Derived Poincaré–Birkhoff–Witt theorems

Abstract

We propose a new general formalism that allows us to study Poincaré–Birkhoff–Witt type phenomena for universal enveloping algebras in the differential graded context. Using it, we prove a homotopy invariant version of the classical Poincaré–Birkhoff–Witt theorem for universal envelopes of Lie algebras. In particular, our results imply that all the previously known constructions of universal envelopes of $L_\infty$-algebras (due to Baranovsky, Lada and Markl, and Moreno-Fernández) represent the same object of the homotopy category of differential graded associative algebras. We also extend Quillen’s classical quasi-isomorphism ${\mathscr {C}} \longrightarrow BU$ from differential graded Lie algebras to $L_\infty$-algebras; this confirms a conjecture of Moreno-Fernández.

Models of operads via homological perturbation by Vladimir Dotsenko

This section records an instance of a general homological perturbation argument which allows one to obtain, in a range of cases, a resolution of a filtered object from the one of its associated graded objects. A similar argument for a perturbative construction of a resolution of a shuffle operad with a Gröbner basis is featured in [23, Th. 4.1]. We keep the assumption on the characteristic of the ground field.

Let us consider a (non-dg) symmetric operad $Q={\mathscr {T}}(X)/(R)$ generated by a finite-dimensional $\Sigma$-module X concentrated in arities greater than one, subject to a finite-dimensional space of relations R. Suppose that the $\Sigma$-module X is equipped with a non-negative weight grading,

$$\begin{aligned}X=\bigoplus _{n\ge 0}X_{(n)}.\end{aligned}$$

This weight grading gives rise to a weight grading of the free operad ${\mathscr {T}}(X)$, and hence, an increasing filtration $F^\bullet {\mathscr {T}}(X)$ such that $F^k{\mathscr {T}}(X)$ is spanned by all elements of weight at most k. This filtration gives rise to a filtration on each $\Sigma$-submodule of ${\mathscr {T}}(X)$. In particular, we may consider the operad $P={\mathscr {T}}(X) /({{\textsf{g}}}{{\textsf{r}}}_F R)$. In general, there is an isomorphism of $\Sigma$-modules between Q and ${{\textsf{g}}}{{\textsf{r}}}_F Q={{\textsf{g}}}{{\textsf{r}}}_F{\mathscr {T}}(X) / {{\textsf{g}}}{{\textsf{r}}}_F(R)$, and a surjection of $\Sigma$-modules from P onto Q.

Theorem

Suppose that the underlying $\Sigma$-modules of the operads P and Q are isomorphic. Consider the minimal quasi-free resolution $P_\infty \longrightarrow P$ in the category of weight graded operads. There exists a quasi-free resolution $Q_\infty \longrightarrow Q$ whose underlying free operad is the same and the differential is obtained from the differential of $P_\infty$ by a perturbation that lowers the weight grading.

Proof

Assume that $P_\infty$ is of the form $({\mathscr {T}}(W),d)$, where W is an $\Sigma$-module that is bi-graded, by weight and by homological degree. We shall prove that there exists a quasi-free resolution $Q_\infty =({\mathscr {T}}(W),d+d')$ of Q so that $d'$ is strictly weight decreasing.

Since the resolution $P_\infty =({\mathscr {T}}(W),d)$ is minimal, it follows in particular that $W_0= X$, $W_1 =\Bbbk s\otimes {{\textsf{g}}}{{\textsf{r}}}_F R$. The operad ${\mathscr {T}}(W)$ can be mapped to both the operad P and the operad Q: One may project it onto its part of homological degree 0, the latter is isomorphic to the free operad ${\mathscr {T}}(X)$ which admits obvious projection maps to P and to Q. Let us choose splittings for those projections; this amounts to exhibiting two idempotent endomorphisms ${\bar{\pi }}$ and $\pi$ of ${\mathscr {T}}(W)$ such that both of which annihilate all elements of positive homological degree, and such that the former annihilates the ideal $({{\textsf{g}}}{{\textsf{r}}}_F R)\subset {\mathscr {T}}(X)={\mathscr {T}}(W)_0$ and the latter annihilates the $(R)\subset {\mathscr {T}}(X)={\mathscr {T}}(W)_0$.

Since P is finitely generated and has no generators of arity 1, components of the free operad ${\mathscr {T}}(W)$ are finite-dimensional, and there exists a weight graded contracting homotopy $h:{\mathscr {T}}(W) \longrightarrow {\mathscr {T}}(W)$ such that $h^2=0$ and $[d,h] = 1 - {\bar{\pi }}$.

We are going to define a derivation $D:{\mathscr {T}}(W) \longrightarrow {\mathscr {T}}(W)$ of degree $-1$ and a contracting homotopy $H:\ker (D) \rightarrow {\mathscr {T}}(W)$ of degree $+1$. Note that, a derivation is fully determined by the images of generators, and that $D|_{W_0}=0$ since ${\mathscr {T}}(W)$ has no elements of negative homological degree. For each element x of ${\mathscr {T}}(W)$ of a certain homological degree, we call the “leading term” of x the homogeneous part of x of maximal possible weight grading; we denote it by ${\widehat{x}}$.

We shall prove by induction on k that one can define the values of D on generators of homological degree $k+1$ and the values of H on elements of $\ker (D)$ of homological degree k so that the following five conditions hold:

• (1) for all elements $x\in {\mathscr {T}}(W)$, the leading term of the difference $D(x)-d(x)$ is of weight lower than that of x,

• (2) we have $D^2=0$ on generators of homological degree $k+1$,

• (3) the leading term of the difference $H(x)-h({\widehat{x}})$ is of weight lower than that of x,

• (4) we have $DH=1-\pi$ on elements of $\ker (D)$ of homological degree k.

As a basis of induction, we shall choose a basis of $W_1 = \Bbbk s\otimes {{\textsf{g}}}{{\textsf{r}}}_F R$, and set $D(s\otimes r')=r$ where r is some element of R for which ${\widehat{r}}=r'$. We note that $D(s r')-d(s r')=r-r'$ has smaller weight than $r'$, so Condition (1) is satisfied. Condition (2) is satisfied for degree reasons, as there are no elements of negative homological degree. Condition (1) together with the fact that ${\bar{\pi }}=\pi$ on elements of weight zero implies that the leading term of $Dh({\widehat{x}})$ is

$$\begin{aligned} dh({\widehat{x}})=(1-{\bar{\pi }})({\widehat{x}})=(1-\pi )({\widehat{x}})={\widehat{x}}-\pi ({\widehat{x}}), \end{aligned}$$

and so we may define H on elements of homological degree zero by induction on weight as follows. On elements x of weight zero, we put $H(x)=h(x)$, and on elements x of positive weight, we put

$$\begin{aligned} H(x)=h({\widehat{x}})+H(x-\pi (x)-Dh({\widehat{x}})). \end{aligned}$$

Both Condition (3) and Condition (4) are proved by induction on weight. For former one, the inductive argument is almost trivial; we shall show how to prove the latter. On elements of weight zero, we have $H(x)=h(x)$ and ${\bar{\pi }}=\pi$, so Condition (4) is true:

$$\begin{aligned} DH(x)=dh(x)=[d,h](x)=(1-{\bar{\pi }})(x)=(1-\pi )(x). \end{aligned}$$

For elements of positive weight, we have, by induction,

$$\begin{aligned} DH(x)&=Dh({\widehat{x}})+DH(x-\pi (x)-Dh({\widehat{x}})) \\&=Dh({\widehat{x}})+(1-\pi )((1-\pi )x-Dh({\widehat{x}}))\\&= (1-\pi )^2(x)+\pi (Dh({\widehat{x}}))\\&=(1-\pi )(x), \end{aligned}$$

since $\pi$ vanishes on the image of $D=(R)$ and $1-\pi$ is a projector. To carry the inductive step, we proceed in a similar way. To define the image under D of a generator of homological degree $k+1>1$, we put $D(x)=d(x)-HDd(x)$. Condition (1) now easily follows by induction. For Condition (2), we note that

$$\begin{aligned} D^2(x)&=D(d(x)-HDd(x)) \\&=Dd(x)-DH(Dd(x))\\&=Dd(x)-(1-\pi )Dd(x)=\pi (Dd(x))=0, \end{aligned}$$

since $Dd(x)\in \ker (D)$, and $\pi$ vanishes on the image of D. From that, we see that whenever $x\in \ker (D)$, we have $x-Dh({\widehat{x}})\in \ker (D)$. Using Condition (1) and the fact that ${\bar{\pi }}$ vanishes on elements of positive homological degree, we see that the leading term of $Dh({\widehat{x}})$ is $dh({\widehat{x}})=(1-{\bar{\pi }})({\widehat{x}})={\widehat{x}}$, so the leading term of $x-Dh({\widehat{x}})$ is of weight lower than that of x. Consequently, we may define H on elements of $\ker (D)$ of homological degree $k>0$ by the same inductive argument: On elements x of weight zero, we put $H(x)=h(x)$, and on elements x of positive weight, we put

$$\begin{aligned} H(x)=h({\widehat{x}})+H(x-Dh({\widehat{x}})). \end{aligned}$$

Once again, a simple inductive argument shows that Conditions (3) and (4) are satisfied, which completes the construction of D and H.

We conclude that D makes ${\mathscr {T}}(W)$ a dg operad, that the homology of that operad is isomorphic to Q, and that differential D is obtained from d by a perturbation $d'$ that lowers the weight grading, as required. $\square$