components T n(L)=L nT_n(L) = L^{\otimes n}:

π n:L nL n\pi_n: L^{\otimes n} \to L^{\otimes n}
v 1v n1n! σS nv σ(1)v σ(n).v_1 \otimes \ldots \otimes v_n \mapsto \frac1{n!} \sum_{\sigma \in S_n} v_{\sigma(1)} \otimes \ldots \otimes v_{\sigma(n)}.

We obtain a composite of maps of filtered spaces

S(L)T(L)U(L)S(L) \to T(L) \to U(L)

and the PBW theorem concerns the application of the associated graded space functor to this composite (denoted ϕ\phi):

Theorem (PBW)

The graded map ϕ gr\phi_{gr} induces an isomorphism between U gr(L)U_{gr}(L) and S gr(L)S_{gr}(L) as graded spaces.

If L(X)L(X) is the free Lie algebra on XX, then U(L(X))T(X)U(L(X)) \cong T(X) (as algebras even) by an adjoint functor argument. Assembling some prior notation, it follows from PBW that there exists an isomorphism of analytic functors on Vect kVect_k:

11X(expL)(X)\frac1{1-X} \cong (\exp \circ L)(X)

which in turn determines a species-isomorphism, which componentwise is an isomorphism of S nS_n-representations:

11X[n](expL)[n].\frac1{1-X}[n] \cong (\exp \circ L)[n].

Notice both sides makes sense as VV-species. A guiding idea behind the species methodology is that the components of such species are structural analogues of coefficients of formal power series. This analogy can be made precise. Let V[[x]]V[ [x]] denote the rig (ring without additive inverses) of isomorphism classes of VV-species, with ++ given by coproduct and \cdot given by \otimes. Let [[x]]\mathbb{N}[ [x]] denote the rig of formal power series

n0a nx nn!\sum_{n \geq 0} \frac{a_n x^n}{n!}

where the coefficients a na_n are natural numbers. This may be equivalently defined to be the rig of sequences a na_n of natural numbers where two sequences are multiplied according to the rule

(ab) n= p(np)a pb np(a \cdot b)_n = \sum_p \binom{n}{p} a_p b_{n-p}

The rigs V[[x]]V[ [x]] and [[x]]\mathbb{N}[ [x]] also have a partially defined composition operation \circ, where fgf \circ g is defined whenever g(0)=0g(0) = 0. In the case of V[[x]]V[ [x]], it is of course the operation which is descended from the substitution product by passing to isomorphism classes.

Proposition

(See reference to Joyal’s first paper on species) The function dim:V[[x]][[x]]\dim: V[ [x]] \to \mathbb{N}[ [x]], sending FF to the sequence of coefficients a n=dim(F[n])a_n = \dim(F[n]), is a rig homomorphism which preserves the \circ operation.

It follows from the proposition and the preceding species isomorphism that dim(L[n])=(n1)!\dim(L[n]) = (n-1)!, since dim(L)(x)\dim(L)(x) is the formal power series expansion of log(1x)-\log(1-x). We are interested in finding an appropriate lift of log(1x)[[x]]-\log(1-x) \in \mathbb{N}[ [x]] to a species log(1X)-\log(1-X) in V[[x]]V[ [x]], and hence an identification between log(1X)-\log(1-X) and the Lie species l[]l[-].

Virtual species

Before we construct the species log(1X)\log(1-X), it is convenient to complete the rig V[[x]]V[ [x]] to a ring. One proceeds exactly as in K-theory, where one passes from vector bundles to virtual bundles.

Definition

V\mathbf{V} is the category of 2\mathbb{Z}_2-graded finite-dimensional vector spaces, with monoidal product given by the formula

(VW) p= m+n=pV mW n(V \otimes W)_p = \sum_{m+n = p} V_m \otimes W_n

and symmetry given by the formula

σ(x my n)=(1) mny nx m\sigma(x_m \otimes y_n) = (-1)^{m n} y_n \otimes x_m

for x mV mx_m \in V_m and y nW ny_n \in W_n.

Definition

Let FF and GG be V\mathbf{V}-species. Then FGF \sim G (FF and GG are virtually equivalent) if F 0G 1F 1G 0F_0 \oplus G_1 \cong F_1 \oplus G_0 as VV-species.

Lemma

The realtion \sim is an equivalence relation.

Proof

Transitivity follows from cancellation: FHGHF \oplus H \cong G \oplus H implies FGF \cong G. This in turn follows from the remark.

It is of course crucial here to work with finite-dimensional vector spaces throughout, in order to avoid the Eilenberg swindle.

Lemma

The relation \sim is respected by \oplus, \otimes, and \circ.

The proof is left to the reader; see also references to Joyal and Yeh.

A virtual species is a virtual equivalence class of V\mathbf{V}-species. The ring of equivalence classes is denoted V[[x]]\mathbf{V}[ [x]].

Many of our calculations refer to manipulations in the ring V[[x]]\mathbf{V}[ [x]] of virtual species, but methodologically it is useful to distinguish the various ways in which virtual equivalences arise. Part of the philosophy behind species is that clarity is promoted and calculations are under good combinatorial control when power series operations ++, \cdot, and \circ can be viewed as arising from categorified functorial operations \oplus, \otimes, and \circ. Put differently, passage from V ℱℬ\mathbf{V}^{\mathcal{FB}} to V[[x]]\mathbf{V}[ [x]] loses categorical information, and it helps to recognize when a virtual equivalence FGF \sim G comes from an isomorphism FGF \cong G in V ℱℬ\mathbf{V}^{\mathcal{FB}}.

In practice, many virtual equivalences which do not come from isomorphisms in V ℱℬ\mathbf{V}^{\mathcal{FB}} come about by applying the following remark.

This principle can be quite powerful. Its application does not commit one to any particular choice of differential structure on CC, so that one is enabled to choose differentials to suit the local occasion. The downside is that because there is no canonical way to split exact sequences, it is sometimes harder to give precise formulas that exhibit such virtual equivalences CH(C)C \sim H(C).

Logarithmic species

One of our goals is to lift the inversion

11x=exp(L(x))implieslog(1x)=L(x)\frac1{1-x} = \exp(L(x)) \qquad implies \qquad -\log(1-x) = L(x)

from the ring [[x]]\mathbb{Z}[ [x]] to the ring V[[x]]\mathbf{V}[ [x]]. In either ring, a necessary condition for F(x)F(x) to have an inverse F 1(x)F^{-1}(x) (with respect to \circ) is that the 0 th0^{th} coefficient F[0]F[0] be 00. Thus, instead of inverting exp(X)\exp(X), we invert exp(X)1\exp(X)-1. Suppose then that log(1+X)\log(1+X) is a virtual inverse of exp(X)1\exp(X)-1:

log(1+X)(exp(X)1)X(exp(X)1)log(1+X).\log(1+X) \circ (\exp(X)-1) \sim X \sim (\exp(X)-1) \circ \log(1+X).
Proposition

If FF and GG are V\mathbf{V}-species such that F[0]=0=G[0]F[0] = 0 = G[0], then exp(FG)exp(F)exp(G)\exp(F \oplus G) \cong \exp(F) \otimes \exp(G).

Proof

It is immediate that exp(F)= n0F n/S n\exp(F) = \sum_{n \geq 0} F^{\otimes n}/S_n is the free commutative monoid in (V ℱℬ,)(\mathbf{V}^{\mathcal{FB}}, \otimes) generated from FF, and the assertion says that the left adjoint exp\exp preserves coproducts.

Defining log((1+F)(1+G))\log((1+F) \otimes (1+G)) to be

log(1+X)(F+F+FG),\log(1+X) \circ (F + F + F \otimes G),

it follows that log((1+F)(1+G))log(1+F)+log(1+G)\log((1+F) \otimes (1+G)) \sim \log(1+F) + \log(1+G). In particular,

log(11X)log(1X)\log(\frac1{1-X}) \sim -\log(1-X)

where of course (F 0,F 1)-(F_0, F_1) is defined to be (F 1,F 0)(F_1, F_0). The species log(1X)\log(1-X) is easily obtained from log(1+X)\log(1+X) by the following result.

Proposition

F(X)[S](1) SF[S]Λ[S]F(-X)[S] \cong (-1)^{|S|} F[S] \otimes \Lambda[S], where Λ[S]\Lambda[S] denotes the top exterior power Λ S(kS)\Lambda^{|S|}(k S) of the vector space kSk S freely generated from SS.

Proof

The V\mathbf{V}-species XX, which by definition is the unit with respect to \circ, is given by (X[1] 0,X[1] 1)=(k,0)(X[1]_0, X[1]_1) = (k, 0) and X[n]=0X[n] = 0 otherwise. Thus (X[1] 0,X[1] 1)=(0,k)(-X[1]_0, -X[1]_1) = (0, k) and X[n]=0-X[n] = 0 otherwise. Hence

(F(X))[S] = REq(S)F[S/R] xS/R(X)[π 1(x)] F[S](X)[1] S\array{ (F \circ (-X))[S] & = & \sum_{R \in Eq(S)} F[S/R] \otimes \bigotimes_{x \in S/R} (-X)[\pi^{-1}(x)] \\ & \cong & F[S] \otimes (-X)[1]^{\otimes |S|} }

Now (X)[1] n(-X)[1]^{\otimes n} is 1-dimensional and is concentrated in degree n(mod2)n \pmod 2 (whence the sign (1) S(-1)^{|S|}). A transposition in S nS_n induces a sign change in (X[1] n) n(mod2)(-X[1]^{\otimes n})_{n \pmod 2}, by definition of symmetry: σ(x 1y 1)=y 1x 1\sigma(x_1 \otimes y_1) = - y_1 \otimes x_1. This proves the claim.

We proceed to compute the inverse log(1+X)\log(1+X) to exp(X)1\exp(X)-1. Recalling an earlier remark, F(X)=exp(X)1F(X) = \exp(X)-1 is the operad such that algebras of the monad FF \circ - (acting on V\mathbf{V}-species GG such that G[0]=0G[0] = 0) are commutative algebras without unit.

The underlying species FF satisfies F[0]=0F[0] = 0, F[1]=1F[1]= 1 (i.e., =(k,0)= (k, 0)). Joyal gives a general method due to G. Labelle for inverting such species. Introduce an operator

δ F:V[[x]]V[[x]]\delta_F: \mathbf{V}[ [x]] \to \mathbf{V}[ [x]]
\,
HHFHH \mapsto H \circ F - H

so that (1+δ F)(H)=HF(1 + \delta_F)(H) = H \circ F; here 11 denotes the identity functor. Observe that δ F\delta_F preserves sums, because F:V ℱℬV ℱℬ- \circ F: \mathbf{V}^{\mathcal{FB}} \to \mathbf{V}^{\mathcal{FB}} is the restriction of a cocontinuous monoidal functor which thus preserves coproducts.

Let 0^\widehat{0} denote the bottom element of the lattice eq(S)eq(S) ordered by inclusion of equivalence relations. This 0^\widehat{0} is the discrete equivalence relation on SS, so that S/0^SS/\widehat{0} \cong S. We have

δ F(H)[S]=( REq(S)H[S/R] xS/RF[π 1(x)])H[S]\delta_F(H)[S] = (\sum_{R \in Eq(S)} H[S/R] \otimes \bigotimes_{x \in S/R} F[\pi^{-1}(x)]) - H[S]

and since H[S]H[S/0^] xS/0^F[1]H[S] \cong H[S/\widehat{0}] \otimes \bigotimes_{x \in S/\widehat{0}} F[1] by our assumptions on FF, we may rewrite the right-hand side (up to virtual equivalence) as

0^<RH[S/R] xS/RF[π 1(x)].\sum_{\widehat{0} \lt R} H[S/R] \otimes \bigotimes_{x \in S/R} F[\pi^{-1}(x)].

Define the V\mathbf{V}-species δ F(H)\delta_F(H) by the above expression, so that δ F\delta_F will be used to denote a functor on V\mathbf{V}-species HH, in addition to an operator on V[[x]]\mathbf{V}[ [x]]. In particular, when F(X)=exp(X)1F(X) = \exp(X)-1, we have

δ F(H)= 0^<RH[S/R].\delta_F(H) = \sum_{\widehat{0} \lt R} H[S/R].

In general, the n thn^{th} iterate δ F n(H)[S]\delta_{F}^{n}(H)[S] is a sum of the form

0^<R 1<<R nterms\sum_{\widehat{0} \lt R_1 \lt \ldots \lt R_n} terms

where terms are indexed by strictly increasing chains of equivalence relations on SS. As soon as nSn \geq |S|, there are no chains of that length, so this sum will be empty. In this way, for each finite SS, δ F n(H)[S]=0\delta_{F}^{n}(H)[S] = 0 for all sufficiently large nn, and so the expression

(1+δ F) 1(H) n0(1) nδ F n(H)(1 + \delta_F)^{-1}(H) \coloneqq \sum_{n \geq 0} (-1)^n \delta_{F}^{n}(H)

makes sense as a functor on V\mathbf{V}-species HH.

We may now construct the inverse species F 1(X)F^{-1}(X):

F 1(X)=(1+δ F) 1(X) n0(1) nδ F n(X)F^{-1}(X) = (1 + \delta_F)^{-1}(X) \coloneqq \sum_{n \geq 0} (-1)^n \delta_{F}^{n}(X)
Proposition

(F 1F)(X)X(F^{-1} \circ F)(X) \sim X.

Proof

We have

(F 1F)(X) (1+δ F)(F 1)(X)) = (1+δ F)( n0(1) nδ F n(X)) n0(1) nδ F n(X)+ n0(1) nδ F n+1(X)\array{ (F^{-1} \circ F)(X) & \sim & (1 + \delta_F)(F^{-1})(X)) & = & (1 + \delta_F)(\sum_{n \geq 0} (-1)^n \delta_{F}^{n}(X)) \\ & & & &\sim \sum_{n \geq 0} (-1)^n \delta_{F}^{n}(X) + \sum_{n \geq 0} (-1)^n \delta_{F}^{n+1}(X) }

which telescopes down to δ F 0(X)=X\delta_{F}^{0}(X) = X.

A bar construction

When FF carries an operad structure, this construction of F 1F^{-1} admits a more categorical interpretation. Observe that there is an embedding

δ F n(X)FF=F n.\delta_{F}^{n}(X) \hookrightarrow F \circ \ldots \circ F = F^{\circ n}.

Let us regard the operad FF as a monoid with multiplication m:FFFm: F \circ F \to F and unit u:XFu: X \to F. There is a (necessarily unique) operad map ε:FX\varepsilon: F \to X, called an augmentation, and this may be used to turn XX into a left FF-module and also into a right FF-module, in the usual way. We may thus form a two-sided bar construction B(X,F,X)B(X, F, X), whose component in dimension nn is isomorphic to F nF^{\circ n}.

The bar construction B(X,F,X)B(X, F, X) is a simplicial object in an additive category, and hence gives rise to a \mathbb{Z}-graded chain complex, where each differential is a signed sum of face maps of the form

i:F (n+1)F (ni)mF (i1)F n.\partial_i: F^{\circ (n+1)} \stackrel{F^{\circ (n-i)} m F^{\circ (i-1)}}{\to} F^{\circ n}.

By reduction of the grading, we may regard B(X,F,X)B(X, F, X) as a 2\mathbb{Z}_2-graded chain complex, provided that the two components are taken as species valued in the category of (possibly infinite-dimensional) vector spaces.

However, since we are dealing with virtual species, we want to cut back to V\mathbf{V}-valued species, where the components are finite-dimensional. To this end, notice that each of the maps i\partial_i restricts to a map

δ F n+1(X)δ F n(X)\delta_{F}^{n+1}(X) \to \delta_{F}^{n}(X)

and the δ F n(X)\delta_{F}^{n}(X) form a chain subcomplex. We regard this chain complex as our preferred bar construction for FF, or more precisely a right bar construction B r(F)B_r(F), as we now explain in more detail.

There is an exact sequence

0δ F()F()X0 0 \to \delta_F \to (-) \circ F \to (-) \circ X \to 0

making the functor δ F\delta_F, for an operad FF, analogous to tensoring on the right with an augmentation ideal IGI G of a group ring G\mathbb{Z}G, sitting in an exact sequence

0IGG0.0 \to I G \to \mathbb{Z}G \to \mathbb{Z} \to 0.

Here δ F\delta_F (resp. IG- \otimes I G) is regarded as a formal or virtual difference between F- \circ F and X- \circ X (resp. G- \otimes \mathbb{Z}G and - \otimes \mathbb{Z}). In forming F 1(X)F^{-1}(X) as

X+(δ F)(X)+(δ F) 2(X)+,X + (-\delta_{F})(X) + (-\delta_F)^{2}(X) + \ldots,

we interpret the additive inverse δ F-\delta_F as the result of composing δ F\delta_F with the degree 1 shift operator Σ\Sigma acting on 2\mathbb{Z}_2-graded species, which plays the structural role of additive inverse. Hence the construction for F 1(X)F^{-1}(X) above is analogous to the normalized bar resolution

+Σ(IG)+(Σ(IG)) 2+\mathbb{Z} + \Sigma(I G) + (\Sigma (I G))^{\otimes 2} + \ldots

originally introduced by Eilenberg-Mac Lane (reference).

To give further weight to the sense in which B r(F)B_r(F) is a bar construction, we assemble the components δ F n(X)F\delta_{F}^{n}(X) \circ F of F 1FF^{-1} \circ F into an acyclic chain complex E r(F)E_r(F) of V\mathbf{V}-species that will be a (right) FF-free resolution of the unit operad XX. Indeed, there is an embedding

δ F n(X)FF (n+1)\delta_{F}^{n}(X) \circ F \hookrightarrow F^{\circ (n+1)}

where now F (n+1)F^{\circ (n+1)} is the degree nn component of the two-sided bar resolution B(X,F,F)B(X, F, F) (as simplicial object). Again, the face maps

i:F (n+1)FF nF\partial_i: F^{\circ (n+1)} \circ F \to F^{\circ n} \circ F

restrict to maps

δ F n+1(X)Fδ F n(X)F\delta_{F}^{n+1}(X) \circ F \to \delta_{F}^{n}(X) \circ F

so that the δ F n(X)F\delta_{F}^{n}(X) \circ F are components of a subcomplex E r(F)E_r(F). This endows the 2\mathbb{Z}_2-graded species F 1FF^{-1} \circ F with a dg-structure.

Then, we may regard the virtual equivalence

(F 1F)(X)X(F^{-1} \circ F)(X) \sim X

as arising from a homotopy equivalence between dg-species:

δ F n+1(X) δ F n(X) δ F 0(X) 0 id 0 0 X 0\array{ \ldots & \delta_{F}^{n+1}(X) & \to & \delta_{F}^n(X) & \to \ldots \to & \to \delta_{F}^0(X) & \to 0 \\ & \downarrow & & \downarrow & & \downarrow \mathrlap{id} \\ & 0 & & 0 & & X & 0 }

Indeed, we have an augmentation map B(X,F,F)XB(X, F, F) \to X between simplicial objects (regarding XX on the right as a constant simplicial object). This restricts to an augmentation map E r(F)XE_r(F) \to X which, by the Dold-Kan correspondence, corresponds to a map between chain complexes as displayed above. To check that the augmentation E r(F)XE_r(F) \to X is a homotopy equivalence, one checks that the standard contracting homotopy on B(X,F,F)B(X, F, F), with components

F nu:F nXF nF,F^{\circ n} \circ u: F^{\circ n} \circ X \to F^{\circ n} \circ F,

restricts to a contracting homotopy on E r(F)E_r(F). In more detail, each component δ F n(X)F\delta_{F}^{n}(X) \circ F breaks up as a coproduct

The Lie species

We now return to our example F(X)=exp(X)1F(X) = \exp(X)-1 and the construction of the inverse F 1(X)=log(1+X)F^{-1}(X) = \log(1+X). By iterating the functor

δ F:H(S 0^<RH[S/R]),\delta_F: H \mapsto (S \mapsto \sum_{\widehat{0} \lt R} H[S/R]),

we derive an expression of δ F n(X)[S]\delta_{F}^{n}(X)[S] as the space whose basis elements are nn-fold chains of strict inclusions of equivalence relations

0^<R 1<R n=1^\widehat{0} \lt R_1 \lt \ldots R_n = \widehat{1}

where 1^\widehat{1} denotes the top element of the lattice Eq(S)Eq(S), namely the indiscrete equivalence relation with one equivalence class. Each such chain may be identified with an (n1)(n-1)-fold chain in the poset Eq(S){0^,1^}Eq(S) - \{\widehat{0}, \widehat{1}\}, in other words as cells of dimension n2n-2 in the simplicial complex underlying the nerve of this poset. Let C i[S]C_i[S] denote the set of cells of dimension ii (or rather the vector space it generates). Then for S>2|S| \gt 2, we may identify the complex B r(F)[S]B_r(F)[S],

0δ F S1(X)[S]δ F 2(X)[S]δ F 1(X)[S]X[S]0,0 \to \delta_{F}^{|S|-1}(X)[S] \to \ldots \to \delta_{F}^{2}(X)[S] \to \delta_{F}^{1}(X)[S] \to X[S] \to 0,

with

0C S3[S]C 0[S]k000 \to C_{|S|-3}[S] \to \ldots \to C_0[S] \to k \to 0 \to 0

It follows that log(1+X)[S]\log(1+X)[S] is virtually equivalent to reduced homology of the simplicial complex CC associated with Eq(S)Eq(S).

Now the lattice of equivalence relations is a geometric lattice, and a result of Folkman is that the reduced homology of such a lattice is trivial except in top degree; more exactly, Folkman’s result implies that the nerve of the poset Eq(S){0^,1^}Eq(S) - \{\widehat{0}, \widehat{1}\} is a bouquet of spheres of dimension S3|S|-3, and the number of spheres is given by the value μ(0^,1^)\mu(\widehat{0}, \widehat{1}) of Rota’s Möbius function on the lattice Eq(S)Eq(S).

Theorem

log(1+X)[S](1) S1H S1(B r(F)[S]).\log(1+X)[S] \sim (-1)^{|S|-1} H_{|S|-1}(B_r(F)[S]).

Proof

By the structural Euler formula, F 1(X)[S]=log(1+X)F^{-1}(X)[S] = \log(1+X) as given by the 2\mathbb{Z}_2-graded chain complex B r(F)[S]B_r(F)[S] is virtually equivalent to its homology, which in turn is equivalent to the reduced homology of C[S]C[S]. By Folkman’s theorem, the reduced homology of C[S]C[S] is concentrated in top degree S3|S|-3, corresponding to the homology of B r(F)[S]B_r(F)[S] sitting in degree S1|S|-1.

Theorem

The Lie species L[S]L[S] is isomorphic to H S1(B r(F)[S])Λ[S]H_{|S|-1}(B_r(F)[S]) \otimes \Lambda[S].

Proof

Starting from the PBW theorem

11Xexp(L(X)),\frac1{1-X} \cong \exp(L(X)),

we were led to the virtual equivalence

log(1X)L(X).-\log(1-X) \sim L(X).

Applying Joyal’s rule of signs to the previous theorem, log(1X)[S]-\log(1-X)[S] is virtually equivalent to

(1) S(1) S1H S1(B r(F)[S])Λ[S]-(-1)^{|S|}(-1)^{|S|-1}H_{|S|-1}(B_r(F)[S]) \otimes \Lambda[S]

which boils down to H S1(B r(F)[S])Λ[S]H_{|S|-1}(B_r(F)[S]) \otimes \Lambda[S]. As virtual species, both this and the Lie species value L[S]L[S] are concentrated in degree 0(mod2)0 \pmod 2. But if two V\mathbf{V}-species concentrated in degree 0 are virtually equivalent, they are isomorphic.

Revised on February 28, 2011 17:33:12 by Todd Trimble