On Weak Lie 3-Algebras

Strongly homotopy Lie algebras or $\mathrm{L}_\infty$-algebras are a generalization of (dg) Lie algebras with better homotopy properties. In particular, a version of the homotopy transfer theorem holds. This is, however, only true when working over a field of characteristic $0.$ When working over an arbitrary unital commutative ring, one must generalize the definition even further, relaxing the skew-symmetry of the brackets up to homotopy in addition to the Jacobi identity.

I just finished my article “On weak Lie $3$-algebras,” in which I consider precisely this problem. I describe a general step by step approach and give explicit definitions for the case of underlying $3$-term complexes. In this post, I’ll try to give a rough idea of the approach used and a quick overview of the main results.

If you want all the details right away, go ahead and read my article. If you have any comments, criticism, or suggestions, don’t hesitate to contact me!

Introduction

The category of reduced augmented dg operads over a unital commutative ring $\mathbf{k}$ admits a model category structure where the weak equivalences are the quasi-isomorphisms and the fibrations are the degree-wise epimorphisms. In this context, we are interested in cofibrant replacements $\mathcal{Q}\overset{\sim}{\longrightarrow}\mathcal{P}$ for a given operad $\mathcal{P}$. Algebras over a cofibrant operad $\mathcal{Q}$ satisfy a version of the Boardman–Vogt homotopy invariance property. Given a cofibrant replacement $\mathcal{Q}\overset{\sim}{\longrightarrow}\mathcal{P}$, we call $\mathcal{Q}$-algebras homotopy $\mathcal{P}$-algebras.

One way to obtain such a cofibrant replacement for an operad is by means of the cobar-bar adjunction $$ \Omega : \textsf{conil dg Coop} \rightleftharpoons \textsf{aug dg Op} : \mathrm{B} . $$ The counit of this adjunction gives functorial cofibrant resolutions $\Omega\mathrm{B}\mathcal{P}\overset{\sim}{\longrightarrow}\mathcal{P}$ for operads $\mathcal{P}$ that are $\mathbb{S}$-cofibrant, i.e. for which $\mathcal{P}(n)$ consist of projective $\mathbf{k}[\mathbb{S}_n]$-modules for all arities $n$ and in all degrees.

Assume for a moment that $\mathbf{k}$ is a field of characteristic $0$. The group algebras $\mathbf{k}[\mathbb{S}_n]$ are then semisimple for all $n$ by Maschke’s theorem, and hence all $\mathbf{k}[\mathbb{S}_n]$-modules are projective. As a result, the cobar-bar resolution $\Omega\mathrm{B}\mathcal{P}\overset{\sim}{\longrightarrow}\mathcal{P}$ is cofibrant. An algebra over this resolution is equivalently a twisting morphism \begin{align} &\alpha\in \operatorname{Hom}_\textsf{$\mathbb{S}$-Mod}(\mathrm{B}\mathcal{P},\operatorname{End}_V)_{-1} , &&\partial\alpha + \alpha\star\alpha = 0 . \end{align} Making this definition explicit in terms of structure maps is a tedious task because of the size of $\mathrm{B}\mathcal{P}$. Furthermore, from an algebraic perspective it is desirable to give a definition of a homotopy $\mathcal{P}$-algebra by a small (possibly minimal) set of structure maps. Operadically, this amounts to replacing the cobar-bar resolution by a smaller cofibrant resolution. This is where Koszul duality comes into play.

Given a dg operad $\mathcal{P}$ by a quadratic presentation, its Koszul dual (dg) cooperad $\mathcal{P}^¡$ is defined by a dual presentation. It is naturally a dg subcooperad $\mathcal{P}^¡\subset\mathrm{B}\mathcal{P}$ of the bar construction on $\mathcal{P}.$ The presentation is called Koszul if the inclusion is a quasi-isomorphism. In this case, the composition $$ \Omega\mathcal{P}^¡ \overset{\sim}{\longrightarrow} \Omega\mathrm{B}\mathcal{P} \overset{\sim}{\longrightarrow} \mathcal{P} $$ provides us with a (typically) much smaller resolution. If $\mathcal{P}^¡$ is $\mathbb{S}$-cofibrant, this Koszul resolution is a cofibrant resolution of $\mathcal{P}$. However, unless $\mathbf{k}$ is a field of characteristic $0$, this condition is generally not satisfied.

To deal with operads over arbitrary unital commutative rings, we developed, in joint work with Bruno Vallette [3], what we called a higher cobar-bar adjunction $$ \widetilde{\Omega} : \textsf{conil higher dg Coop} \rightleftharpoons \textsf{nu dg Op} : \widetilde{\mathrm{B}} . $$ We show that the augmented counit gives functorial cofibrant resolutions $\mathrm{I}\oplus\widetilde{\Omega}\widetilde{\mathrm{B}}\bar{\mathcal{P}}\overset{\sim}{\longrightarrow}\mathcal{P}$ for operads $\mathcal{P}$ that are $\mathbf{k}$-cofibrant, i.e. for which $\mathcal{P}(n)$ consist of projective $\mathbf{k}$-modules for all arities $n$ and in all degrees. In addition, we show that the this resolution is isomorphic to a classical construction: $\mathrm{I}\oplus\widetilde{\Omega}\widetilde{\mathrm{B}}\bar{\mathcal{P}}\cong\Omega\mathrm{B}(\mathcal{P}\otimes\mathcal{E})$. For more details, see my post on symmetric homotopy theory for operads and its references.

Assuming a given operad $\mathcal{P}$ is $\mathbf{k}$-cofibrant, the resolution $\Omega\mathrm{B}(\mathcal{P}\otimes\mathcal{E})\overset{\sim}{\longrightarrow}\mathcal{P}$ is cofibrant. Homotopy $\mathcal{P}$-algebras are then equivalent to twisting morphisms \begin{align} &\alpha\in \operatorname{Hom}_\textsf{$\mathbb{S}$-Mod}(\mathrm{B}(\mathcal{P}\otimes\mathcal{E}),\operatorname{End}_V)_{-1} , &&\partial\alpha + \alpha\star\alpha = 0 . \end{align} Again, it is desirable to reduce $\mathrm{B}(\mathcal{P}\otimes\mathcal{E})$ to a dg subcooperad. In the context of this higher cobar-bar adjunction, however, a Koszul duality approach is not (yet) available.

Homotopy Lie algebras

Our goal in this article is to find a cofibrant resolution for the $\mathrm{Lie}$ operad over any unital commutative ring $\mathbf{k}.$ Now it is a standard result that the operad $\mathrm{Lie}$ admits a Koszul presentation [9, §13.2]. This leads to a small resolution $g_\kappa\colon\mathrm{L}_\infty=\Omega\mathrm{Lie}^¡\overset{\sim}{\longrightarrow}\mathrm{Lie}$ of the $\mathrm{Lie}$ operad, and algebras over this resolution are called $\mathrm{L}_\infty$-algebras. From the introduction we know that when $\mathbf{k}$ is a field of characteristic $0$, this Koszul resolution is cofibrant and the $\mathrm{L}_\infty$-algebras are homotopy Lie algebras, while over an arbitrary unital commutative ring $\mathbf{k}$ the Koszul resolution $\mathrm{L}_\infty$ is not cofibrant.

Our approach in this article is, to construct an $\mathbb{S}$-free resolution $\psi\colon\mathrm{Lie}^\diamond\overset{\sim}{\longrightarrow}\mathrm{Lie}^¡$ of the Koszul dual cooperad for Lie and verify that the composition $$ g_{\kappa\circ\psi} = (g_\kappa\circ\Omega\psi) \colon \Omega\mathrm{Lie}^\diamond \longrightarrow \Omega\mathrm{Lie}^¡ \longrightarrow \mathrm{Lie} $$ is a quasi-isomorphism. Since $\mathrm{Lie}^\diamond$ is $\mathbb{S}$-cofibrant, this is then a cofibrant resolution. Unfortunately, we do not have a method to construct a small and complete $\mathbb{S}$-free resolution for $\mathrm{Lie}^¡$ at this point. Instead, we proceed degree-wise to construct a dg cooperad $\mathrm{Lie}^\diamond_3$ that satisfies the desired conditions in low degrees. By the following remark, this allows us to define the category of algebras at least on underlying $3$-term complexes.

Remark. Let $(V,\mathrm{d})$ be a $k$-term complex (in degrees $0,\dotsc,k-1$.) Its endomorphism operad $\operatorname{End}_V$ is then concentrated in degrees $≤k-1.$ Now assume $\mathcal{C}$ is a differential non-negatively graded cooperad. Since the Maurer–Cartan equation is of degree $-2$, twisting morphisms $\mathcal{C}\to\operatorname{End}_V$ are completely determined by their restriction to the degree $≤k+1$ truncation of $\mathcal{C}$. We denote this truncation by $\mathcal{C}_k:=\tau_{≤k+1}\mathcal{C}.$

Note, however, that, while we use the notation $\mathrm{Lie}^\diamond_3$, we give no proof that it can be extended to a cofibrant dg cooperad $\mathrm{Lie}^\diamond.$

Comparison with Roytenberg’s weak Lie 2-algebras

In [11], Roytenberg introduces a category of weak Lie $2$-algebras using a Baez–Crans style [1] categorification approach. This approach, however, does not extend easily to $3$-term complexes and beyond. While our approach is completely different, we recover Roytenberg’s definitions of weak Lie 2-algebras, their weak morphisms, and his homotopy transfer and skew-symmetrization results when restricting to underlying $3$-term complexes.

The Koszul dual cooperads of the Lie and Leibniz operads

We recall the Koszul dual cooperads $\mathrm{Lie}^¡$ and $\mathrm{Leib}^¡$ of the operads $\mathrm{Lie}$ and $\mathrm{Leib}$, respectively. The history of $\mathrm{L}_\infty$-algebras goes back to at least Drinfeld’s 1988 letter to Schechtman [4], in which they were called dg Sugawara–Lie algebras. Koszul duality of Leibniz algebras was first considered by Loday [8]. For explicit definitions of $\mathrm{L}_\infty$-algebras, $\mathrm{Leib}_\infty$-algebras, and their homotopy morphisms, see e.g. [5,6].

The Koszul dual cooperad for $\mathrm{Lie}$ is the $\mathbb{S}$-module given by $\mathrm{Lie}^¡(0)=0$ and $\mathrm{Lie}^¡(n)=(\bar{l}_n\cdot\mathbf{k}\cdot\mathrm{sgn}_n)[n-1]$ for $n\geq 1$, equipped with the decomposition map \begin{align} \Delta(\bar{l}_n) &= \sum_{\substack{1\leq j\leq n\cr \llap{i_1}+\dotsb+\rlap{i_j=n}}} \varepsilon \cdot \sum_{\sigma\in\bar{\mathrm{Sh}}^{-1}(i_1,\dotsc,i_j)} (-1)^{|\sigma|} \cdot \bar{l}_j \circ (\bar{l}_{i_1},\dotsc,\bar{l}_{i_j})^\sigma , \end{align} where the missing sign is given by $$ \varepsilon = (-1)^{(j-1)(n-j)} \cdot (-1)^{\sum_{p=1}^j(p-1)(i_p-1)} . $$ Leibniz algebras are essentially non-commutative Lie algebras, and the analogue statement is true for $\mathrm{Leib}_\infty$-algebras. The Koszul dual cooperad for $\mathrm{Leib}$ is the $\mathbb{S}$-module defined by $\mathrm{Leib}^¡(0)=0$ and $\mathrm{Leib}^¡(n)=(l_n\cdot\mathbf{k}[\mathbb{S}_n])[n-1]$ for $n\geq 1$, equipped with the decomposition map given as for $\mathrm{Lie}^¡$ but substituting $l_n$ for $\bar{l}_n.$

There is an obvious morphism of dg cooperads \begin{align} &\psi\colon \mathrm{Leib}^¡ \longrightarrow \mathrm{Lie}^¡ , &&\psi(l_n^\sigma) = (-1)^\sigma \cdot \bar{l}_n . \end{align} Since $\mathrm{Leib}^¡$ is $\mathbb{S}$-free and $\psi$ is surjective, this provides us with a good starting point for our resolution of $\mathrm{Lie}^¡.$

Continuing the resolution

In general, the approach is as follows. First we construct an $\mathbb{S}$-free resolution $\mathrm{Lie}^\diamond_k$ of the $\mathbb{S}$-module $\mathrm{Lie}^¡$ up to degree $k+1$, i.e. such that $\operatorname{H}_r(\psi)=0$ for $r≤k.$ In a second step, we equip the dg $\mathbb{S}$-module $\mathrm{Lie}^\diamond_k$ with a decomposition map and show that our definition turns it into a dg cooperad. All computations can be done over the integers, since $\mathbb{Z}$ is the initial object in the category of unital commutative rings.

We construct, for each arity $n≥1$, an exact augmented complex $0\leftarrow\mathrm{Lie}^¡(n)_{n-1}\leftarrow\mathrm{Lie}^\diamond_k(n)_\bullet$ in $\mathbf{k}[\mathbb{S}_n]$-modules up to degree $k+1.$ As indicated in the previous section, we choose $\mathrm{Lie}^\diamond_k(n)_{n-1}:=\mathrm{Leib}^¡(n)_{n-1}$ and the augmentation map to be $\psi$, i.e. our complexes are of the following shape: \begin{align} 0 \longleftarrow \mathrm{Lie}^¡(n)_{n-1} \overset{\psi}{\longleftarrow} \mathrm{Leib}^¡(n)_{n-1} \overset{\mathrm{d}_n}{\longleftarrow} \mathrm{Lie}^\diamond_k(n)_n \overset{\mathrm{d}_{n+1}}{\longleftarrow} \dots \overset{\mathrm{d}_{k+1}}{\longleftarrow} \mathrm{Lie}^\diamond_k(n)_{k+1} . \end{align}

The usual approach to constructing free resolution of modules applies, i.e. we construct the higher degrees of the complex $\mathrm{Lie}^\diamond_k(n)_\bullet$ by successively applying the following steps for $r=n-1,\dotsc,k$:

  1. compute $\ker(\mathrm{d}_r),$
  2. choose generators $\{x_i\}$ for $\ker(\mathrm{d}_r)$ as a $\mathbf{k}[\mathbb{S}_n]$-module, and
  3. define $\mathrm{Lie}^\diamond_k(n)_{r+1}:=\langle\hat{x}_i\rangle$ to be the free $\mathbf{k}[\mathbb{S}_n]$-module generated by these symbols and the differential by $\mathrm{d}_{r+1}(\hat{x}_i):=x_i.$

To turn the $\mathbb{S}$-module $\mathrm{Lie}^\diamond_k$ into a dg cooperad, we equip it with a decomposition map $\Delta$ as follows. For $r=n-1$ we define the decomposition to be as for $\mathrm{Leib}^¡$. Now note that for any decomposition map, this diagram must commute: $$ \require{AMScd} \begin{CD} \mathrm{Lie}^\diamond_k(n)_{r+1} @>{\Delta}>> (\mathrm{Lie}^\diamond_k\circ\mathrm{Lie}^\diamond_k)(n)_{r+1} \cr @V{\mathrm{d}}VV @V{\mathrm{d}}VV \cr \mathrm{Lie}^\diamond_k(n)_r @>{\Delta}>> (\mathrm{Lie}^\diamond_k\circ\mathrm{Lie}^\diamond_k)(n)_r . \end{CD} $$ We use this condition as follows: in each arity $n$, proceed degree-wise for $r=n-1,\dotsc,k+1$. For a $\mathbf{k}[\mathbb{S}_n]$-basis $\{x_i\}$ for $\mathrm{Lie}^\diamond_k(n)_{r+1},$ solve the equations $\mathrm{d}(y_i)=\Delta(\mathrm{d}x_i)$ for $y_i$ and define $\Delta$ by $\Delta(x_i):=y_i.$ Since we work over $\mathbb{Z}$, this amounts to solving systems of linear diophantine equations. Finally, it remains to check that the decomposition map defined in this way is in fact coassociative. This is not automatic from the approach we use.

We make the approach described above explicit for $k=3.$

The twisted composite product

Assume we had a full $\mathbb{S}$-free resolution $\psi\colon\mathrm{Lie}^\diamond\to\mathrm{Lie}^¡.$ Its composition $\kappa\circ\psi$ with the canonical Koszul twisting morphism $\kappa\colon\mathrm{Lie}^¡\to\mathrm{Lie}$ is again a twisting morphism, and it corresponds to the morphism $$ g_{\kappa\circ\psi} = (g_\kappa\circ\Omega\psi) \colon \Omega\mathrm{Lie}^\diamond \longrightarrow \Omega\mathrm{Lie}^¡ \longrightarrow \mathrm{Lie} $$ of dg operads. One way to show that $g_{\kappa\circ\psi}$ is a quasi-isomorphism is by proving acyclicity of the twisted composite product $\mathrm{Lie}^\diamond\circ_{\kappa\circ\psi}\mathrm{Lie}.$ Since we only have a resolution $\psi\colon\mathrm{Lie}^\diamond_3\to\mathrm{Lie}^¡$ in low degrees, we show the following truncated statement. This is, of course, a neccessary condition if we hope to extend our dg cooperad $\mathrm{Lie}^\diamond_3$.

Proposition. The twisted composite product $\mathrm{Lie}^\diamond_3\circ_{\kappa\circ\psi}\mathrm{Lie}$ satisfies $$ \operatorname{H}_r\big( (\mathrm{Lie}^\diamond_3\circ_{\kappa\circ\psi}\mathrm{Lie})(n) \big) = 0 , $$ for all $r≤3$ in all arities $n.$

Category of weak Lie 3-algebras

We give explicit definitions for $3$-term $\mathrm{EL}_\infty$-algebras a.k.a. weak Lie $3$-algebras. Since $\mathrm{Leib}^¡_3\subset\mathrm{Lie}^\diamond_3$, weak Lie $3$-algebras and their homotopy morphisms are particular $3$-term $\mathrm{Leib}_\infty$-algebras and homotopy morphisms of such.

Weak Lie $3$-algebras

A weak Lie $3$-algebra structure on a $3$-term complex $(L,\mathrm{d})$ is a twisting morphism [9, §6.4f.], i.e. a degree $-1$ solution to the Maurer–Cartan equation: \begin{align} &\lambda\colon \mathrm{Lie}^\diamond_3 \to \mathrm{End}_L , &&\partial(\lambda) + \lambda\star\lambda = 0 . \end{align} Restricting $\lambda$ to $\mathrm{Leib}^¡_3\subset\mathrm{Lie}^\diamond_3$, we obtain the definition of Leibniz $3$-algebras. We make both definitions explicit below by evaluating the Maurer–Cartan equation on generators, using the notation $\lambda_*=\lambda(l_*).$

Definition. A Leibniz $3$-algebra $(L,\mathrm{d},\lambda)$ consists of a $3$-term chain complex $$ (L,\mathrm{d}) = L_0 \overset{\mathrm{d}}{\longleftarrow} L_1 \overset{\mathrm{d}}{\longleftarrow} L_2 , $$ equipped with structure maps \begin{align} &\lambda_2\colon L^{\otimes 2} \longrightarrow L , &&\lambda_3\colon L^{\otimes 3} \longrightarrow L[1] , &&\lambda_4\colon L^{\otimes 4} \longrightarrow L[2] , \end{align} satisfying the following generalized Jacobi identities: \begin{align} \partial(\lambda_2) &= 0, \cr \partial(\lambda_3) &= - \lambda_2\circ_1\lambda_2 + \lambda_2\circ_2\lambda_2 - (\lambda_2\circ_2\lambda_2)^{(12)} , \cr \partial(\lambda_4) &= \lambda_2\circ_1\lambda_3 + \lambda_2\circ_2\lambda_3 - (\lambda_2\circ_2\lambda_3)^{(12)} + (\lambda_2\circ_2\lambda_3)^{(123)} \cr &\quad- \lambda_3\circ_1\lambda_2 + \lambda_3\circ_2\lambda_2 - (\lambda_3\circ_2\lambda_2)^{(12)} \cr &\quad- \lambda_3\circ_3\lambda_2 + (\lambda_3\circ_3\lambda_2)^{(23)} - (\lambda_3\circ_3\lambda_2)^{(132)} , \cr 0 &= \lambda_2\circ_1\lambda_4 - \lambda_2\circ_2\lambda_4 + (\lambda_2\circ_2\lambda_4)^{(12)} - (\lambda_2\circ_2\lambda_4)^{(123)} \cr &\quad+ (\lambda_2\circ_2\lambda_4)^{(1234)} + \lambda_3\circ_1\lambda_3 + \lambda_3\circ_2\lambda_3 - (\lambda_3\circ_2\lambda_3)^{(12)} \cr &\quad+ (\lambda_3\circ_2\lambda_3)^{(123)} + \lambda_3\circ_3\lambda_3 - (\lambda_3\circ_3\lambda_3)^{(23)} + (\lambda_3\circ_3\lambda_3)^{(132)} \cr &\quad+ (\lambda_3\circ_3\lambda_3)^{(234)} - (\lambda_3\circ_3\lambda_3)^{(1342)} + (\lambda_3\circ_3\lambda_3)^{(13)(24)} \cr &\quad+ \lambda_4\circ_1\lambda_2 - \lambda_4\circ_2\lambda_2 + (\lambda_4\circ_2\lambda_2)^{(12)} + \lambda_4\circ_3\lambda_2 \cr &\quad- (\lambda_4\circ_3\lambda_2)^{(23)} + (\lambda_4\circ_3\lambda_2)^{(132)} - \lambda_4\circ_4\lambda_2 \cr &\quad+ (\lambda_4\circ_4\lambda_2)^{(34)} - (\lambda_4\circ_4\lambda_2)^{(243)} + (\lambda_4\circ_4\lambda_2)^{(1432)} . \end{align}

Definition. A weak Lie $3$-algebra $(L,\mathrm{d},\lambda)$ is a Leibniz $3$-algebra equipped with additional structure maps \begin{align} \lambda_{2;1}&\colon L^{\otimes 2} \longrightarrow L[1] , &\lambda_{2;1,1}&\colon L^{\otimes 2} \longrightarrow L[2] , \cr & &\lambda_{3;1}&\colon L^{\otimes 3} \longrightarrow L[2] , \cr & &\lambda_{3;2}&\colon L^{\otimes 3} \longrightarrow L[2] . \end{align} We require these to satisfy the following identities: \begin{align} \partial(\lambda_{2;1}) &= \lambda_2 + \lambda_2^{(12)} , \cr \partial(\lambda_{2;1,1}) &= \lambda_{2;1} - \lambda_{2;1}^{(12)} , \cr \partial(\lambda_{3;1}) &= \lambda_3 + \lambda_3^{(12)} + \lambda_2\circ_1\lambda_{2;1} , \cr \partial(\lambda_{3;2}) &= \lambda_3 + \lambda_3^{(23)} + \lambda_{2;1}\circ_1\lambda_2 - \lambda_2\circ_2\lambda_{2;1} + (\lambda_{2;1}\circ_2\lambda_2)^{(12)} , \cr \lambda_{2;1,1} + \lambda_{2;1,1}^{(12)} &= 0 , \cr \lambda_{3;1} - \lambda_{3;1}^{(12)} &= \lambda_2\circ_1\lambda_{2;1,1} , \cr \lambda_{3;1} - \lambda_{3;2}^{(12)} & + \lambda_{3;1}^{(132)} - \lambda_{3;2} + \lambda_{3;1}^{(23)} - \lambda_{3;2}^{(123)} \cr &= \lambda_{2;1}\circ_1\lambda_{2;1} + \lambda_{2;1}\circ_2\lambda_{2;1} + (\lambda_{2;1}\circ_2\lambda_{2;1})^{(12)} \cr &\quad+ (\lambda_{2;1,1}\circ_2\lambda_2)^{(132)} , \cr \lambda_{3;2} - \lambda_{3;2}^{(23)} &= \lambda_{2;1,1}\circ_1\lambda_2 - \lambda_2\circ_2\lambda_{2;1,1} + (\lambda_{2;1,1}\circ_2\lambda_2)^{(12)} , \cr \lambda_4 + \lambda_4^{(12)} &= \lambda_2\circ_1\lambda_{3;1} + (\lambda_2\circ_2\lambda_{3;1})^{(123)} + \lambda_3\circ_1\lambda_{2;1} - \lambda_{3;1}\circ_3\lambda_2 , \cr \lambda_4 + \lambda_4^{(23)} &= \lambda_2\circ_1\lambda_{3;2} + \lambda_2\circ_2\lambda_{3;1} - \lambda_{3;1}\circ_1\lambda_2 \cr &\quad- \lambda_3\circ_2\lambda_{2;1} - (\lambda_{3;1}\circ_2\lambda_2)^{(12)} - (\lambda_{3;1}\circ_3\lambda_2)^{(132)} , \cr \lambda_4 + \lambda_4^{(34)} &= \lambda_{2;1}\circ_1\lambda_3 + \lambda_2\circ_2\lambda_{3;2} - (\lambda_2\circ_2\lambda_{3;2})^{(12)} + (\lambda_{2;1}\circ_2\lambda_3)^{(123)} \cr &\quad- \lambda_{3;2}\circ_1\lambda_2 + \lambda_{3;2}\circ_2\lambda_2 - (\lambda_{3;2}\circ_2\lambda_2)^{(12)} \cr &\quad+ \lambda_3\circ_3\lambda_{2;1} + (\lambda_{3;2}\circ_3\lambda_2)^{(23)} - (\lambda_{3;2}\circ_3\lambda_2)^{(132)} . \end{align}

Homotopy morphisms of weak Lie $3$-algebras

A homotopy morphism of weak Lie $3$-algebras $f\colon(L,\mathrm{d},\lambda)\to(L’,\mathrm{d’},\lambda’)$ is a kind of twisting morphism, specifically a degree $0$ solution to the Maurer–Cartan equation: \begin{align} &f\colon \mathrm{Lie}^\diamond_3 \to \mathrm{End}^L_{L’} , &&\partial(f) - f\ast\lambda + \lambda’\circledast f = 0. \end{align} Restricting $f$ in the definition to $\mathrm{Leib}^¡_3\subset\mathrm{Lie}^\diamond_3$, we obtain the definition of homotopy morphisms for Leibniz $3$-algebras. We make both definitions explicit below, again by evaluating the Maurer–Cartan equation on generators using the notation $f_*=f(l_*).$

Definition. A weak morphism or homotopy morphism of Leibniz $3$-algebras $(L,\mathrm{d},\lambda)\to(L’,\mathrm{d’},\lambda’)$ consists of maps \begin{align} &f_1\colon L \to L’ , &&f_2\colon L^{\otimes 2} \to L’[1] , &&f_3\colon L^{\otimes 3} \to L’[2] , \end{align} satisfying the equations \begin{align} \partial(f_1) &= 0 , \cr \partial(f_2) &= f_1\circ\lambda_2 - \lambda’_2\circ(f_1,f_1) , \cr \partial(f_3) &= f_1\circ\lambda_3 - f_2\circ_2\lambda_2 + f_2\circ_1\lambda_2 + (f_2\circ_2\lambda_2)^{(12)} \cr &\quad- \lambda’_3\circ(f_1,f_1,f_1) - \lambda’_2\circ(f_1,f_2) + \lambda’_2\circ(f_2,f_1) \cr &\quad+ \lambda’_2\circ(f_1,f_2)^{(12)} , \cr \hspace{2em}&\hspace{-2em} f_1\circ\lambda_4 - \lambda’_4\circ(f_1,f_1,f_1,f_1) \cr &= f_2\circ_1\lambda_3 + f_2\circ_2\lambda_3 - (f_2\circ_2\lambda_3)^{(12)} + (f_2\circ_2\lambda_3)^{(123)} \cr &\quad- f_3\circ_1\lambda_2 + f_3\circ_2\lambda_2 - (f_3\circ_2\lambda_2)^{(12)} - f_3\circ_3\lambda_2 \cr &\quad+ (f_3\circ_3\lambda_2)^{(23)} - (f_3\circ_3\lambda_2)^{(132)} + \lambda’_2\circ(f_3,f_1) + \lambda’_2\circ(f_1,f_3) \cr &\quad- \lambda’_2\circ(f_1,f_3)^{(12)} + \lambda’_2\circ(f_1,f_3)^{(123)} - \lambda’_2\circ(f_2,f_2) \cr &\quad+ \lambda’_2\circ(f_2,f_2)^{(23)} - \lambda’_2\circ(f_2,f_2)^{(132)} + \lambda’_3\circ(f_2,f_1,f_1) \cr &\quad- \lambda’_3\circ(f_1,f_2,f_1) + \lambda’_3\circ(f_1,f_2,f_1)^{(12)} + \lambda’_3\circ(f_1,f_1,f_2) \cr &\quad- \lambda’_3\circ(f_1,f_1,f_2)^{(23)} + \lambda’_3\circ(f_1,f_1,f_2)^{(132)} . \end{align} Let $L\overset{f}{\rightarrow}L’\overset{f’}{\rightarrow}L^{\prime\prime}$ be weak morphisms of Leibniz $3$-algebras. Their composition is defined by the following components: \begin{align} (f’\circ f)_1 &:= f’_1\circ f_1 , \cr (f’\circ f)_2 &:= f’_2\circ(f_1,f_1) + f’_1\circ f_2 , \cr (f’\circ f)_3 &:= f’_3\circ(f_1,f_1,f_1) - f’_2\circ(f_2,f_1) + f’_2\circ(f_1,f_2) \cr &\quad- f’_2\circ(f_1,f_2)^{(12)} + f’_1\circ f_3 . \end{align}

Definition. A weak morphism or homotopy morphism of weak Lie $3$-algebras $(L,\mathrm{d},\lambda)\to(L’,\mathrm{d’},\lambda’)$ is a homotopy morphism of Leibniz $3$-algebras with an additional structure map $$ f_{2;1}\colon L^{\otimes 2} \to L’[2] , $$ satisfying the following equations: \begin{align} \partial(f_{2;1}) &= - f_2 - f_2^{(12)} + f_1\circ\lambda_{2;1} - \lambda’_{2;1}\circ(f_1,f_1) , \cr f_1\circ\lambda_{2;1,1} - \lambda’_{2;1,1}\circ(f_1,f_1) &= f_{2;1} - f_{2;1}^{(12)} , \cr f_1\circ\lambda_{3;1} - \lambda’_{3;1}\circ(f_1,f_1,f_1) &= f_3 + f_3^{(12)} + f_2\circ_1\lambda_{2;1} + \lambda’_{2}\circ(f_{2;1},f_1) , \cr f_1\circ\lambda_{3;2} - \lambda’_{3;2}\circ(f_1,f_1,f_1) &= f_3 + f_3^{(23)} - f_2\circ_2\lambda_{2;1} + f_{2;1}\circ_1\lambda_{2} \cr &\quad+ (f_{2;1}\circ_2\lambda_{2})^{(12)} - \lambda’_{2}\circ(f_1,f_{2;1}) \cr &\quad- \lambda’_{2;1}\circ(f_2,f_1) - \lambda’_{2;1}\circ(f_1,f_2)^{(12)} . \end{align} Let $L\overset{f}{\rightarrow}L’\overset{f’}{\rightarrow}L^{\prime\prime}$ be weak morphisms of weak Lie $3$-algebras. Their composition is defined as the composition of the underlying weak morphisms of Leibniz $3$-algebras, with the additional structure map given by $$ (f’\circ f)_{2;1} := f’_{2;1}\circ(f_1,f_1) + f’_1\circ f_{2;1} . $$

Homotopy transfer for weak Lie $3$-algebras

While we do not have a full cofibrant resolution of the Lie operad—and therefore no automatic homotopy transfer result—we prove the following version of the homotopy transfer theorem by explicit construction.

Theorem (HTT). Let $(L,\mathrm{d},\lambda)$ be a weak Lie $3$-algebra and let $(L’,\mathrm{d’})$ be a deformation retract of $(L,\mathrm{d}),$ i.e. $$ h \circlearrowright (L,\mathrm{d}) \underset{i}{\overset{p}{\rightleftarrows}} (L’,\mathrm{d’}), \quad \text{such that } \begin{cases} \mathrm{id}_L - i\circ p = [\mathrm{d},h] , \cr \mathrm{id}_{L’} - p\circ i = 0 . \end{cases} $$ Then $(L’,\mathrm{d’})$ can be equipped with a transferred weak Lie $3$-algebra structure in such a way, that the map $i$ admits an extension to a weak morphism of weak Lie $3$-algebras.

Proof. We construct the transferred structure $\lambda’$ and the extension of $i$ explicitly, following the same pattern as decribed in [9, §10.3] for the transfer of $\Omega\mathcal{P}_\infty$-algebras.

Skew-symmetrization

Since $\mathrm{L}_\infty$-algebras are, by definition, $\mathrm{Leib}_\infty$-algebras with fully skew-symmetric structure maps, it seems natural to try and construct $\mathrm{L}_\infty$-algebras by skew-symmetrizing the structure maps of $\mathrm{Leib}_\infty$-algebras. In operadic terms, this means we’d like to construct a morphism $\phi$ as in \begin{align} & \psi: \mathrm{Leib}^¡ \rightleftarrows \mathrm{Lie}^¡ :\phi , &\text{such that $\psi\circ\phi=\mathrm{id}$.} \end{align} Assume for a moment that we had such a morphism. In this case, given any $\mathrm{Leib}_\infty$-algebra $(L,\mathrm{d},\lambda)$, we obtain an $\mathrm{L}_\infty$-algebra $(L,\mathrm{d},\bar{\lambda})$ by precomposing the corresponding twisting morphism with $\phi$, i.e. $\bar{\lambda}:=\lambda\circ\phi.$

For a naive attempt at defining such a morphism $\phi,$ we let $$ \phi(\bar{l}_n) := \frac{1}{n!} \sum_{\sigma\in\mathbb{S}_n} (-1)^{|\sigma|} \cdot l_n^\sigma . $$ Provided that the $1/n!$ exist, this gives a well-defined morphism of dg $\mathbb{S}$-modules and satisfies $\psi\circ\phi=\mathrm{id}.$ Now a simple computation shows that the map $\phi$ defined in this way does not commute with the decomposition map $\Delta.$ However, when viewing $\mathrm{Leib}^¡\subset\mathrm{Lie}^\diamond$ as a dg subcooperad, the defect is actually a coboundary \begin{align} (\Delta\circ\phi-\phi\circ\Delta)(\bar{l}_3) &= \frac{1}{12} \sum_{\sigma\in\mathbb{S}_3} (-1)^{|\sigma|} (l_2\circ(l_2,1)+l_2\circ(1,l_2))^\sigma \cr &= \frac{1}{12} \sum_{\sigma\in\mathbb{S}_3} (-1)^{|\sigma|} \mathrm{d}l_{2;1}\circ(l_2,1)^\sigma . \end{align} This shows that while homotopy Leibniz algebras in general do not admit a skew-symmetrization functor, for weak Lie $3$-algebras such a construction may exist. It also shows that $\phi$ as defined above is not enough and we should extend it by higher degree terms to obtain a homotopy morphism of dg cooperads. We give an explicit construction of such a morphism $\Phi$ and thereby prove the following result.

Lemma. The morphism $\Omega\psi$ admits a right inverse, i.e. a morphism $\Phi$ of dg operads \begin{align} &\Omega\psi : \Omega\mathrm{Lie}^\diamond_3 \rightleftarrows \Omega\mathrm{Lie}^¡_3 : \Phi , &\text{such that $\Omega\psi\circ\Phi=\mathrm{id}.$} \end{align}

Given a weak Lie $3$-algebra $g_\lambda\colon\Omega\mathrm{Lie}^\diamond_3\to\mathrm{End}_L$, the above lemma can be used to construct a (semi-strict) Lie $3$-algebra by precomposition with $\Phi$, i.e. $$ g_\bar{\lambda} = \left( \Omega\mathrm{Lie}^¡_3 \overset{\Phi}{\longrightarrow} \Omega\mathrm{Lie}^\diamond_3 \overset{g_\lambda}{\longrightarrow} \mathrm{End}_L \right) . $$ Explicitly, this lead to the following definition.

Definition. Let $(L,\mathrm{d},\lambda)$ be a weak Lie $3$-algebra. We define its skew-symmetrization to be the (semi-strict) Lie $3$-algebra $(L,\mathrm{d},\bar{\lambda})$ given by the following structure maps: \begin{align} \bar{\lambda}_2 &:= \frac{1}{2} \sum_{\sigma\in\mathbb{S}_2} (-1)^{|\sigma|} \cdot \lambda_2^\sigma , \cr \bar{\lambda}_3 &:= \frac{1}{6} \sum_{\sigma\in\mathbb{S}_3} (-1)^{|\sigma|} \cdot \lambda_3^\sigma - \frac{1}{24} \sum_{\sigma\in\mathbb{S}_3} (-1)^{|\sigma|} \cdot (\lambda_{2;1}\circ_1\lambda_2 + \lambda_{2;1}\circ_2\lambda_2)^\sigma , \cr \bar{\lambda}_4 &:= \frac{1}{24} \sum_{\sigma\in\mathbb{S}_4} (-1)^{|\sigma|} \cdot \lambda_4^\sigma \cr &\quad+ \frac{1}{48} \sum_{\sigma\in\mathbb{S}_4} (-1)^{|\sigma|} \cdot \left(\begin{aligned} \lambda_{2;1}\circ_1\lambda_3 - \lambda_{3;1}\circ_1\lambda_2 + \lambda_{3;2}\circ_2\lambda_2 \cr - \lambda_{2;1}\circ_2\lambda_3 - \lambda_{3;1}\circ_2\lambda_2 + \lambda_{3;2}\circ_3\lambda_2 \end{aligned}\right)^\sigma . \end{align}

The situation for weak morphisms $(L,\mathrm{d},\lambda)\to(L’,\mathrm{d’},\lambda’)$ of weak Lie $3$-algebras is slightly more complicated: such a morphism is easiest described by a type of twisting morphism $\mathrm{Lie}^¡_3\to\mathrm{End}^L_{L’}.$ There is, however, no clear way to pull such a twisting morphism back along $\Phi.$ Instead, we give an ad hoc definition of skew-symmetrization for morphisms of weak Lie $3$-algebras and prove that—while not strictly functorial—it is in a sense functorial up to homotopy.

Applications

In the final section of this article, we discuss two applications of our theory.

Higher symplectic geometry

In [2,10], the notion of an $n$-plectic manifold $(M,\omega)$ was introduced as a smooth manifold $M$ with a closed, non-degenerate $(n+1)$-form $\omega\in\Omega^{n+1}(M).$ An $(n-1)$-form $\alpha\in\Omega^{n-1}(M)$ is called Hamiltonian, if there exists a vector field $v_\alpha\in\mathfrak{X}(M)$ such that $\mathrm{d}\alpha=-\iota(v_\alpha)\omega.$

Now given such an $n$-plectic manifold $(M,\omega),$ one can consider the truncated de Rham complex $$ \Omega^{n-1}_\mathrm{Ham}(M) \longleftarrow \dots \longleftarrow \Omega^1(M) \longleftarrow \Omega^0(M) = C^\infty(M) , $$ with the grading such that Hamiltonian forms are of degree $0$ and functions are of degree $n-1.$ In [2,10], two algebraic structures were introduced on this complex: an $\mathrm{L}_\infty$-algebra $\mathrm{L}_\infty(M,\omega)$ and a (dg) Leibniz algebra $\mathrm{Leib}(M,\omega).$ The bracket of the Leibniz algebra $\mathrm{Leib}(M,\omega)$ is shown to be skew-symmetric up to an exact form.

In the case of $3$-plectic manifolds, $\mathrm{Leib}(M,\omega)$ is actually a weak Lie $3$-algebra. We prove the following result.

Proposition. Let $(M,\omega)$ be a $3$-plectic manifold. Then $\mathrm{L}_\infty(M,\omega)$ and $\mathrm{Leib}(M,\omega)$ are isomorphic as weak Lie $3$-algebras.

This is the obvious extension of the analogous result for $2$-plectic manifolds which was shown by Rogers [10].

Higher Courant algebroids

In [7], Liu–Sheng introduce the notion of an LWX $2$-algebroid as a higher analogue of a Courant algebroid and construct a (semi-strict) Lie $3$-algebra for any LWX 2-algebroid a [7, Theorem 3.10]. In this article, I give a construction of a weak Lie $3$-algebra for any LWX $2$-algebroid and show that their semi-strict Lie $3$-algebra is in fact the skew-symmetrization of this weak Lie $3$-algebra.

Consider a $2$-term dg vector bundle $E=(E_0\overset{\partial}{\leftarrow}E_1)$ over a base manifold $M$ equipped with a morphism $\rho\colon{}E\to{}TM$ of dg vector bundles, a graded bilinear map $(␣\circ␣)\colon\Gamma{}E\otimes\Gamma{}E\to\Gamma{}E$ which is skew-symmetric on $\Gamma{}E_0\otimes\Gamma{}E_0$, a graded $3$-form $\Omega\colon\Gamma{}E\otimes\Gamma{}E\otimes\Gamma{}E\to\Gamma{}E[1],$ and a non-degenerate symmetric bilinear form $S\colon\Gamma{}E\otimes\Gamma{}E\to{}C^\infty(M).$ Using these data, one defines a map $\mathcal{D}\colon{}C^\infty(M)\to\Gamma{}E_1$ by requiring $S(e,\mathcal{D}f)=\rho(e)(f),$ for all $e\in\Gamma{}E.$ The tuple $\mathcal{E}=(E,\rho,\circ,\Omega,S)$ is called an LWX $2$-algebroid if it satisfies a number of technical conditions which we omit here.

Proposition. For any LWX $2$-algebroid, there is an associated complex $$ (L,\mathrm{d}) := \left( \Gamma E_0 \overset{\partial}{\longleftarrow} \Gamma E_1 \overset{\mathcal{D}}{\longleftarrow} C^\infty(M) \right) , $$ which, when equipped with structure maps \begin{align} \lambda_2 &= (␣\circ␣) + S\circ_2\mathcal{D} , &\lambda_{2;1} &= S , &\lambda_{2;1,1} &= 0 , \cr & &\lambda_3 &= \Omega , &\lambda_{3;1} &= 0 , \cr &&& &\lambda_{3;2} &= 0 , \cr &&& &\lambda_4 &= 0 , \cr \end{align} defines a weak Lie $3$-algebra.

Applying our skew-symmetrization construction for weak Lie $3$-algebras to this particular structure, we obtain the (semi-strict) Lie $3$-algebra defined in [7, Theorem 3.10], as promised.

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