CHAPTER: TENSOR HIERARCHIES AS HIGHER GAUGE ALGEBRAS

\(E_2L_\infty\)-algebras

§ REVIEW OF TENSOR HIERARCHIES

Before analyzing the algebraic structure underlying tensor hierarchies in more detail, let us briefly review the physical context. Consider the Lagrangian of ungauged Einstein–Maxwell-scalar theory in $d$ dimensions,

\begin{equation}\label{Lth}
\mathcal{L}_{\text{ungauged}} = \star R +\tfrac{1}{2} g_{xy} \rmd \varphi^x \wedge \star \rmd \varphi^y  -\tfrac{1}{2} a_{ij} F^{i} \wedge \star F^{j} +\cdots
\end{equation}

with scalars $\varphi^x$ mapping spacetime to a scalar manifold $\caM$ and 1-form abelian gauge potentials $A^i$ with field strengths $F^{i}=\rmd A^{i}$. Here, $g_{xy}(\varphi)$ and $a_{ij}(\varphi)$ are symmetric and positive-definite on the entire scalar manifold $\caM$. The ellipsis denotes possible deformations, such as a scalar potential $\mathcal{V}(\varphi)$ or topological terms such as, e.g., ${d}_{ijk}F^{i} \wedge  F^{j}\wedge A^{k}$ for $d=5$, as familiar from supergravity. The set of constant tensors controlling these deformation terms, which includes those appearing in the tensor hierarchies that do not enter (<ref>), will be referred to as deformation tensors.

We assume that there is a global symmetry group $\sfG$ acting on scalars and 1-form potentials such that the undeformed action (<ref>) is invariant under this action. In particular, the total gauge potential one-form $A$ takes value in a representation $V_{-1}$ of $\sfG$, which is isomorphic to the fibers of the tangent space of the scalar manifold. In the presence of deformations, we assume that there is a non-abelian subgroup $\sfK\subseteq \sfG$ leaving the full action invariant.

Infinitesimally, the corresponding Lie algebra actions of $\frg=\sfLie(\sfG)$ on the scalars and the gauge potential are given by

\begin{equation}
\delta_\lambda \varphi^x = \lambda^{\alpha} k_{\alpha}{}^{x}(\varphi)~,~~~ \delta_\lambda A^{i} =  \lambda^{\alpha} t_{\alpha}{}^{i}{}_{j} A^{j}~,
\end{equation}

where  $t_{\alpha}{}^{i}{}_{j}, \alpha=1,2,\ldots,\dim \frg$, are the generators of $\frg$ in the representation $V_{-1}$ with respect to some bases $e_\alpha$ of $\frg$ and $e_i$ of $V_{-1}$. Invariance under $\sfG$ requires that $k(\phi)$ be  Killing vectors of the scalar manifold and $\mathcal{L}_{k_\alpha}a_{ij}=-2\ t_{\alpha}{}^{k}{}_{(i} a_{j)k}$.

In order to gauge[That is, we promote a global symmetry $\sfH$ to a local one by adding a principal $\sfH$-bundle on our spacetime and consider (a part of) the one-form potential as a connection on this bundle.] a subgroup $\sfH\subseteq \sfK\subseteq \sfG$ with Lie algebra $\frh=\sfLie(\sfH)$, we first note that we can trivially regard the pair $(V_{-1},\frg$) as a differential graded Lie algebra

\begin{equation}\label{eq:trivialdgLA}
\end{equation}

with evident Lie bracket on $V_0$ and the Lie bracket $[-,-]:V_{0}\times V_{-1}\rightarrow V_{-1}$ given by the action of $\frg$ on $V_{-1}$. Because the gauge potential takes values in $V_{-1}$, it does not make sense to gauge a Lie subalgebra of $\frg$ which is larger than $V_{-1}$. Therefore, we can identify the subalgebra $\frh$ with the image of a linear map

\begin{equation}
\begin{aligned}
\Theta: V_{-1}&\twoheadrightarrow \frh~~~~~~\subset \frg~,
\\
e_i&\mapsto \Theta_i{}^\alpha e_\alpha~.
\end{aligned}
\end{equation}

The $(\Theta_{i}{}^{\alpha} e_\alpha)$ then form a spanning set[but not necessarily a basis] of the Lie algebra $\frh$. Moreover, we have an induced action of $\frh$ on $V_{-1}$, given by

\begin{equation}
(\Theta_{i}{}^{\alpha} e_\alpha) \acton e_j=\Theta_{i}{}^{\alpha} t_{\alpha j}{}^k e_k\eqqcolon X_{ij}{}^k e_k
\end{equation}

\begin{equation}
t_{\alpha j}{}^k=-t_{\alpha}{}^k{}_j\eand t_{\alpha i}{}^jt_{\beta j}{}^k-t_{\beta i}{}^jt_{\alpha j}{}^k=-f_{\alpha\beta}{}^\gamma t_{\gamma i}{}^k~.
\end{equation}

In order to guarantee closure of the Lie bracket on $\frh$ and consistency of the action, we can assume that we can incorporate $\Theta$ into (<ref>) such that

\begin{equation}\label{eq:ThetadgLA}
\end{equation}

is again a differential graded Lie algebra. To jump ahead of the story, note that this guarantees the existence of a higher gauge algebra via <ref>, which we anticipate as part of the construction of a higher gauge theory. The fact that $\Theta$ is a derivation for the graded Lie bracket then implies the quadratic closure constraint

\begin{equation}\label{eq:quad_closure}
f_{\alpha\beta}{}^{\gamma}\Theta^{\alpha}_{i}\Theta^{\beta}_{j}=\Theta^{\gamma}_{k}X_{ij}{}^{k}~~~\Leftrightarrow~~X_{im}{}^\ell X_{j\ell}{}^n-X_{jm}{}^\ell X_{i\ell}{}^n=-X_{ij}{}^\ell X_{\ell m}{}^n~.
\end{equation}

It is well known that this quadratic closure constraint implies that the $X_{ij}{}^k$ form the structure constants of a Leibniz algebra on $V_{-1}$,

\begin{equation}\label{eq:action_V_1_V_1}
e_i\circ e_j\coloneqq X_{ij}{}^k e_k~,
\end{equation}

cf. e.g. <cit.>. This is unsatisfactory given that the 1-form gauge potentials $A$ will take values in $V_{-1}$ and $V_{-1}$ should therefore have some Lie structure. As noted in <cit.>, and as evident from <ref>, this Leibniz algebra can be promoted to an $\ophLie$-algebra. Moreover, the fact that we have the differential graded algebra (<ref>) guarantees that we will have an $\ophLie$-algebra via <ref> (or, if preferred, the corresponding $L_\infty$-algebra obtained from <ref>). This will turn out to be indeed the higher gauge algebra underlying the tensor hierarchies.

But let us continue with the tensor hierarchy from the physicists' perspective. The quadratic closure constraint (<ref>) allows us to introduce a consistent combination of a covariant derivative on the scalar fields and local transformations parameterised by $\Lambda_{(0)}\in C^\infty(M)\otimes V_{-1}$:

\begin{equation}\label{eq:first_gauge}
\begin{gathered}
\nabla\varphi^i\coloneqq  \rmd \varphi^i + \Theta^{\alpha}_{j} A^{j} k_{\alpha}{}^{i}(\varphi)~,
\\
\delta_{\Lambda_{(0)}} \varphi^i\coloneqq  {\Lambda_{(0)}}^{i}\Theta^{\alpha}{}_{j}  k_{\alpha}{}^{j}(\varphi)~,~~~
\delta_{\Lambda_{(0)}} A^i\coloneqq \rmd {\Lambda_{(0)}}^i+X_{jk}{}^iA^j{\Lambda_{(0)}}^k~.
\end{gathered}
\end{equation}

Note that the action (<ref>) of $V_{-1}$ on $V_{-1}$ is usually not faithful, and the parameterisation by $\Lambda^i$ is thus usually highly degenerate.

In light of our above discussion of the higher Lie algebra arising from the Leibniz algebra (<ref>), it is not surprising that the naive gauge transformation (<ref>) of the gauge potential $A$ does not render the naive definition of curvature $\rmd A^{i}+\tfrac12 X_{jk}{}^{i}A^{j}\wedge A^{k}$ covariant. This is remedied by introducing a second $\sfG$-module $V_{-2}\subset \text{Sym}^2(V_{-1})$, where $r=1,2\ldots\dim V_{-2}$ for some basis $(e_{r})$ together with a map

\begin{equation}
\begin{aligned}
Z: V_{-2}&\rightarrow V_{-1}~,
\\
e_r&\mapsto Z^i{}_r e_i~.
\end{aligned}
\end{equation}

This allows us to introduce a $V_{-2}$-valued 2-form potential $B$ and a $V_{-2}$-valued 1-form gauge parameter $\Lambda_{(1)}$ to generalised gauge transformations and curvatures as usual in higher gauge theory:

\begin{equation}
\begin{gathered}
\delta A^{i} = \rmd \Lambda^i_{(0)}+X^i_{jk}A^j\Lambda^k_{(0)} +Z^{i}{}_{r}\Lambda^{r}_{(1)}~,~~~
\delta B^r= \nabla\Lambda^{r}_{(1)}+\ldots~,
\\
F^{i}=\rmd A^{i}+\tfrac12 X^{i}{}_{jk}A^{j}\wedge A^{k}+Z^{i}{}_{r}B^{r}~,~~~H^r=\nabla B^r+\ldots~,
\end{gathered}
\end{equation}

where here $\nabla$ is the covariant derivative given by the natural action of $\frh$ on $V_{-2}$ and the ellipses refer to covariantizing terms that are needed to complete the kinematical data to that of an adjusted higher gauge theory. In particular, the latter will include terms involving the various deformation tensors. This process is then iterated in a reasonably obvious fashion until the full kinematical data of an adjusted higher gauge theory is obtained[The fact that this iteration terminates is guaranteed because spacetime is finite-dimensional.].

In the gauged supergravity literature there is also often a  linear representation constraint

\begin{equation}
P_\Theta \Theta =\Theta~,
\end{equation}

where $P_\Theta$ is the projector onto the representation contained in $V_{-1}^*\otimes \mathfrak{g}$  carried by $\Theta$, which  will be denoted $\rho_\Theta$. This can be understood as a requirement of supersymmetry <cit.>, the mutual locality of the action <cit.> or anomaly cancellation <cit.>.

A final important ingredient is now that the electromagnetic duality contained in U-duality needs to link potential $p$-forms to potential $d-p-2$-forms, and correspondingly the $\sfG$-modules $V_{-p}$ and $V_{p+2-d}$ have to be dual to each other in the lowest degrees that involve physical gauge potentials.

The above constraints restrict severely the choices of representations $V_{-2}$, $V_{-3}$. In <ref> we listed some important concrete examples of maximal supergravities, in which $\sfK=\sfG$. In this case, there is a tensor hierarchy dgLa determined by the U-duality group <cit.>, with graded vector space described in <ref> and derivation given by the action of $\Theta$. Also, the electromagnetic duality is visibly reflected in the duality of representations in the cases $d=5,6,7$.

$d$    $ \sfG                           $   $ V_{-1}            $   $ V_{-2}           $   $ V_{-3}           $   $ V_{-4}            $   $ V_{-5}       $   $ V_{-6}       $

7      $ \sfSL(5,\IR )                  $   $ \rep{10}_c    $   $ \rep{5}     $   $ \rep{5}_c    $   $ \rep{10} $   $ \rep{24} $   $ \rep{15}_c\oplus\rep{40} $

6      $ \sfSO(5,5)                $   $ \rep{16}_c     $   $ \rep{10}    $   $ \rep{16} $   $ \rep{45} $   $ \rep{144} $   $ \rep{10\oplus126\oplus320} $

5      $ \sfE_{6(6)}              $   $ \rep{27}_c     $   $ \rep{27} $   $ \rep{78} $   $ \rep{351}_c $   $ \rep{27\oplus1728} $

4      $ \sfE_{7(7)}              $   $ \rep{56} $   $ \rep{133} $   $ \rep{912} $   $ \rep{133\oplus8645} $

3      $ \sfE_{8(8)}              $   $ \rep{248} $   $ \rep{1\oplus 3875} $   $ \rep{3875\oplus147250} $

Global symmetry groups $\sfG$ of maximal supergravity in $3\leq d\leq 7$ spacetime dimensions (ignoring discrete factors). The $\sfG$ representations $V_{-p}$ are carried by  $p$-forms in the tensor hierarchy. The scalars ($0$-forms) are valued in $\caM\coloneqq \sfG/\sfG_0$,  where $\sfG_0\subset \sfG$ is the maximal compact subgroup.

We note that in the presence of generic deformations, the differential graded Lie algebra constructed in the maximally supersymmetric case is actually insufficient and needs to be extended further by at least one step in both directions. We shall explain this below, when discussing the example $d=5$.

§ HLIE-ALGEBRAS AND EILH-ALGEBRAS

We start with the definition of the two Koszul-dual concepts of $\ophLie$-algebras and $\opEilh$-algebras that underlie our definition of $E_2L_\infty$-algebras.
Koszul duality was briefly reviewed in <ref>; we stress, however, that more than an intuitive understanding of Koszul duality is not required for our discussion.

§.§ hLie-algebras

In this section, we define a generalisation of differential graded Lie algebras that contains hemistrict Lie 2-algebras as defined in <cit.>. Besides a differential, Lie 2-algebras contain a binary operator of degree $0$ and an alternator, i.e. a binary operator of degree $-1$. In the case of Lie 3-algebras, we also obtain a higher alternator of degree $-2$. It is thus natural to allow for higher alternators of arbitrary degree, and to minimally extend the algebraic relations known from Lie 2- and Lie 3-algebras in such a way that they are compatible with the differential. This leads to the following definition.

An $\ophLie$-algebra is a graded vector space $\frE$ together with a differential $\eps_1$ and a collection of binary products $\eps_2^i$ of degree $i$ for $i\in\{0,1\}$,

\begin{equation}
\begin{aligned}
\eps_1&: \frE\rightarrow \frE~,~~~|\eps_1|=1~,
\\
\eps^i_2&: \frE\otimes \frE\rightarrow \frE~,~~~|\eps^i_2|=-i~,
\end{aligned}
\end{equation}

that satisfy the following relations:

\begin{equation}\label{eq:hLie-relations}
\begin{aligned}
\eps_1(\eps_1(x_1))&=0~,
\\
\eps_1(\eps^i_2(x_1,x_2))&=(-1)^i\big(\eps^i_2(\eps_1(x_1),x_2)+(-1)^{|x_1|}\eps^i_2(x_1,\eps_1(x_2))\big)
\\
\\
\eps^i_2(\eps_2^i(x_1,x_2),x_3)&=(-1)^{i(|x_1|+1)}\eps^i_2(x_1,\eps^i_2(x_2,x_3))-(-1)^{(|x_1|+i)|x_2|}\eps^i_2(x_2,\eps_2^i(x_1,x_3))
\\
\\
\eps^{j}_2(\eps^{i}_2(x_1,x_2),x_3)&=
\\
\eps^{i}_2(\eps^{j}_2(x_1,x_2),x_3)&=
\\
\end{aligned}
\end{equation}

for all $i,j\in \IN$ and $j<i$, where we regard $\eps_2^i=0$ for $i\not\in\{0,1\}$, and for all $x_1,x_2,x_3\in \frE$.

We note that the first three relations in (<ref>) for $i=0$ amount to the relations for a differential graded Leibniz algebra with differential $\eps_1$ and Leibniz product $\eps_2^0$. If $\eps_2^1$ vanishes, then $\eps_2^0$ is graded antisymmetric, and the Leibniz bracket becomes Lie. If we restrict to the case $\eps_2^i=0$ for $i\neq 0$, we simply recover differential graded Lie algebras. If we restrict ourselves to 2-term $\ophLie$-algebras, i.e. $\ophLie$-algebras concentrated in degrees $-1$ and $0$, then only $\eps_2^0$ and $\eps_2^1$ can be non-trivial for degree reasons, and we obtain the hemistrict Lie 2-algebras of <cit.> with a graded symmetric $\eps_2^1$. The latter map is a chain homotopy sometimes called the alternator, capturing the failure of $\eps_2^0(x_1,x_2)$ to be antisymmetric:

\begin{equation}
\begin{aligned}
\\
\end{aligned}
\end{equation}

We close this section with two results on $\ophLie$-algebras that allow us to construct new $\ophLie$-algebras from existing ones. First, we can create a larger $\ophLie$-algebra by tensoring $\ophLie$-algebras with differential graded commutative algebras, just as in the case of Lie algebras,

The tensor product of a differential graded commutative algebra and an $\ophLie$-algebra carries a natural $\ophLie$-algebra structure.

On the tensor product of a differential graded commutative algebra $\frA$ and an $\ophLie$-algebra $\frE$, we define

\begin{equation}
\begin{aligned}
\hat \frE\coloneqq \frA\otimes \frE=\oplus_{k\in \IN}(\frA\otimes \frE)_k~,~~~(\frA\otimes \frE)_k=\sum_{i+j=k} \frA_i\otimes \frE_j~,
\\
\hat\eps_1(a_1\otimes x_1)=(\rmd a_1)\otimes x_1+(-1)^{|a_1|}a_1\otimes \eps_1(x_1)~,
\\
\hat\eps^i_2(a_1\otimes x_1,a_2\otimes x_2)=(-1)^{i(|a_1|+|a_2|)}(a_1a_2)\otimes \eps^i_2(x_1,x_2)~,
\end{aligned}
\end{equation}

One then readily verifies the axioms (<ref>).

Second, we note that the axioms (<ref>) of $\ophLie$-algebras have some translation invariance built in, which allows us to construct new $\ophLie$-algebras by shifting degrees.

Given an $\ophLie$-algebra $(\frE,\eps_1,\eps_2^i)$, there is an $\ophLie$-algebra structure on the grade-shifted complex $\tilde \frE=s\frE$ with

\begin{equation}\label{eq:shifted_hLie_relations}
\begin{aligned}
\tilde \eps_1(sa)&=-s\eps_1(a)~,
\\
\tilde \eps^i_2(sa,sb)&=(-1)^{|a|+i}s\eps^{i-1}_2(a,b)~.
\end{aligned}
\end{equation}

Here, $s$ is the shift isomorphism $s:\frE\rightarrow \frE[-1]$ with that $|sa|=|a|+1$.

The proof is a straightforward check of the relations (<ref>) for $\tilde \frE$. The fact that $\tilde \eps_1$ is a differential is evident. The remaining relations follow rather directly by replacing all arguments $x_i$ in (<ref>) by $sx_i$, shifting all $|x_i|$ by $1$, and pulling out factors of $s$ using the rules (<ref>). The resulting equations are then satisfied due to (<ref>).

§.§ Eilh-algebras

For our constructions, it will be convenient to have the Koszul-dual concept to that of $\ophLie$-algebras at our disposal.

An $\opEilh$-algebra is a differential graded vector space endowed with binary operations $\oslash_i$ of degree[Note that $\oslash_i$ should really be regarded as a binary function of degree $i$ in order to identify the correct Koszul signs. For example,

\begin{equation}
t_1 \oslash_i t_2\coloneqq \oslash_i(t_1,t_2)~,~~~(-\oslash_i (-\oslash_j-))(t_1,t_2,t_3)=(-1)^{j|t_1|}t_1\oslash_i (t_2\oslash_j t_3)~.
\end{equation}
] $i\in \{0,1\}$ that satisfy the following quadratic identities:

\begin{equation}\label{eq:Eilh-relations}%\label{eq:2-term-Eilh-relations}
\begin{aligned}
a\oslash_0(b\oslash_0 c)&=(a\oslash_0 b+(-1)^{|a|\,|b|} b\oslash_0 a)\oslash_0 c~,
\\
-(-1)^{|a|}  a\oslash_1(b\oslash_1 c)&=(a\oslash_1 b+(-1)^{|a|\,|b|}b\oslash_1 a) \oslash_1 c~,
\\
a\oslash_0 (b\oslash_1 c)&=(-1)^{|a|}(a \oslash_0 b)\oslash_1 c~,
\\
a\oslash_1(b\oslash_0 c)&=
(-1)^{|a|\,|b|}(b \oslash_0 a)\oslash_1 c~,
\end{aligned}
\end{equation}

and the differential satisfies the evident deformed Leibniz rule given by

\begin{equation}\label{eq:def_Leibniz}%\label{eq:mod_Leibniz_simp}
\begin{aligned}
Q(a\oslash_0 b)&=(Qa)\oslash_0 b+(-1)^{|a|}a\oslash_0 Qb+ a \oslash_1 b-(-1)^{|a|\,|b|}b\oslash_1a~,
\\
Q(a\oslash_1 b)&=-\big((Qa)\oslash_1 b+(-1)^{|a|}a\oslash_1 Qb\big)~.
\end{aligned}
\end{equation}

The relations (<ref>) and (<ref>) are equivalent (up to some signs) to those defining the operad (an abstraction of the axioms of an algebraic strutcure) $\caR^!$ introduced in Squires <cit.> to capture 2-term $E_2L_\infty$-algebras. The free algebra over $\caR^!$ is dual to our notion of 2-term $E_2L_\infty$-algebra with a graded symmetric alternator $\sfalt(x,y)=(-1)^{|x|\,|y|}\sfalt(y,x)$.

The relation between $\opEilh$-algebras and  $\ophLie$-algebras is equivalent to that between Lie algebras and commutative algebras: it provides another example of Koszul duality, cf. <ref>; see also <cit.> for details. Instead of going into further details, let us construct the Chevalley–Eilenberg algebra of an $\ophLie$-algebra by constructing the corresponding differential semifree $\opEilh$-algebra.

We start from an $\ophLie$-algebra $\frE$ which we assume for clarity of the discussion to admit a nice basis $(\tau_\alpha)$ and whose underlying graded vector space can be dualised degreewise. We will make this assumption for all the graded vector spaces from here on, mostly for pedagogical reasons: it allows us to give very explicit formulas. More generally, one may want to work with (graded) pseudocompact vector spaces, cf. the discussion in <cit.>.

Consider the graded vector space[Our convention for degree shift is the common one, $V[k]\coloneqq \bigoplus_i V[k]_i$ with $V[k]_i\coloneqq V_{k+i}$, implying that $V[k]$ is the graded vector space $V$ shifted by $-k$.] $V\coloneqq \frE[1]^*$ and together with its free, non-associative tensor algebra $\bigoslash^\bullet_\bullet V$ with respect to all the products $\oslash_i$ simultaneously. The quadratic identities (<ref>) allow us to rearrange all tensor products such that they are nested from left to right. We thus define

\begin{equation}
\begin{aligned}
{\caE(V)}&\coloneqq \IR~\oplus~V~\oplus~\bigoplus_{i\in \IN}V\oslash_i V~\oplus~\bigoplus_{i,j\in \IN}(V\oslash_i V)\oslash_j V~\oplus~\dotsb
\\
\end{aligned}
\end{equation}

We also define the reduced tensor algebra

\begin{equation}\label{eq:red_ten_algebra}
\overline{\caE(V)}\coloneqq V~\oplus~\bigoplus_{i\in \IN}V\oslash_i V~\oplus~\bigoplus_{i,j\in \IN}(V\oslash_i V)\oslash_j V~\oplus~\dotsb~,
\end{equation}

which is, in fact, sufficient for the description of $\ophLie$-algebras.

We call an $\opEilh$-algebra of the form $(\caE(V),Q)$ for some graded vector space $V$ without any restrictions on the products $\oslash_i$ beyond (<ref>) semifree.

Semifree $\opEilh$-algebras yield the Chevalley–Eilenberg description of $\ophLie$-algebras:

The Chevalley–Eilenberg algebra $\sfCE(\frE)$ of an $\ophLie$-algebra $\frE$ whose differential and binary products are given by

\begin{equation}
\begin{aligned}
\eps_1&: \frE\rightarrow \frE~,~~~&\tau_\alpha&\mapsto m^\beta_\alpha \tau_\beta~,~~~&&|\eps_1|=1~,
\\
\eps^i_2&: \frE\otimes \frE\rightarrow \frE~,~~~&\tau_\alpha\otimes \tau_\beta&\mapsto m^{i,\gamma}_{\alpha\beta}\tau_\gamma~,~~~&&|\eps^i_2|=i
\end{aligned}
\end{equation}

for some $m^\beta_\alpha$ and $m^{i,\gamma}_{\alpha\beta}$ taking values in the underlying ground field is the $\opEilh$-algebra $\caE(V)$ with $V=\frE[1]^*$ and the differential

\begin{equation}\label{eq:Q_hLie}
Q t^\alpha=-(-1)^{|\beta|}m^\alpha_\beta t^\beta-(-1)^{i(|\beta|+|\gamma|)+|\gamma|(|\beta|-1)}\,m^{i,\alpha}_{\beta\gamma}\,t^\beta \oslash_i t^\gamma~.
\end{equation}

Here, $|\beta|$ is shorthand for $|t^\beta|$, the degree of $t^\beta$ in $V$.

In the case of Lie algebras and $L_\infty$-algebras, the (homotopy) Jacobi relations are equivalent to a nilquadratic differential in the corresponding Chevalley–Eilenberg algebra, and this is also the case here:

The equation $Q^2=0$ for the differential of the Chevalley–Eilenberg algebra of an $\opEilh$-algebra that is of the form (<ref>) is equivalent to the relations (<ref>).

The proof is again a straightforward but tedious verification of the axioms, which is better left to a computer algebra program.

Just as we can consider restricted $\ophLie$-algebras in which only a subset of the binary products $\eps_2^i$ are non-vanishing, we can also restrict the image of $Q t^\alpha$ in $\caE(V)$. In the case of $\opEilh$-algebras, we can often further restrict the tensor products $\oslash_i$ appearing in $\caE(V)$ from $i\in \IN$ to $i\in I\subset \IN$, resulting in a tensor algebra $\caE_I(V)$. We then obtain the following restriction theorem:

The equation $Q^2=0$ on a semifree $\opEilh$-algebra $\caE(V)$ is equivalent to $Q^2V=0$. If the differential $Q$ closes on $\caE_I(V)$, then $Q^2 V=0$ is equivalent to $Q^2=0$ on $\caE(V)$.

The condition $Q^2=0$ on $\caE_I(V)$ is equivalent to the condition $Q^2=0$ on $V\subset \caE_I(V)$, which follows by direct computation, using the deformed Leibniz rule (<ref>):

\begin{equation}
Q^2(a\oslash_i b)=(Q^2 a)\oslash_i b+a\oslash_i (Q^2 b)~.
\end{equation}

The same holds for $Q^2=0$ on $\caE_I(V)$, and combining both we obtain the desired result.
Note, however, that we cannot simply put the products $\oslash_i=0$ for $i\notin I$ if $I$ does not contain $0$. The deformed Leibniz rule (<ref>) would render the lowest non-vanishing product graded symmetric or antisymmetric and the quadratic relations (<ref>) would then imply severe restrictions on the products at cubic order. Restrictions of $\caE(V)$ to $\caE_I(V)$ that are evidently sensible are of the form $I=\{0,\ldots,n\}$.

§.§ Cohomology of semifree Eilh-algebras

An important tool in studying Lie algebras is Lie algebra cohomology, and we consider the generalisation to $\ophLie$-algebras in the following. As we saw above, this amounts to the cohomology of semifree $\opEilh$-algebras. Due to the deformation of the usual Leibniz rule to (<ref>), a subtle and important point arises. For ordinary differential graded algebras, the cohomology again carries the structure of a differential graded algebra of the same type, with product induced from the product on the original algebra. In particular, the product of two cocyles is again a cocycle. The deformation of the Leibniz rules can now evidently break this connection.

To start, let us consider the cohomology of the semifree $\opEilh$-algebra $(\caE(V),Q_0)$ with the most trivial differential $Q_0$ satisfying

\begin{equation}
\begin{aligned}
\\
Q_0(a\oslash_i b)&=(-1)^i\big((Q_0a)\oslash_i b+(-1)^{|a|}a\oslash_i Q_0b\big)
\\
&\hspace{1cm}+(-1)^i (a \oslash_{i+1}b)-(-1)^{|a|\,|b|} (b\oslash_{i+1} a)~.
\end{aligned}
\end{equation}

for all $v\in V$ and $a,b\in \caE(V)$.

The $Q_0$-cohomology of $(\caE(V),Q_0)$ is the vector space[Here, $\odot$ denotes the symmetrised tensor product.] $\sfE_0(\bigodot^\bullet V)$, with the embedding map $\sfE_0:\bigodot^\bullet V\mapsto \caE(V)$ defined by

\begin{equation}
\sfE_0(v_1\odot \dotsb \odot v_n)\coloneqq \sum_{\sigma\in S_n}(\dotsb(v_{\sigma(1)}\oslash_0 v_{\sigma(2)})\oslash_0\dotsb )\oslash_0 v_{\sigma(n)}~.
\end{equation}

This vector space carries the evident structure of a differential graded commutative algebra.

It is clear that the image of $\sfE_0$ forms the kernel of $Q_0$ inside $\caE_{\{0\}}(V)$ and that $\caE_{\{0\}}(V)$ cannot contain any coboundaries.  Moreover, it is easy to show that all cocycles are coboundaries in the case of elements of $V\oslash_\bullet V$: here, the kernel of $Q_0$ consists of elements of the form

\begin{equation}
v_1\oslash_i v_2+(-1)^{i+|v_1|\,|v_2|}v_2\oslash_i v_1=(-1)^{i-1}Q(v_1\oslash_{i-1}v_2)~.
\end{equation}

For elements that are cubic and of higher order in $V$, the proof is subtle and lengthy, and we only sketch it here.
We first restrict to elements $a\in\caE(V)$ of a particular degree and order in $V$, as both the deformed Leibniz rule (<ref>) and the algebra relations (<ref>) preserve these. We then split $Q a\in \caE(V)$ into polynomials of the same monomial type, i.e. the same order and type of products $\oslash_i$. These terms have to vanish separately. The condition $Q a=0$ requires the application of the relations (<ref>) $d-2$ times, where $d$ is the degree of $a$ in $V$. This application enforces a particular symmetry structure on $a$ which forces it to lie in the image of $V$.

<Ref> together with the usual arguments underlying a general Hodge–Kodaira decomposition[see e.g. <cit.>] then yield the following theorem:

Consider the trivial semifree $\opEilh$-algebra from <ref>. Then we have the following contracting homotopy between $\caE(V)$ and the differential graded commutative algebra $(\bigodot^\bullet V,0)$:

\begin{equation}\label{eq:contracting_hom_Q0}
\begin{tikzcd}
\ar[loop,out=194,in= 166,distance=20,"\sfH_0"](\caE(V),Q_0)\arrow[r,shift left]{}{\sfP_0} & (\bigodot^\bullet V,0) \arrow[l,shift left]{}{\sfE_0}~,
\end{tikzcd}
\end{equation}

with the projection

\begin{equation}
\begin{aligned}
\sfP_0(v_1)&=v_1
\\
\sfP_0\big(((v_0\oslash_{i_1} v_2)\oslash_{i_2} v_3)\dotsb \oslash_{i_n} v_n\big)&=\begin{cases}
\tfrac{1}{(n+1)!}v_0\odot v_1\odot\dotsb\odot v_n& i_1=\dotsb=i_n=0~,
\\
0 & \mbox{else}
\end{cases}
\end{aligned}
\end{equation}

for all $v_0,\ldots, v_n\in V$, such that

\begin{equation}\label{eq:homotopy_transfer_relations}
\begin{gathered}
\sfH_0\circ \sfH_0= \sfH_0\circ  \sfE_0=0~,~~~ \sfP_0\circ  \sfH_0=0~,
\\
\sfid_{\caE(V)}- \sfE_0\circ  \sfP_0= \sfH_0\circ  Q_0+ Q_0\circ  \sfH_0~.
\end{gathered}
\end{equation}

As usual, the map $\sfH_0$ is not unique.

We note that such a contracting homotopy for ordinary differential graded algebras often induces an algebra morphism $\Phi\coloneqq \sfE_0\circ \sfP_0$. This is not the case here, as clearly $\Phi(a)\oslash_i\Phi(b)\neq \Phi(a\oslash_i b)$ in general. We shall return to this point in <ref>.

The contracting homotopy (<ref>) has a number of important generalisations and applications. Here, we note that it evidently extends to differentials $Q_{\rm lin}=Q_0+\rmd$, where $\rmd:V\rightarrow V$ and $\rmd$ satisfies the ordinary Leibniz rules on $\caE(V)$ and $\bigodot^\bullet V$:

\begin{equation}\label{eq:homotopy_for_transfer_EL_to_L}
\begin{tikzcd}
\ar[loop,out=194,in= 166,distance=20,"\sfH_0"](\caE(V),Q_0+\rmd)\arrow[r,shift left]{}{\sfP_0} & (\bigodot\nolimits^\bullet V,\rmd) \arrow[l,shift left]{}{\sfE_0}~.
\end{tikzcd}
\end{equation}

Moreover, if we have a differential $\rmd:\bigodot^\bullet V\rightarrow \bigodot^\bullet V$ non-linear in $\odot$, we still have a corresponding contracting homotopy

\begin{equation}
\begin{tikzcd}
\ar[loop,out=194,in= 166,distance=20,"\sfH_0"](\caE(V),Q_0+Q_1)\arrow[r,shift left]{}{\sfP_0} & (\bigodot\nolimits^\bullet V,\rmd) \arrow[l,shift left]{}{\sfE_0}
\end{tikzcd}
\end{equation}

\begin{equation}
Q_1=\sfE_0\circ \rmd\circ \sfP_0~.
\end{equation}

We therefore arrive at the following statement:

Any semifree differential graded commutative algebra $(\bigodot^\bullet V,\rmd)$ gives rise to the semifree $\opEilh$-algebra $(\caE(V),Q_0+\sfE_0\circ \rmd\circ \sfP_0)$.

We will pick up this thread of our discussion again later.

§ EL-INFINITY-ALGEBRAS

§.§ EL-infinity-algebras and their morphisms

We define $E_2L_\infty$-algebras to be the homotopy version of $\ophLie$-algebras.

An \(E_2L_\infty\)-algebra is a graded vector space \(\frE\) equipped with a nilpotent differential on the semi-free \(\opEilh\)-algebra $\sfCE(\frE)\coloneqq \caE(V)$ for $V=\frE[1]^*$.

The differential $Q$ is fully specified by its action on $V$. With respect to a basis $(t^\alpha)$ on $V$, we encode this action in structure constants $m$ taking values in the ground field as follows:

\begin{equation}\label{eq:structure_constants_ELinfty}
Q t^\alpha=\pm m^\alpha\pm m^\alpha_\beta t^\beta~\pm~m^{i_1,\alpha}_{\beta_1\beta_2}~t^{\beta_1}\oslash_{i_1} t^{\beta_2}~\pm~m^{i_1i_2,\alpha}_{\beta_1\beta_2\beta_3}~(t^{\beta_1}\oslash_{i_1} t^\gamma)\oslash_{i_2} t^\delta+\ldots~.
\end{equation}

Here the choice of signs $\pm$ is a convention[For $L_\infty$-algebras, we follow the conventions of <cit.> for the structure constants and the differential in the Chevalley–Eilenberg algebra.] and we shall be more explicit below, cf.also (<ref>). The structure constants $m$ define higher products,

\begin{equation}
\begin{gathered}
\eps^I_n:\frE^{\otimes n}\rightarrow \frE~,
\\
\eps_0=m^\alpha \tau_\alpha~,~~~
\eps_1(\tau_\alpha)=m^\beta_\alpha \tau_\beta~,~~~\eps_2^i(\tau_\alpha,\tau_\beta)=m^{i,\gamma}_{\alpha\beta}\tau_\gamma~,~~~\ldots
\\
\eps^I_n(\tau_{\alpha_1},\ldots,\tau_{\alpha_n})=m_{\alpha_1\ldots\alpha_n}^{I,\beta} \tau_\beta~,
\end{gathered}
\end{equation}

where $I$ is a multi-index consisting of $n-1$ indices $i_1,i_2,\ldots,\in \{0,1\}$ and we define $|I|\coloneqq i_1+i_2+\ldots$. The products $\eps^I_n$ have degree $-|I|$.

It is useful to identify the following special cases:

Let $(\frE,\eps_k^I)$ be an $E_2L_\infty$-algebra. If $\eps_0\neq 0$, we call $\frE$ curved, otherwise $\frE$ is uncurved. An uncurved $E_2L_\infty$-algebra is called hemistrict, if $\eps_k^I=0$ for $k\geq 3$. It is called strict if it is hemistrict and $\eps_2^i=0$ for $i>0$. Finally, $\frE$ is called semistrict if $\epsilon_n^I=0$ for $I\neq (0,\ldots,0)$.

In the following, all our $E_2L_\infty$-algebras will be uncurved.
Clearly, hemistrict $E_2L_\infty$-algebras are simply $\ophLie$-algebras, and therefore $E_2L_\infty$-algebras subsume differential graded Lie algebras.

As an example, let us consider a family of weak models of the string Lie 2-algebra which we will use to exemplify many of our constructions in the following. We consider the graded vector space $V=(\frg\oplus \IR[1])[1]^*$, where $\frg$ is a finite-dimensional quadratic (i.e. metric) Lie algebra with structure constants $f^\alpha_{\beta\gamma}$ and Cartan–Killing form $\kappa_{\alpha\beta}$ with respect to a basis $(\tau_\alpha)$. On $V$, we introduce basis vectors $t^\alpha\in \frg[1]^*$ and $r\in \IR[2]^*$ of degrees $1$ and $2$, respectively. The differential is given by

\begin{equation}
\begin{aligned}
Q t^\alpha&=-f^\alpha_{\beta\gamma}\,t^\beta \oslash_0 t^\gamma+(1-\vartheta)\kappa_{\alpha\delta} f^\delta_{\beta\gamma}\,(t^\beta \oslash_0 t^\gamma) \oslash_0 t^\delta-2\vartheta \kappa_{\beta\gamma}\,t^\beta \oslash_0 t^\gamma~,
\\
Q r &=0
\end{aligned}
\end{equation}

with $\vartheta\in \IR$, and a direct calculation verifies $Q^2=0$. This defines the family of $E_2L_\infty$-algebras $\frstring^{\rmwk,\vartheta}_\rmsk(\frg)$ with the following underlying graded vector space and higher products:

\begin{equation}\label{eq:skeltal_model_string_Lie_2_algebra}
\begin{gathered}
\frstring^{\rmwk,\vartheta}_\rmsk(\frg)~\coloneqq ~(\IR[1]\xrightarrow{~0~}\frg)~,
\\
\eps_2(x_1,x_2)=[x_1,x_2]~,~~~\eps_2^1(x_1,x_2)=2\vartheta(x_1,x_2)
\\
\eps^{00}_3(x_1,x_2,x_3)=(1-\vartheta)(x_1,[x_2,x_3])~,
\end{gathered}
\end{equation}

for $x\in \frg$ and $y\in \IR$. All other higher products vanish. We notice that $\frstring^{\rmwk,\vartheta}_\rmsk(\frg)$ is an uncurved $E_2L_\infty$-algebra for each $\vartheta\in\IR$. It becomes semistrict for $\vartheta=0$ and hemistrict for $\vartheta=1$.

Another very general and useful example is the $E_2L_\infty$-algebra of inner derivations $\frinn(\frE)$ of another $E_2L_\infty$-algebra $\frE$. This is obtained as a straightforward generalisation of the definition of the (unadjusted) Weil algebra of an $L_\infty$-algebra.

The (unadjusted) Weil algebra of an $E_2L_\infty$-algebra $\frE$ is the $\opEilh$-algebra

\begin{equation}
\sfW(\frE)\coloneqq \Big(~\bigoslash\nolimits^\bullet~(\frE[1]^*\oplus \frE[2]^*)~,~Q_\sfW~\Big)~,
\end{equation}

where the Weil differential is defined by the relations

\begin{equation}
Q_\sfW=Q_\sfCE+\sigma~,~~~Q_\sfW\sigma=-\sigma Q_\sfW~,
\end{equation}

in which \(Q_\sfCE\) is the differential on \(\sfCE(\frE)\subset\sfW(\frE)\) and
$\sigma\colon\frE[1]^*\rightarrow \frE[2]^*$ is the shift isomorphism, trivially extended to $\sfW(\frE)$ by the (undeformed) Leibniz rule, i.e.

\begin{equation}
\sigma(a\oslash_i b)=(-1)^i\big(\sigma a\oslash_i b+(-1)^{|a|}a \oslash_i \sigma b\big)~.
\end{equation}

The $E_2L_\infty$-algebra dual to $\sfW(\frE)$ is the inner derivation $E_2L_\infty$-algebra $\frinn(\frE)$ of $\frE$.

In terms of semifree $\opEilh$-algebras, the notion of a morphism of $E_2L_\infty$-algebras becomes evident.

A morphism of $E_2L_\infty$-algebras $\phi:\frE\rightarrow \tilde \frE$ is a morphism dual to the corresponding morphism $\Phi:\sfCE(\frE)\rightarrow \sfCE(\tilde \frE)$ of $\opEilh$-algebras. For $\sfCE(\frE)=({\caE(V)}, Q)$ and $\sfCE(\tilde \frE)=({\caE(\tilde V)},\tilde Q)$, such a morphism is compatible with the differential,

\begin{equation}
Q\circ \Phi=\Phi\circ \tilde Q~,
\end{equation}

and the product structure,

\begin{equation}
\Phi(x\oslash_i y)=\Phi(x)\oslash_i\Phi(y)
\end{equation}

for all $x,y\in \sfCE(\frE)$ and $i\in\{0,1\}$.

Recall that the appropriate notion of equivalence for homotopy algebras is that of a quasi-isomorphism, which we make explicit in the following definition.

Two $E_2L_\infty$-algebras $\frE$ and $\tilde \frE$ are called quasi-isomorphic if there is a morphism of $E_2L_\infty$-algebras $\phi:\frE\rightarrow \tilde \frE$ such that the contained chain map $\phi_1$, the dual to the linear component of the dual morphism of $\opEilh$-algebras $\Phi$, descends to an isomorphism between the cohomologies of $\frE$ and $\tilde \frE$.

We note that the cohomology $H^\bullet_{\eps_1}(\frE)$ of an $E_2L_\infty$-algebra $\frE$ is dual to the cohomology with respect to the linear part of the Chevalley–Eilenberg differential $Q$. In the case of the Weil algebra, the included shift isomorphism $\sigma$ renders the cohomology $H^\bullet_{\eps_1}(\frinn(\frE))$ trivial. We therefore obtain a quasi-isomorphism between $\frinn(\frE)$ and the trivial $E_2L_\infty$-algebra, extending the situation for $L_\infty$-algebras.

In the future, it may be useful to have an inner product structure on an $E_2L_\infty$-algebra. The appropriate notion here is a simple generalisation of cyclic structures on $L_\infty$-algebras.

An $E_2L_\infty$-algebra $\frE$ is called cyclic if it is equipped with a non-degenerate graded-symmetric bilinear form $\langle-,-\rangle:\frE\times \frE\rightarrow K$, where $K$ is the ground field, such that

\begin{equation}\label{eq:cyclicity}
\langle e_1,\eps_i(e_2,\ldots,e_{i+1})\rangle=(-1)^{i+i(|e_1|+|e_{i+1}|)+|e_{i+1}|\sum_{j=1}^{i}|e_j|}\langle e_{i+1},\eps_i(e_1,\ldots,e_{i})\rangle
\end{equation}

for all $e_i\in \frE$.

§.§ Homotopy transfer and minimal model theorem

A minimal characteristic of homotopy algebras is that they provide a nice notion of homotopy transfer. The latter will be important for all our future constructions, and we therefore develop a form of the homological perturbation lemma below.

We start from deformation retract between two differential graded complexes $(\frE,\eps_1)$ and $(\tilde \frE,\tilde \eps_1)$. That is, we have chain maps $\sfp$ and $\sfe$, together with a map $\sfh$ of degree $-1$, which fit into the diagram

\begin{equation}
\begin{tikzcd}
\ar[loop,out=194,in= 166,distance=20,"\sfh"](\frE,\eps_1)\arrow[r,shift left]{}{\sfp} & (\tilde \frE,\tilde \eps_1) \arrow[l,shift left]{}{\sfe}~,
\end{tikzcd}
\end{equation}

and satisfy the relations

\begin{equation}\label{eq:homotopy_transfer_relations_dual}
\begin{gathered}
\sfp\circ \sfe=\sfid_{\tilde \frE}~,~~~\sfid_{\frE}-\sfe\circ \sfp=\sfh\circ \eps_1+\eps_1\circ \sfh~,
\\
\sfh\circ\sfh=0~,~~~\sfh\circ \sfe=0~,~~~\sfp\circ \sfh=0~.
\end{gathered}
\end{equation}

We now want to consider the homological perturbation lemma for the semifree $\opEilh$-algebras on ${\caE(V)}$ and ${\caE(\tilde V)}$ with differentials $Q$ and $\tilde Q$, respectively, defined by

\begin{equation}
V=\frE[1]^*~,~~~Q=\eps_1^*\eand \tilde V=\tilde \frE[1]^*~,~~~\tilde Q=\tilde \eps_1^*~.
\end{equation}

The deformation retract (<ref>) lifts to the deformation retract

\begin{equation}
\begin{tikzcd}
\ar[loop,out=194,in= 166,distance=20,"\sfH_0"]({\caE(V)},Q)\arrow[r,shift left]{}{\sfP_0} & ({\caE(\tilde V)},\tilde Q) \arrow[l,shift left]{}{\sfE_0}~,
\end{tikzcd}
\end{equation}

with[The notation is chosen to match more closely the formulas of <ref>.]

\begin{equation}
\begin{aligned}
\sfE_0(v)&=\sfp^*(v)~,~~~&\sfE_0(x\oslash_i y)&=\sfE_0(x)\oslash_i\sfE_0(y)~,
\\
\sfP_0(v)&=\sfe^*(v)~,~~~&\sfP_0(x\oslash_i y)&=\sfP_0(x)\oslash_i\sfP_0(y)~,
\end{aligned}
\end{equation}

for all $v\in V$ and $x,y\in {\caE(V)}$. The dual to the contracting homotopy is continued by a modification of the tensor trick,

\begin{equation}
\begin{aligned}
\sfH_0(v)&=\sfh^*(v)~,
\\
\sfH_0(x\oslash_i y)&=(-1)^i\sfH_0(x)\oslash_i\sfE_0(\sfP_0(y))+(-1)^{i+|x|}x\oslash_i\sfH_0(y)
\\
\end{aligned}
\end{equation}

These maps satisfy the relations

\begin{equation}\label{eq:homotopy_transfer_relations_0}
\begin{gathered}
\sfP_0\circ\sfE_0=\sfid_{\caE(\tilde V)}~,~~~\sfid_{\caE(V)}-\sfE_0\circ \sfP_0=\sfH_0\circ Q+Q\circ \sfH_0~,
\\
\sfH_0\circ\sfH_0=0~,~~~\sfH_0\circ \sfE_0=0~,~~~\sfP_0\circ \sfH_0=0~.
\end{gathered}
\end{equation}

The proof of (<ref>) is a straightforward computation for elements in $\oslash^2_\bullet V$. The general case follows then by iterating the algebra relations and applying (<ref>).

The higher products $\eps_i$ with $i>1$ on $\frE$ can now be regarded as a perturbation of the differential. Dually, we have a perturbation of $Q$,

\begin{equation}
Q\rightarrow \hat Q\coloneqq Q+Q_\delta~.
\end{equation}

We can then use the homological perturbation lemma <cit.> to transfer these structures over to higher products $\tilde \eps_i$ on $\tilde \frE$, or, dually, to a perturbed differential $\tilde Q$ on $\caE(\tilde V)$. The formulas for this are the usual ones, cf. <cit.>.

Starting from the deformation retract (<ref>), we can construct another deformation retract

\begin{equation}
\begin{tikzcd}
\ar[loop,out=194,in= 166,distance=20,"\sfH"]({\caE(V)},\hat Q)\arrow[r,shift left]{}{\sfP} & ({\caE(\tilde V)},\hat{\tilde Q}) \arrow[l,shift left]{}{\sfE}~,
\end{tikzcd}
\end{equation}

\begin{equation}
\begin{gathered}
\hat Q\coloneqq Q+Q_\delta~,~~~\hat{\tilde Q}=\tilde Q+ \sfP\circ Q_\delta \circ \sfE_0~,
\\
\sfP\coloneqq \sfP_0\circ(\sfid+Q_\delta\circ\sfH_0)^{-1}~,~~~\sfE\coloneqq \sfE_0-\sfH\circ Q_\delta\circ \sfE_0~,~~~\sfH=\sfH_0\circ(\sfid+Q_\delta\circ \sfH_0)^{-1}~,
\end{gathered}
\end{equation}

where inverse operators are defined via the evident geometric series, such that

\begin{equation}\label{eq:homotopy_transfer_relations2}
\begin{gathered}
\sfP\circ \sfE=\sfid_{\caE(\tilde V)}~,~~~
\sfid_{\caE(V)}-\sfE\circ \sfP=\sfH\circ \hat Q+\hat Q\circ \sfH~,
\\
\sfH_0\circ\sfH_0=\sfH_0\circ \sfE=0~,~~~\sfP\circ \sfH=0~,
\end{gathered}
\end{equation}

The lemma follows from the usual perturbation lemma, cf. <cit.>, with the specialisation that $\sfE$ and $\sfP$ are here morphisms of $\opEilh$-algebras. To see this, we note that $Q_\delta$ acts as a derivation,

\begin{equation}
Q_\delta(x\oslash_i y)=(-1)^i\big((Q_\delta x)\oslash_i y+(-1)^{|x|}x\oslash_i Q_\delta y\big)~.
\end{equation}

Moreover, in the non-vanishing terms of

\begin{equation}
\begin{aligned}
\sfP(x\oslash_i y)&=(\sfP_0\circ(\sfid-Q_\delta \circ \sfH_0+Q_\delta\circ \sfH_0\circ Q_\delta\circ \sfH_0-\ldots)\big)(x\oslash_i y)
\\
\end{aligned}
\end{equation}

all the $\sfH_0$ are precomposed by a $Q_\delta$, as otherwise the map $\sfP_0$, which is precomposed to all summands, would annihilate the term due to $\sfP_0\circ \sfH_0=0$. The relation

\begin{equation}
\sfP(x\oslash_i y)=\sfP(x)\oslash_i\sfP(y)
\end{equation}

follows then by a direct computation. The same holds for $\sfE$.

We note that for small perturbations $Q_\delta$, the homological perturbation <ref> implies that

\begin{equation}\label{eq:HPL_geom_series}
\hat{\tilde Q}=\tilde Q+\sfP_0\circ Q_\delta\circ \sfE_0-\sfP_0\circ Q_\delta\circ \sfH_0\circ Q_\delta \circ \sfE_0+\sfP_0\circ Q_\delta\circ \sfH_0\circ Q_\delta\circ \sfH_0\circ Q_\delta \circ \sfE_0-\dotsb~.
\end{equation}

A direct consequence of homotopy transfer is the existence of minimal models for homotopy algebras. Consider the deformation retract (<ref>) with $(\tilde \frE,\tilde \eps_1=0)=H^\bullet_{\eps_1}(\frE)$ the cohomology of $(\frE,\eps_1)$ as well as $\sfp$ and $\sfe$ the projection onto the cohomology and a choice of embedding, respectively. Then the homotopy transfer yields the structure of a quasi-isomorphic $E_2L_\infty$-algebra on the cohomology of $(\frE,\eps_1)$. This implies the minimal model theorem.

Any $E_2L_\infty$-algebra $\frE$ comes with a quasi-isomorphic $E_2L_\infty$-algebra structure on its cohomology $H^\bullet_{\eps_1}(\frE)$. We call this a minimal model of $\frE$.

§.§ L-infinity-algebras as EL-infinity-algebras

As expected, $L_\infty$-algebras are special cases of $E_2L_\infty$-algebras.

A semistrict $E_2L_\infty$-algebra $\frE$ is an $L_\infty$-algebra. Conversely, any $L_\infty$-algebra is a (semistrict) $E_2L_\infty$-algebra. Dually, the data contained in a differential $Q$ in a semifree $\opEilh$-algebra $({\caE(V)},Q)$ with $\rmIm(Q|_V)\subset \bigoslash_0^\bullet V$ is equivalent to the data contained in a differential $\tilde Q$ on the semifree differential graded commutative algebra $(\bigodot^\bullet V,\tilde Q)$.

It suffices to show the dual statement, which is a direct consequence of <ref>.

Concretely, given an $L_\infty$-algebra $\frL$ with higher products $\mu_k$ this yields a semistrict $E_2L_\infty$-algebra with higher products

\begin{equation}
\eps^I_k=\begin{cases}
\mu_k & \mbox{for~$|I|=0$}~,
\\
0 & \mbox{else}~.
\end{cases}
\end{equation}

Dually, we have agreement in the structure constants in the corresponding Chevalley–Eilenberg differentials $Q_{\rm L}$ and $Q_{\rm EL}$ for the $L_\infty$-algebra and its trivial lift to $E_2L_\infty$-algebra up to combinatorial factors:

\begin{equation}
\begin{aligned}
Q_{\rm L} t^\alpha&= q^\alpha_\beta t^\beta+\tfrac12 q^\alpha_{\beta\gamma}t^\beta t^\gamma+\tfrac1{3!}q^\alpha_{\beta\gamma\delta}t^\beta t^\gamma t^\delta+\ldots~,
\\
Q_{\rm EL} t^\alpha&= q^\alpha_\beta t^\beta+q^\alpha_{\beta\gamma}t^\beta \oslash_0t^\gamma+q^\alpha_{\beta\gamma\delta}(t^\beta\oslash_0 t^\gamma)\oslash_0 t^\delta+\ldots~.
\end{aligned}
\end{equation}

Conversely, any semistrict $E_2L_\infty$-algebra $\frE$ is an $L_\infty$-algebra with higher products $\mu_k=\eps^0_k$. As an example, consider the $\vartheta=0$ case of the family of weak string Lie 2-algebra models (<ref>). This is a semistrict $E_2L_\infty$-algebra and therefore an $L_\infty$-algebra.

For consistency, we obviously expect the following.

Any $L_\infty$-algebra morphism $\phi:\frL\rightarrow \tilde \frL$ lifts to an $E_2L_\infty$-algebra morphism $\hat \phi:\hat \frL\rightarrow \hat{\tilde \frL}$, where $\hat \frL$ and $\hat{\tilde \frL}$ are the $L_\infty$-algebras $\frL$ and $\tilde \frL$, regarded as $E_2L_\infty$-algebras.

We prove this statement again from the dual perspective. Let $(\bigodot^\bullet V,Q)$ and $(\bigodot^\bullet \tilde V,\tilde Q)$ be the Chevalley–Eilenberg algebras of $\frL$ and $\tilde \frL$, respectively. The Chevalley–Eilenberg algebras of $\hat \frL$ and $\hat{\tilde \frL}$ are then

\begin{equation}
(\caE(V),\hat Q=Q_0+\sfE\circ Q\circ \sfP)
\eand
(\caE(\tilde V),\hat{\tilde Q}=Q_0+\sfE\circ \tilde Q\circ \sfP)~,
\end{equation}

cf. <ref>. The dual of the morphism $\phi$,

\begin{equation}
\Phi:\sfCE(\frL)\rightarrow \sfCE(\tilde \frL)~,
\end{equation}

trivially lifts to the following dual of an $E_2L_\infty$-algebra morphism

\begin{equation}
\hat \Phi:\sfCE(\hat\frL)\rightarrow \sfCE(\hat{\tilde \frL})\ewith\hat \Phi(v)\coloneqq \sfE(\Phi(v))~,
\end{equation}

and we note that $\hat \Phi \circ \sfE=\sfE\circ \Phi$. It then follows that

\begin{equation}
\begin{aligned}
(\hat \Phi\circ \sfE\circ Q)(v)&=(\sfE\circ \Phi\circ Q)(v)
\\
\hat \Phi(\hat Qv)&=(\sfE\circ \tilde Q\circ \sfP\circ \sfE\circ\Phi)(v)~,
\\
\hat \Phi(\hat Qv)&=\hat{\tilde Q}\hat \Phi(v)~,
\end{aligned}
\end{equation}

and $\hat\Phi$ is the dual to the desired morphism of $E_2L_\infty$-algebras $\hat \phi$.

§.§ 2-term EL-infinity-algebras

Having identified $L_\infty$-algebras within $E_2L_\infty$-algebras, let us also make contact with the 2-term $E_2L_\infty$-algebras of <cit.>.

In <cit.>, Baez–Crans introduced semistrict Lie 2-algebras, which are linear categories equipped with a strictly antisymmetric  bilinear functor,  the categorified Lie bracket, that is only required to satisfy the Jacobi identity up to a coherent trilinear natural transformation, the Jacobiator.  In <cit.>  semistrict Lie 2-algebras were fully categorified  to weak Lie 2-algebras by also relaxing  antisymmetry  to hold only up to a coherent natural transformation, the alternator.  Of course, a weak Lie 2-algebra with trivial alternator is a semistrict Lie 2-algebra. Similarly,  a weak Lie 2-algebra  with trivial Jacobiator is referred to as a  hemi-strict Lie 2-algebra. If both the alternator and Jacobiator are trivial, it is a strict Lie 2-algebra.

By passing to its normalised chain complex, we transition from the categorical description containing many redundancies to a more convenient description in terms of differential graded algebras. In particular, a semistrict Lie 2-algebra is seen to be equivalent to a 2-term $L_\infty$-algebra <cit.>, i.e. an $L_\infty$-algebra with underlying graded vector space concentrated in degrees $-1$ and $0$. Analogously,   by    passing to its normalised chain complex, any weak Lie 2-algebra is seen to be equivalent to a  2-term $E_2L_\infty$-algebra in the sense of <cit.>, where the letter $E$ was added to indicate that `everything' is relaxed up to homotopy.

An $E_2L_\infty$-algebra structure on a two-term complex concentrated in degrees $-1$ and $0$, $\frE:\frE_{-1}\xrightarrow{~\eps_1~}\frE_0$ has only three non-trivial higher products,

\begin{equation}
\begin{aligned}
\eps_2\coloneqq \eps_2^0~&:~\frE_i\otimes \frE_j\rightarrow \frE_{i+j}~,
\\
\eps_3\coloneqq \eps_3^{00}~&:~\frE_0\otimes \frE_0\otimes \frE_0\rightarrow \frE_{-1}~,
\\
\sfalt\coloneqq \eps_2^1~&:~\frE_0\otimes \frE_0\rightarrow \frE_{-1}~.
\end{aligned}
\end{equation}

The map $\eps_2$ is a chain map, and the maps $\sfalt$ and $\eps_3$ are chain homotopies[Here, $\sigma_{12}$ denotes the obvious permutation.]

\begin{equation}
\begin{aligned}
\sfalt&\colon\eps_2(-,-)+\eps_2(-,-)\circ \sigma_{12} \rightarrow 0~,\\
\eps_3&\colon\eps_2(-,\eps_2(-,-))-\eps_2(\eps_2(-,-),-)-\eps_2(-,\eps_2(-,-))\circ \sigma_{12}\rightarrow 0~.
\end{aligned}
\end{equation}

In addition, the higher products satisfy the relations

\begin{equation}\label{eq:EL:axioms}
\begin{aligned}
\\
\\
\\
\end{aligned}
\end{equation}

for all $x_i\in \frE_0$.

Perhaps the easiest way of proving the above relations is to consider the corresponding Chevalley–Eilenberg algebra (and we again assume, for simplicity, that the degree-wise duals of $\frE$ are nice, cf. <ref>). That is, we consider the tensor algebra $\caE(V)$ for $V=\frE[1]^*$. The differential $Q$ is determined by its action on the basis elements $t^\alpha$. Since the latter can be of degree $1$ or $2$, it follows that $Qt^\alpha$ is of degree $2$ or $3$, and therefore it has to be of the form

\begin{equation}\label{eq:form_EL_infty_Q}
\begin{aligned}
Q t^\alpha&=-(-1)^{|\beta|}m^\alpha_\beta t^\beta-(-1)^{|\gamma|(|\beta|-1)}\,m^{0,\alpha}_{\beta\gamma}\,t^\beta \oslash_0 t^\gamma
\\
&~~~-(-1)^{|\beta|+|\gamma|+|\delta|+|\gamma|(|\beta|-1)+|\delta|(|\beta|+|\gamma|-2)}\,m^{00,\alpha}_{\beta\gamma\delta}\,(t^\beta \oslash_0 t^\gamma) \oslash_0 t^\delta
\\
&~~~-(-1)^{|\beta|+|\gamma|+|\gamma|(|\beta|-1)}\,m^{1,\alpha}_{\beta\gamma}\,t^\beta \oslash_1 t^\gamma~,
\end{aligned}
\end{equation}

where the $(t^\alpha)$ form a basis on $V$. The above formula can be reduced further when we split $(t^\alpha)=(r^a,s^i)$, where $|r^a|=1$ and $|s^i|=2$:

\begin{equation}\label{eq:form_2-term-EL_infty_Q}
\begin{aligned}
Q r^a&=-m^a_a s^a-m^a_{bc}\,r^b \oslash_0 r^c~,
\\
Q s^i&=-m^i_{a j}\,r^a\oslash_0 s^j+m^a_{ja}\,s^j\oslash_0 r^a\\
&~~~+m^i_{abc}\,(r^a\oslash_0 r^b) \oslash_0 r^c-n^i_{ab}\,r^a\oslash_1 r^b~.
\end{aligned}
\end{equation}

\begin{equation}
\begin{gathered}
\eps_1(\tau_\alpha)=m^\beta_\alpha\tau_\beta,~~\eps_2(\tau_\alpha,\tau_\beta)=m^{0,\gamma}_{\alpha\beta}\tau_\gamma,~~\eps_3(\tau_\alpha,\tau_\beta,\tau_\gamma)=m^{00,\delta}_{\alpha\beta\gamma}\tau_\delta,~~\sfalt(\tau_\alpha,\tau_\beta)=m^{1,\gamma}_{\alpha\beta}\tau_\gamma
\end{gathered}
\end{equation}

with respect to the basis $(\tau_\alpha)$ of $\frE$, which is shifted-dual to the basis $(t^\alpha)$ of $V=\frE[1]^*$, we readily verify that $Q^2=0$ corresponds to the equations (<ref>).

The properties (<ref>), with a slightly weaker condition on $\sfalt$, were given as the defining axioms in the definition of a 2-term $E_2L_\infty$-algebra in the sense of <cit.>.

An $E_2L_\infty$-algebra of the form considered in <ref> is a 2-term $E_2L_\infty$-algebra in the sense of <cit.> with a graded symmetric alternator.

Our additional condition of graded symmetric alternator is, in fact, very natural. It guarantees a rectification to semistrict Lie 2-algebras, and we shall return to this point in <ref>.

§.§ EL-infinity-algebras as L-infinity-algebras and the rectification theorem

In <ref>, we identified $L_\infty$-algebras with semistrict $E_2L_\infty$-algebras, which suggests that there should be a rectification of $E_2L_\infty$-algebras to $L_\infty$-algebras. This is indeed the case, as we show below.

We start with a projection of $E_2L_\infty$-algebras onto $L_\infty$-algebras, extending results of <cit.>.

Any $E_2L_\infty$-algebra $(\frE,\eps_i)$ induces an $L_\infty$-algebra structure on the graded vector space $\frE$. This $L_\infty$-algebra structure is induced by homotopy transfer using the homotopy (<ref>).

The proof is readily obtained by applying the homological perturbation lemma to the contracting homotopy

\begin{equation}
\begin{tikzcd}
\ar[loop,out=204,in= 156,distance=70,"\sfH_0"](\caE(V),Q_0+Q_1+Q_\delta)\arrow[r,shift left]{}{\sfP_0} & (\odot^\bullet V,Q_L) \arrow[l,shift left]{}{\sfE_0}~,
\end{tikzcd}
\end{equation}

cf. (<ref>). Consider the Chevalley–Eilenberg algebra $\sfCE(\frE)$ of $\frE$, and split the differential $Q=Q_0+Q_1+Q_\delta$ into $Q_0$, a linear part $Q_1$, and a perturbation $Q_\delta$. Then homotopy transfer yields a differential

\begin{equation}\label{eq:HPL_formula_Q_EL_to_L}
Q_L=Q_1+\sfP_0\circ Q_\delta\circ\sfE_0-\sfP_0\circ Q_\delta\circ \sfH_0\circ Q_\delta\circ \sfE_0+\sfP_0\circ Q_\delta\circ \sfH_0\circ Q_\delta\circ \sfH_0\circ Q_\delta\circ \sfE_0+\ldots
\end{equation}

on $\odot^\bullet(\frE[1]^*)$. By construction, $Q_L^2=0$.
Moreover, $Q_L$ satisfies the Leibniz rule on $\odot^\bullet(\frE[1]^*)$: the deformation terms in the Leibniz rule (<ref>) are graded antisymmetric, and this graded antisymmetry is preserved by subsequent applications of $\sfH$ and $Q_\delta$. The final projector $\sfP_0$ then eliminates these terms.

As an example, we can compute the antisymmetrisation of a 2-term $E_2L_\infty$-algebra and reproduce[Due to different conventions, there is a relative minus sign between our $\mu_3$ and that of <cit.>.] <cit.>.

Given a 2-term $E_2L_\infty$-algebra $\frE$, there is an $L_\infty$-algebra structure on the graded vector space $\frE$ with higher products

\begin{equation}\label{eq:antisymmetrisation_2-term}
\begin{aligned}
\mu_1(y)&\coloneqq \eps_1(y)~,\\
\mu_2(x_1,x_2)&\coloneqq \tfrac12(\eps_2(x_1,x_2)-\eps_2(x_2,x_1))~,\\
\mu_2(x_1,y)=-\mu_2(y,x_1)&\coloneqq \tfrac12(\eps_2(x_1,y)-\eps_2(y,x_1))~,\\
\mu_3(x_1,x_2,x_3)&\coloneqq \tfrac{1}{3!}\sum_{\sigma\in S_3}\chi(\sigma;x_1,x_2,x_3)\Big(\eps_3(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)})\\
&\hspace{4.5cm}+\tfrac12 \sfalt(\eps_2(x_{\sigma(1)},x_{\sigma(2)}),x_{\sigma(3)})\Big)
\end{aligned}
\end{equation}

for all $x\in \frE_0$ and $y\in \frE_{-1}$.

We start from the Chevalley–Eilenberg algebra $\sfCE(\frE)$. As always, we assume for convenience that there is a basis, explicitly given by elements $r^\alpha$, $s^a$ of degrees $1$ and $2$ of $\frE[1]^*$. The differential then reads as

\begin{equation}\label{eq:CE_differential_2-term}
\begin{aligned}
Q r^a&=-m^a_a s^a-m^a_{bc}\,r^b \oslash_0 r^c~,
\\
Q s^i&=-m^i_{a j}\,r^a\oslash_0 s^j+m^a_{ja}\,s^j\oslash_0 r^a\\
&~~~+m^i_{abc}\,(r^a\oslash_0 r^b) \oslash_0 r^c-n^i_{ab}\,r^a\oslash_1 r^b~,
\end{aligned}
\end{equation}

cf. (<ref>). Evaluate formula (<ref>) yields

\begin{equation}
\begin{aligned}
Q_L r^a &=-m^a_i s^i-\tfrac12 m^a_{bc}\,r^b \odot r^c~,
\\
Q_L s^i&=-(m^i_{a j}+m^i_{ja})r^a \odot s^j+\tfrac1{3!}(m^i_{abc}+\tfrac12 n^i_{dc}m^d_{ab})r^a\odot r^b\odot r^c~.
\end{aligned}
\end{equation}

This is the differential for the Chevalley–Eilenberg algebra of $(\frE,\mu_i)$ with the higher products (<ref>).

As observed already in <cit.>, the antisymmetrisation map is functorial for 2-term $E_2L_\infty$-algebras, and any morphism of $E_2L_\infty$-algebra induces a morphism between the corresponding antisymmetrised $L_\infty$-algebras. The antisymmetrisation map fails, however, to be functorial for 3-term $E_2L_\infty$-algebras. It is natural to conjecture that the antisymmetrisation is $n$-functorial for an $n$-term $E_2L_\infty$-algebra. We will not need this result and refrain from going further into these technicalities.

A new result is that this antisymmetrisation map lifts indeed to a quasi-isomorphism of $E_2L_\infty$-algebras.

Let $\frE$ be an $E_2L_\infty$-algebra, and let $\frE'$ be the $L_\infty$-algebra induced by <ref>, regarded as an $E_2L_\infty$-algebra. Then there is a quasi-isomorphism $\phi:\frE\rightarrow \frE'$.

We prove this statement again using the Chevalley–Eilenberg algebras $\sfCE(\frE)$ and $\sfCE(\frE')$. Note that as graded vector spaces, $\frE[1]^*=\frE'[1]^*$. If $Q=Q_0+Q_1+Q_\delta$ is the differential on $\sfCE(\frE)$, then the differential on $\sfCE(\frE')$ reads as

\begin{equation}\label{eq:lifted_differential}
Q'v=Q_1v+\sfE_0\circ \sfP_0\circ (\sfid+Q_\delta\circ \sfH_0)^{-1}\circ Q_\delta v
\end{equation}

for all $v\in \frE[1]^*$. We must construct an invertible $\opEilh$-algebra morphism $\Phi:\caE(\frE[1]^*)\rightarrow \caE(\frE[1]^*)$ satisfying $Q\Phi=\Phi Q'$. The desired morphism on $\frE[1]^*$ is[This morphism implements a coordinate transformation such that the image of $Q$ on $\tilde v=\Phi(v)$ has no component in the subspace $Q_0\sfH_0\caE(\frE[1]^*)$. This then implies that it has no component in $\sfH_0 Q_0\caE(\frE[1]^*)$ either. The only remaining component is in $\sfE_0\sfP_0\caE(\frE[1]^*)$, which implies that $Q$ is the Chevalley–Eilenberg differential of an $L_\infty$-algebra, trivially regarded as an $E_2L_\infty$-algebra.]

\begin{equation}
\Phi(v)=(\sfid-\sfH_0\circ Q_\delta+\sfH_0\circ Q_\delta\circ \sfH_0\circ Q_\delta-\ldots)(v)=(\sfid+\sfH_0\circ Q_\delta)^{-1}(v)~,
\end{equation}

and using

\begin{equation}
\begin{aligned}
Q_0\circ \sfH_0&=\sfid-\sfE_0\circ \sfP_0-\sfH_0\circ Q_0~,
\\
Q_0Q_\delta&=-Q_\delta^2-Q_\delta Q_0~,
\end{aligned}
\end{equation}

one readily verifies that $Q\Phi v=\Phi Q'v$ for all $v\in \frE[1]^*$, which is sufficient. Since the morphism is clearly invertible, this is a quasi-isomorphism.

We can now combine <ref>, <ref> as well as the strictification theorem for $L_\infty$-algebras to obtain the following.

Any $E_2L_\infty$-algebra is quasi-isomorphic to a differential graded Lie algebra, trivially regarded as an $E_2L_\infty$-algebra.

More directly, this follows from the strictification theorem for generic homotopy algebras, see e.g. <cit.>.

<Ref> also shows that the choice of <cit.> not to symmetrise the alternator was perhaps not the best. It leads to a classification of 2-term $E_2L_\infty$-algebras which is generally larger than that of $L_\infty$-algebras <cit.>, contradicting the rectification theorem expected in line with the situation for $L_\infty$-algebras.

A consequence of the strictification theorem and homotopy transfer is the following. Just as for $\ophLie$-algebras, we can also tensor an $E_2L_\infty$-algebra by a differential graded commutative algebra[One may be tempted to replace the differential graded commutative algebra with an $\opEilh$-algebra, but already the product between an $\opEilh$-algebra and an $\ophLie$-algebra does not carry a natural $\ophLie$-algebra structure.]:

The tensor product of an $E_2L_\infty$-algebra and a differential graded commutative algebra carries a natural $E_2L_\infty$-algebra structure.

We can invoke the argument presented in <cit.> for the existence of general tensor products between certain homotopy algebras. That is, by <ref>, $\frE$ is quasi-isomorphic to a hemistrict $E_2L_\infty$-algebra $\frE^{\rm hst}$, and the chain complexes $\frA\otimes \frE^{\rm hst}$ and $\frA\otimes \frE$ are quasi-isomorphic. By <ref>, $\frA\otimes \frE^{\rm hst}$ carries an $\ophLie$-algebra structure, and the homological perturbation lemma allows us to perform a homotopy transfer from $\frA\otimes \frE^{\rm hst}$ to $\frA\otimes \frE$, leading to the desired $E_2L_\infty$-algebra structure.

Instead of using the above elegant but abstract argument, we can also perform a direct computation in the dual Chevalley–Eilenberg picture. This leads to the following explicit formulas for the tensor product $\hat \frE=\frA\otimes \frE$ of a differential graded commutative algebra $\frA$ and an $E_2L_\infty$-algebra $\frE$:

\begin{equation}
\begin{gathered}
\hat \frE\coloneqq \frA\otimes \frE=\oplus_{k\in \IZ}(\frA\otimes \frE)_k~,~~~(\frA\otimes \frE)_k=\sum_{i+j=k} \frA_i\otimes \frE_j~,
\\
\hat\eps_1(a_1\otimes x_1)=(\rmd a_1)\otimes x_1+(-1)^{|a_1|}a_1\otimes \eps_1(x_1)~,
\\
\hat\eps^I_k(a_1\otimes x_1,\ldots,a_k\otimes x_k)=(-1)^{|I|(|a_1|+\ldots+|a_k|)}(a_1\ldots a_k)\otimes \eps^I_k(x_1,\ldots,x_k)~.
\end{gathered}
\end{equation}

§.§ Examples: String Lie algebra models

Let us illustrate the above structure theorems using the important and archetypal examples of 2-term $E_2L_\infty$-algebra models for the string Lie algebra. We have already encountered the $E_2L_\infty$-algebras $\frstring^{\rmwk,\vartheta}_\rmsk(\frg)$ in (<ref>). A short computation using formulas (<ref>) shows that these all antisymmetrise to the skeletal model of the string 2-term $L_\infty$-algebra (previously discussed in <ref>):

\begin{equation}\label{eq:skeletal_string}
\begin{gathered}
\frstring_\rmsk(\frg)~\coloneqq ~(\IR\xrightarrow{~0~}\frg)~,
\\
\mu_2(x_1,x_2)=[x_1,x_2]~,~~~\mu_3(x_1,x_2,x_3)=(x_1,[x_2,x_3])~.
\end{gathered}
\end{equation}

Recall from <ref> that this $L_\infty$-algebra (which is a minimal model for its quasi-isomorphism class) is quasi-isomorphic to the loop model,

\begin{equation}
\begin{gathered}
\frstring_\rmlp(\frg)~\coloneqq ~(L_0\frg\oplus\IR\frg\xrightarrow{~\mu_1~}P_0\frg)~,
\\
\mu_1(\lambda,r)=\lambda~,
\\
\mu_2(\gamma_1,\gamma_2)=[\gamma_1,\gamma_2]~,~~~\mu_2(\gamma_1,(\lambda,r))=([\gamma_1,\lambda],2\int_0^1\rmd \tau~(\dot \gamma_1,\lambda)~,
\\
\mu_3(\gamma_1,\gamma_2,\gamma_3)=0~,
\end{gathered}
\end{equation}

where $L_0\frg$ and $P_0\frg$ are the based path and based loop spaces of $\frg$ defined in <ref>. There are two quasi-isomorphisms,

\begin{equation}
\begin{tikzcd}
\frstring_\rmsk(\frg) \arrow[r,bend left=10]{}{\psi} & \frstring_\rmlp(\frg) \arrow[l,bend left=10]{}{\phi}
\end{tikzcd}~,
\end{equation}

and their explicit forms are found e.g. in <cit.>. This implies that there is a quasi-isomorphic family of $E_2L_\infty$-algebras that antisymmetrise to $\frstring_\rmlp(\frg)$, which is readily found:

\begin{equation}
\begin{gathered}
\frstring^{\rmwk,\vartheta}_\rmlp(\frg)~\coloneqq ~(L_0\frg\oplus \IR\xrightarrow{~\eps_1~}P_0\frg)~,
\\
\eps_1(\lambda,r)=\lambda~,
\\
\eps^0_2(\gamma_1,\gamma_2)=[\gamma_1,\gamma_2]~,~~~\eps^0_2(\gamma_1,(\lambda,r))=\left([\gamma_1,\lambda],2\int_0^1\rmd \tau~(\dot \gamma_1,\lambda)\right)~,
\\
\eps_2^1(\gamma_1,\gamma_2)=\big(0,2\vartheta(\dpar\gamma_1,\dpar\gamma_2)\big)
\\
\eps^{00}_3(\gamma_1,\gamma_2,\gamma_3)=\vartheta(\dpar \gamma_1,[\dpar \gamma_2,\dpar \gamma_3])~.
\end{gathered}
\end{equation}

Altogether, we can summarise the situation in the following commutative diagram:

\begin{equation}
\begin{tikzcd}
\frstring_\rmsk^{\rmwk,\vartheta}(\frg) \arrow[r,bend left=10]{}{\hat\psi} \arrow[d,"\rm asym"] & \frstring_\rmlp^{\rmwk,\vartheta}(\frg) \arrow[d,"\rm asym"] \arrow[l,bend left=10]{}{\hat\phi}
\\
\frstring_\rmsk(\frg) \arrow[r,bend left=10]{}{\psi} & \frstring_\rmlp(\frg) \arrow[l,bend left=10]{}{\phi}
\end{tikzcd}
\end{equation}

The morphisms ${\rm asym}$ are special cases of the antisymmetrisation map (<ref>), and the morphisms $\hat\phi$ and $\hat\psi$ are formed by lifts of the morphisms $\phi$ and $\psi$ as given by <ref>.

Generically, on top of every $L_\infty$-algebra, there is a family of $E_2L_\infty$-algebras that antisymmetrise to it. The additional structure constants contained in the alternators of the $E_2L_\infty$-algebra will turn out to be crucial in the construction of higher gauge theories.

§ RELATIONS TO OTHER ALGEBRAS

In the following, we explain the relation between $E_2L_\infty$-algebras and homotopy Leibniz algebras and, in particular, to differential graded Lie algebras. The latter prepares our interpretation of tensor hierarchies.

§.§ Relation to homotopy Leibniz algebras

Just as Lie algebras are Leibniz algebras that happen to have an antisymmetric Leibniz bracket, $E_2L_\infty$-algebras are $\opLeib_{\infty}$-algebras whose higher Leibniz brackets are antisymmetric up to homotopies. Homotopy Leibniz algebras were defined in <cit.>, and they can be defined as differential free Zinbiel algebras. <cit.> which, as suggested by the name[This nomenclature is a successful joke suggested by J. M. Lemaire. Zinbiel algebras are also known as (commutative) shuffle algebras, and the free Zinbiel algebra over a vector space is the shuffle algebra on its tensor algebra.]

Explicitly, consider the semifree non-associative tensor algebra $\oslash_0^\bullet V$ for a graded vector space $V$ with only the first relation of (<ref>) imposed. A (nilquadratic) differential $Q$ on this algebra which satisfies the ordinary Leibniz rule then defines a homotopy Leibniz algebra. All the additional structure in $\opEilh$ (as well as the resulting additional structure in $E_2L_\infty$-algebras) capture the appropriate notion of symmetry up to homotopy of the higher Leibniz brackets.

Ordinary Leibniz algebras form an interesting source of 2-term $\ophLie$-algebras, which had been observed before:

Any Leibniz algebra induces canonically a hemistrict 2-term $E_2L_\infty$-algebra concentrated in degrees $-1$ and $0$.

Explicitly, let $\frg$ be a Leibniz algebra, and write $\frg^{\rm ann}=[\frg,\frg]$. Then

\begin{equation}
\frE(\frg)=\big(\frE(\frg)_{-1}\xrightarrow{~\eps_1~}\frE(\frg)_{0}\big)\coloneqq \big(\frg^{\rm ann} \xhookrightarrow{~~~~} \frg\big)
\end{equation}

is a differential graded Leibniz algebra, and we promote it to a 2-term $E_2L_\infty$-algebra by

\begin{equation}
\sfalt(e_1,e_2)\coloneqq [e_1,e_2]+[e_2,e_1]\in \frg^{\rm ann}
\end{equation}

for all $e_1,e_2\in \frg$.[We note that this result, together with <ref>, immediately implies that any Leibniz algebra gives rise to a 2-term $L_\infty$-algebra as shown separately in <cit.>.]

§.§ hLie-algebras from differential graded Lie algebras and derived brackets

Given a differential graded Lie algebra $\frg=\bigoplus_{k\in \IZ}\frg_k$, one can construct an associated $L_\infty$-algebra on the grade-shifted partial complex $\frL=\bigoplus_{k\leq 0}\frg[1]$. As explained in <cit.>, this is a corollary to the result of <cit.> that the mapping cone of a morphism between two differential graded Lie algebras carries a natural $L_\infty$-algebra structure. In this section, we present a refinement of this associated $L_\infty$-algebra to an $\ophLie$-algebra. The existence of the $L_\infty$-algebra is then a corollary to the antisymmetrisation <ref>. Our construction extends the construction of $\opLeib_\infty$-algebras from $\opLeib$-algebras in <cit.> as well as the construction of 2-term $E_2L_\infty$-algebras from 3-term differential graded Lie algebras in <cit.>.

Given a differential graded Lie algebra, we readily construct a grade-shifted $\ophLie$-algebra.

Given a differential graded Lie algebra $(\frg,\rmd_\frg,\{-,-\})$ with $\frg=\bigoplus_{k\in \IZ}\frg_k$, we have an associated $\ophLie$-algebra

\begin{equation}
\frE=\bigoplus_{k\leq 0} \frE_k~,~~~\frE_k=\frg_{k-1}
\end{equation}

with higher products

\begin{equation}
\begin{aligned}
\eps_1(x_1)&\coloneqq \begin{cases}
\rmd_\frg x~&\mbox{for}~|x|_\frE<0~,
\\
0 &\mbox{else}~,
\end{cases}
\\
\eps_2^i(x_1,x_2)&\coloneqq \begin{cases}
\{\delta x_1,x_2\} & \mbox{for}~i=0~,
\\
(-1)^{|x_1|_\frE}\{x_1,x_2\} & \mbox{for}~i=1~,
\\
0& \mbox{else}
\end{cases}
\end{aligned}
\end{equation}

for all $x_1,x_2\in \frE$. Here, $\delta\coloneqq \rmd_{\frg}|_{\frg_{-1}}$ and $|x_1|_\frE$ denotes the degree of $x_1$ in $\frE$.

The proof consists of a straightforward verification of the axioms of an $\ophLie$-algebra (<ref>), which is most conveniently done again with a computer algebra program.

Let us discuss the explicit form of the antisymmetrisation in some more detail. We assume, as usual, that $\frE$ admits a nice basis $(\tau_\alpha)$, so that $\frE[1]^*$ has a dual basis $(t^\alpha)$. the Chevalley–Eilenberg differential then reads as

\begin{equation}\label{eq:Q_hLie2}
Q t^\alpha=-(-1)^{|\beta|}m^\alpha_\beta t^\beta-(-1)^{i(|\beta|+|\gamma|)+|\gamma|(|\beta|-1)}\,m^{i,\alpha}_{\beta\gamma}\,t^\beta \oslash_i t^\gamma~,
\end{equation}

and we have the following theorem.

For each $\ophLie$-algebra $(\frE,\eps^i_j)$ (with the above mentioned restrictions), there is an $L_\infty$-algebra $(\frE,\mu_i)$ with first four higher products reading as

\begin{equation}\label{eq:antisymmetrisation_hLie}
\begin{aligned}
\mu_1(x_1)&\coloneqq \eps_1(x_1)~,\\
\mu_2(x_1,x_2)&\coloneqq \tfrac12(\eps^0_2(x_1,x_2)-\eps^0_2(x_2,x_1))~,\\
\mu_3(x_1,x_2,x_3)&\coloneqq \tfrac{1}{3!}\sum_{\sigma\in S_3}\chi(\sigma;x_1,x_2,x_3)\Big(\eps^0_3(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)})\\
&\hspace{1.7cm}+\tfrac14\big( \eps^1_2(\eps^0_2(x_{\sigma(1)},x_{\sigma(2)}),x_{\sigma(3)})
\\
\mu_4(x_1,x_2,x_3,x_4)&\coloneqq 0~,
\end{aligned}
\end{equation}

for all $x_i\in \frE$.

We use again <ref> and determine the Chevalley–Eilenberg differential (<ref>) of $(\frE,\mu_i)$,

which allows us to compute $Q_L$ up to quartic order. This produces the higher products (<ref>).

We can now compose the map from differential graded Lie algebras to $\ophLie$-algebras with the antisymmetrisation <ref>. This reproduces the following propostion of <cit.>, which in turn is a specialisation of <cit.>:

Given a differential graded Lie algebra $(\frg,\rmd,[-,-])$, we have an $L_\infty$-algebra structure on the truncated complex

\begin{equation}
\frg_{\leq 0}=\big(~~
\ldots~\xrightarrow{~\rmd~}~\frg_{-2}~\xrightarrow{~\rmd~}~\frg_{-1}~\xrightarrow{~\rmd~}~\frg_{0}~\xrightarrow{~0~}~*~\xrightarrow{~0~}~\ldots~~)
\end{equation}

\begin{equation}
\begin{aligned}
\mu_1(x_1)&=\begin{cases}
\rmd x_1 & \mbox{for $|x_1|<0$}~,
\\
0 & \mbox{for $|x_1|=0$}
\end{cases}
\\
\mu_k(x_1,\ldots,x_k)&=\frac{(-1)^k}{(k-1)!}B_{k-1}\sum_{\sigma\in S_{k}}\chi(\sigma;x_1,\ldots,x_k)[[\ldots[[\delta x_{\sigma(1)},x_{\sigma(2)}],\ldots],x_{\sigma(k)}]~,
\end{aligned}
\end{equation}

\begin{equation}
\delta(x_1)=\begin{cases}
\rmd x_1 & \mbox{for $|x_1|=0$}~,
\\
0 & \mbox{else}
\end{cases}
\end{equation}

for all $x_i\in \frg_{\leq 0}$. Here, $B_k$ are the Bernoulli numbers[i.e. $B_0,B_1,\ldots=1,-\tfrac12,\tfrac{1}{6},0,-\tfrac{1}{30},0,\tfrac{1}{42},\ldots$].

Altogether, our above constructions suggest the following picture:

\begin{equation}\label{eq:diagram_algebras}
\begin{tikzcd}
\mbox{dg Lie algebra} \arrow[r,"\text{\Cref{thm:ophLie_from_dgLA}}", bend left=30] \arrow[rr,"\text{\Cref{prop:dgLA_to_L_infty}}",swap, bend right=20]& \mbox{$\ophLie$-algebra} \arrow[r,"\text{\Cref{thm:antisym_hLie}}", bend left=30] & \mbox{$L_\infty$-algebra}
\end{tikzcd}
\end{equation}

Our formulas (<ref>) show that this picture is true for differential graded Lie algebras concentrated in degrees $d\geq -3$. For more general differential graded Lie algebras, this picture is still very plausible from the expression for (<ref>). A complete proof, however, would require an explicit expression of the homotopy $\sfH_0$ to all orders, which has not been completed at the time of writing of this thesis.

From <ref> it is also clear that $\mu_4$ in (<ref>) vanishes because $B_3=0$. Similarly, all even higher brackets $\mu_{2i}$ with $i\geq 1$ vanish, as the odd Bernoulli numbers $B_k$ for $k\geq 3$ vanish.

As a simple example, consider a quadratic Lie algebra $\frg$, and construct the differential graded Lie algebra

\begin{equation}
\frG=(~\ldots\xrightarrow{~0~}*\xrightarrow{~0~}\underbrace{\IR}_{\frG_{-2}}\xrightarrow{~0~}\underbrace{\frg}_{\frG_{-1}}\xrightarrow{~\sfid~}\underbrace{\frg}_{\frG_{0}}\xrightarrow{~0~}*\xrightarrow{~0~}\ldots~)~,
\end{equation}

concentrated in degrees $-2$, $-1$, $0$ with differential and Lie brackets

\begin{equation}
\end{equation}

for all $x_1,x_2\in \frG_0\cong\frg$ and $y_1,y_2\in \frG_{-1}\cong \frg$, where $[-,-]$ and $(-,-)$ are the Lie bracket and the Cartan–Killing form on $\frg$. Then the associated $\ophLie$-algebra is

\begin{equation}
\begin{gathered}
\frE=(~\IR\xrightarrow{~0~}\frg~)~,
\\
\eps_1(r)\coloneqq 0~,
\\
\eps^0_2(x_1,x_2)=[x_1,x_2]~,~~~\eps^1_2(x_1,x_2)=2(x_1,x_2)~.
\end{gathered}
\end{equation}

We thus recover the hemistrict $E_2L_\infty$-algebra $\frstring^{\rm wk,1}_{\rm sk}(\frg)$ introduced in <ref>. The antisymmetrisation of this $\ophLie$-algebra then yields the skeletal string Lie 2-algebra model $\frstring_{\rm sk}(\frg)$. Interestingly, a quick consideration of the case leads to the conclusion that there is no differential graded Lie algebra that reproduces the strict string Lie 2-algebra model $\frstring_\rmlp(\frg)=\frstring^{\rmwk,0}_\rmlp(\frg)$. This points towards a possible extension of <ref> producing $E_2L_\infty$-algebras from certain $L_\infty$-algebras.

§ HIGHER GAUGE THEORY WITH EL-INFINITY-ALGEBRAS

In this section, we develop and explore the generalities of higher gauge theory using $E_2L_\infty$-algebras as higher gauge algebras.

§.§ Homotopy Maurer–Cartan theory for EL-infinity-algebras

Recall that given an $L_\infty$-algebra $\frL$ with higher products $\mu_i$, there is a functor $\sfMC(\frL,-)$ taking a differential graded commutative algebra $\fra$ to Maurer–Cartan elements with values in $\fra$, cf. e.g. <cit.>. This functor is represented by the Chevalley–Eilenberg algebra $\sfCE(\frL)$ of the $L_\infty$-algebra.

What we usually call Maurer–Cartan elements in $\frL$ are Maurer–Cartan elements with values in $\IR$, where the latter is regarded as a trivial differential graded algebra $\IR_\fra$ with underlying vector space $\IR$, spanned by a generator $w$ subject to the relation $w^2=w$, and trivial differential.

For concreteness' sake, let us assume that $\frL$ is degree-wise finite, and let $(t^A)$ be the generators of $\frL[1]^*$ dual to some basis $(\tau_A)$ of $\frL$. A Maurer–Cartan element is encoded in a morphism of differential graded commutative algebras $a:\sfCE(\frL)\rightarrow \IR_\fra$, which is fully determined by the image of the generators $(t^\alpha)$ of degree $0$,

\begin{equation}\label{eq:dga_morphism}
a: \sfCE(\frL)\rightarrow \IR~,~~~t^\alpha \mapsto a^\alpha w
\end{equation}

for $a^\alpha\in \IR$. Dually, we have an element $a\coloneqq a^\alpha\tau_\alpha\in \frL_1$, the gauge potential. Compatibility with the differential requires the curvature

\begin{equation}
f\coloneqq \mu_1(a)+\tfrac12 \mu_2(a,a)+\tfrac1{3!}\mu_3(a,a,a)+\dotsb~~\in \frL_2
\end{equation}

to vanish, and the equation $f=0$ is called the homotopy Maurer–Cartan equation. This curvature satisfies the Bianchi identity

\begin{equation}
\sum_{k\geq0}\frac{1}{k!}\mu_{k+1}(a,\ldots,a,f)=0~.
\end{equation}

Infinitesimal gauge transformations are obtained from infinitesimal homotopies between morphisms from $\sfCE(\frL)$ to $\IR$. They are
parameterised by elements $c\in \frg_0$ and act according to

\begin{equation}\label{eq:GaugeTrafo}
\delta_{c} a=\sum_{k\geq 0} \frac{1}{k!}\mu_{k+1}(a,\ldots,a, c)~.
\end{equation}

Higher homotopies yield higher gauge transformations.

Similarly, one defines Maurer–Cartan elements of an $A_\infty$-algebra with values in a differential graded algebra.

In the case of $E_2L_\infty$-algebras, we can still consider tensor products of a base $E_2L_\infty$-algebra $\frE$ and a differential graded commutative algebra $\frA$. However, the Chevalley–Eilenberg algebra $\sfCE(\frE)$ is an $\opEilh$-algebra and not a differential graded commutative algebra. Therefore the homotopy Maurer–Cartan functor cannot be represented by it directly.

There are two loopholes to this obstruction. First, we can lift the differential graded commutative algebra $\frA$, if it is semifree, to an $\opEilh$-algebra $\hat \frA$ as explained in <ref>. We can then consider $\opEilh$-algebra morphisms

\begin{equation}
a:\sfCE(\frE)\rightarrow \hat \frA~.
\end{equation}

Second, we can project $\sfCE(\frE)$ to the Chevalley–Eilenberg algebra of the $L_\infty$-algebra $\frL$ induced by $\frE$ and consider the usual morphisms

\begin{equation}
a:\sfCE(\frL)\rightarrow \frA~.
\end{equation}

A third approach is simply to consider general morphisms of $\opEilh$-algebras. In particular, one may want to replace differential forms with more general objects, cf. also <cit.>.

We note that, in general, the three different types of morphism will give rise to different sets of Maurer–Cartan elements with the first one encompassing the second one. In all the applications we are aware of, however, the second approach is the appropriate one. While the difference between an $E_2L_\infty$-algebra and the corresponding $L_\infty$-algebra obtained by antisymmetrisation is then invisible at the level of homotopy Maurer–Cartan theory, the additional algebraic structure in an $E_2L_\infty$-algebra is important in adjusting non-flat higher gauge theories.

§.§ Firmly adjusted Weil algebras from hLie-algebras

Special cases of adjusted Weil algebras whose corresponding morphisms (<ref>) into differential forms yield adjusted higher gauge theories with closed BRST complex are the following ones:

A firmly adjusted Weil algebra of an $L_\infty$-algebra $\frL$ is a differential graded commutative algebra obtained from the Weil algebra $\sfW(\frL)$ by a coordinate change

\begin{equation}\label{eq:firm_rotation}
\hat t^A\mapsto \hat t'^A\coloneqq \hat t^A+p^A_{B_1B_2\dotso B_mC_1C_2\dotso C_n}\hat t^{B_1}\dotsm\hat t^{B_m}t^{C_1}\dotsm t^{C_n}~,
\end{equation}

where $t^A\in \frL[1]^*$, $\hat t^A\in \frL[2]^*$, $m\ge1$, and $n\ge0$, such that the image of the resulting differential $Q_{\rm fadj}$ on generators in $\frL[2]^*$ contains no generator in $\frL[1]^*$ except for at most one of degree $1$.

We note that putting the generators $(\sigma t^A)$ to zero still recovers the Chevalley–Eilenberg algebra $\sfCE(\frL)$ of $\frL$. In this sense, the coordinate change has not changed the underlying $L_\infty$-algebra. Moreover, note that any Weil algebra is fully contractible in the sense that the cohomology of its linearised differential is trivial. Dually, it is the Chevalley–Eilenberg algebra of an $L_\infty$-algebra which is quasi-isomorphic to the trivial $L_\infty$-algebra. The non-trivial information contained in the Weil algebra is the relation between the generators $(t^A)$ and $(\sigma t^A)$, which translates under the morphism (<ref>) into the relation between gauge potentials and their curvatures. Our coordinate change thus changes the definition of the curvatures and, as partially flat homotopies describe gauge transformations, also the gauge transformations. Firmly adjusted Weil algebras ensure that the corresponding BRST complex closes: the restricted terms govern the Bianchi identities, which fix the gauge transformations of the curvatures. Closure of the latter is what induces the fake curvature conditions. Thus, firmly adjusted Weil algebras are adjusted Weil algebras.

As an example, consider the following firmly adjusted Weil algebra of the string Lie 2-algebra (<ref>):

\begin{equation}\label{eq:ext_twt_string_sk_differential}
\begin{aligned}
Q_{\rm fadj}~&:~&t^\alpha &\mapsto -\tfrac12 f^\alpha_{\beta\gamma} t^\beta  t^\gamma + \hat t^\alpha~,~&r &\mapsto \tfrac{1}{3!} f_{\alpha\beta\gamma} t^\alpha  t^\beta  t^\gamma -\kappa_{\alpha\beta}t^\alpha\hat t^\beta+ \hat r'~,
\\
&&\hat t^\alpha &\mapsto -f^\alpha_{\beta\gamma} t^\beta  \hat t^\gamma~,~
&\hat r'&\mapsto \kappa_{\alpha\beta}\hat t^\alpha\hat t^\beta~,
\end{aligned}
\end{equation}

which is obtained from the coordinate transformation $\hat r\mapsto \hat r'=\hat r+\kappa_{\alpha\beta} \hat t^\alpha t^\beta$. Here, $t\in \frg[1]^*$, $r\in \IR[2]^*$ and $\hat t=\sigma t$, $\hat r=\sigma r$. Under the morphism (<ref>), this firmly adjusted Weil algebra gives rise to the usual string connections

\begin{equation}
\begin{aligned}
a&=A+B\in \Omega^1(M,\frg)\oplus \Omega^2(M,\IR)~,
\\
f&=F+H\in \Omega^2(M,\frg)\oplus \Omega^3(M,\IR)~,
\\
F&=\rmd A+\tfrac12[A,A]~,
\\
H&=\rmd B-\tfrac1{3!}(A,[A,A])+(A,F)=\rmd B+{\rm cs}(A)~.
\end{aligned}
\end{equation}

More generally, consider an $L_\infty$-algebra obtained from an $\ophLie$-algebra by antisymmetrisation. For simplicity, we also assume that the $L_\infty$-algebra has maximally ternary brackets. Its Weil algebra then reads as

\begin{equation}
\begin{aligned}
Q_\sfW t^\alpha&=-(-1)^{|\beta|}m^\alpha_\beta t^\beta-(-1)^{|\gamma|(|\beta|-1)}\tfrac12\,m^{\alpha}_{\beta\gamma}\,t^\beta t^\gamma
\\
&~~~~~-(-1)^{|\beta|(|\gamma|+1)+|\delta|(|\beta|+|\gamma|+1)}\tfrac{1}{3!}\,m^{\alpha}_{\beta\gamma\delta}\,t^\beta t^\gamma t^\delta +\hat t^\alpha~,
\\
Q_\sfW \hat t^\alpha&=(-1)^{|\beta|}m^\alpha_\beta \hat t^\beta+(-1)^{|\gamma|(|\beta|-1)}\,m^{\alpha}_{\beta\gamma}\,\hat t^\beta t^\gamma
\\
&~~~~~+(-1)^{|\beta|(|\gamma|+1)+|\delta|(|\beta|+|\gamma|+1)}\tfrac{1}{2}\,m^{\alpha}_{\beta\gamma\delta}\,\hat t^\beta t^\gamma t^\delta~.
\end{aligned}
\end{equation}

In general, this Weil algebra is clearly not firmly adjusted because of the explicit form of $Q_\sfW\hat t^\alpha$. Let us therefore perform the coordinate change

\begin{equation}
\hat t^\alpha\mapsto \hat t'^\alpha\coloneqq \hat t^\alpha+s^\alpha_{\beta\gamma} \hat t^\beta t^\gamma~.
\end{equation}

The new Weil differential then reads as follows.

\begin{equation}\label{eq:fadjust_Qp}
\begin{aligned}
Q'_\sfW \hat t'^\alpha&=(-1)^{|\beta|}m^\alpha_\beta \hat t'^\beta
+(-1)^{1+|\beta|}s^\alpha_{\beta\gamma}\hat t'^\beta \hat t'^\gamma
+(-1)^{|\gamma|(|\beta|-1)}\,m^{\alpha}_{\beta\gamma}\,\hat t'^\beta t^\gamma
\\
&~~~~~+\big(-(-1)^{|\beta|} m^\alpha_\beta s^\beta_{\gamma\delta}+(-1)^{|\gamma|} s^\alpha_{\beta\delta}m^\beta_\gamma
+(-1)^{|\gamma|+|\delta|} s^\alpha_{\gamma\beta}m^\beta_\delta\big)\hat t'^\gamma t^\delta+\cdots~,
\end{aligned}
\end{equation}

where the ellipsis denotes cubic and higher terms. Let us now further restrict to $\ophLie$-algebras obtained from a differential graded algebra via <ref> with differential $\Theta^\alpha_\beta$ and structure constants $f^\alpha_{\beta\gamma}$. In this case, we have

\begin{equation}
\tfrac12 f^\alpha_{\delta\gamma}\Theta^\delta_\beta & \mbox{if $|\beta|=1$}~,
\\
0 & \mbox{else}~;
\end{cases}
\end{equation}

we also put

\begin{equation}
s^\alpha_{\beta\gamma}=\tfrac12(-1)^{|\beta|(|\gamma|+1)} f^\alpha_{\beta\gamma}~.
\end{equation}

In the above formulas, $|\alpha|,|\beta|,|\gamma|\geq 1$, and $|\delta|=0$. Together with the Jacobi identity for the $f^\alpha_{\beta\gamma}$, one can then easily verify that $Q'$ becomes a firmly adjusted Weil differential,

\begin{equation}\label{eq:fadjust_Q}
Q_{\rm fadj} \hat t'^\alpha=(-1)^{|\beta|}m^\alpha_\beta \hat t'^\beta
+(-1)^{1+|\beta|\,|\gamma|}\tfrac12f^\alpha_{\beta\gamma}\hat t'^\beta \hat t'^\gamma~.
\end{equation}

We conclude with the following theorem.

Given an $L_\infty$-algebra with maximally ternary brackets that is obtained from the antisymmetrisation of a differential graded Lie algebra by <ref>, then there is a corresponding firmly adjusted Weil algebra. The data necessary for an adjustment arises from the alternators in the corresponding $\ophLie$-algebra.

Below, we shall give examples motivated from higher gauge theory. We stress, however, that the definition of an adjustment is also interesting for purely algebraic considerations, as it allows for the definition of a differential graded algebra of invariant polynomials for an $L_\infty$-algebra which is compatible with quasi-isomorphisms of this $L_\infty$-algebra, cf. the discussion in <cit.>.

We also note that our construction highlights the features needed for obtaining a firmly adjusted Weil algebra. In particular, it is not necessary that the $\ophLie$-algebra was obtained from a differential graded Lie algebra; it was sufficient that there be a relation between the parameters $s^\alpha_{\beta\gamma}$ of the coordinate change and the structure constants $f^\alpha_{\beta\gamma}$ of the Lie algebra to ensure that (<ref>) reduces to (<ref>). This is the case, for example, in the tensor hierarchies in non-maximally supersymmetric gauged supergravity.

§.§ Example: (1,0)-gauge structures

As a first more involved example of $E_2L_\infty$-algebras arising in the context of higher gauge theory, let us consider the higher gauge algebra defined in <cit.>, see also <cit.>. This algebra is a specialisation of the general non-abelian algebraic structure identified in <cit.> and can be derived from tensor hierarchies, to which we shall return shortly. The latter had received an interpretation as an $L_\infty$-algebra with some “extra structure” before, cf. <cit.> as well as <cit.>. Here, we show that it is, in fact an $E_2L_\infty$-algebra.

The higher gauge algebra $\hat\frg^\omega$ for $\frg$ a quadratic Lie algebra has underlying graded complex

\begin{equation}\label{eq:ghsk-complex}
\hat\frg^\omega=\left(
\begin{tikzcd}[row sep=0cm,column sep=2cm]
\frg^*_v\arrow[r]{}{\mu_1=\sfid} & \frg^*_u & \IR^*_s \arrow[r]{}{\mu_1=\sfid} & \IR_p^*
\\
& \oplus & \oplus & \oplus
\\
\underbrace{\phantom{\frg^*_v}}_{\hat \frg^\omega_{\rm sk,-3}} & \underbrace{\IR_q}_{\hat \frg^\omega_{\rm sk,-2}} \arrow[r]{}{\mu_1=\sfid} & \underbrace{\IR_r}_{\hat \frg^\omega_{\rm sk,-1}} & \underbrace{\frg_t}_{\hat \frg^\omega_{\rm sk,0}}
\end{tikzcd}\right)~,
\end{equation}

where the subscripts merely help to distinguish between isomorphic subspaces. In <cit.>, this differential complex was extended to an $L_\infty$-algebra $\hat\frg^\omega$ with higher products

\begin{equation}\label{eq:ghsk-brackets}
\begin{aligned}
\mu_2(t_1,t_2)&=[t_1,t_2]\in\frg_t~,\\
\mu_2(t,u)&=u\big([-,t]\big)\in\frg^*_u~,~~~&
\mu_2(t,v)&=v\big([-,t]\big)\in\frg_v^*~,\\
\mu_3(t_1,t_2,t_3)&=(t_1,[t_2,t_3])\in\IR_r~,~~~&
\mu_3(t_1,t_2,s)&= s\big(\,(-,[t_1,t_2])\,\big)\in\frg^*_u~,
\end{aligned}
\end{equation}

where $t\in \frg_t$, etc. Moreover, $[-,-]$ and $(-,-)$ denote the Lie bracket and the quadratic form in $\frg$, respectively. When constructing gauge field strengths based on this $L_\infty$-algebra, the following, additional maps feature as part of the adjustment data:

\begin{equation}
\begin{aligned}
\nu_2(t_1,t_2)&=-2(t_1,t_2)\in \IR_r~,~~~&\nu_2(t,s)&=2s(-,t)\in\frg_u^*~,
\\
\nu_2(t_1,u_1)&=u_1\big([-,t_1]\big)\in \frg_v^*~.
\end{aligned}
\end{equation}

As motivated in more detail later, it is useful to first perform a quasi-isomorphism on $\hat\frg^\omega$ leading to the higher brackets

\begin{equation}\label{eq:ghsk-brackets2}
\begin{aligned}
\mu_2(t_1,t_2)&=[t_1,t_2]\in\frg_t~,\\
\mu_2(t,u)&=\tfrac12u\big([-,t]\big)\in\frg^*_u~,~~~&
\mu_2(t,v)&=\tfrac12v\big([-,t]\big)\in\frg_v^*~,\\
\mu_3(t_1,t_2,t_3)&=(t_1,[t_2,t_3])\in\IR_r~,~~~&
\mu_3(t_1,t_2,s)&= s\big(\,(-,[t_1,t_2])\,\big)\in\frg^*_u~,
\\
\mu_3(t_1,t_2,u)&= \tfrac14v\big(\,(-,[t_1,t_2])\,\big)\in\frg^*_v~.
\end{aligned}
\end{equation}

This is the $L_\infty$-algebra obtained by <ref> from the $\ophLie$-algebra $\frE$ with differential complex (<ref>) with $\eps_1=\mu_1$ and the additional binary products

\begin{equation}
\begin{aligned}
\eps_1&=\mu_1~,
\\
\eps^0_2(t_1,t_2)&=-\eps^0_2(t_2,t_1)=[t_1,t_2]\in \frg_t~,
\\
\eps^0_2(t,u)&=u\big([-,t]\big)\in \frg^*_u~,~~~&
\eps^0_2(t,v)&=v\big([-,t]\big)\in \frg_v^*~,
\\
\eps^1_2(t_1,t_2)&=\eps^1_2(t_2,t_1)=2(t_1,t_2)\in \IR_r~,
\\
\eps^1_2(t,s)&=3!s\big(-,t\big)\in\frg_u^*~,~~~
&\eps^1_2(s,t)&=\eps^1_2(t,s)=3!s\big(-,t\big)\in \frg_u^*~,
\\
\eps^1_2(t,u)&=u\big([-,t]\big)\in \frg_v^*~,~~~&\eps^1_2(u,t)&=\eps^1_2(t,u)=u\big([-,t]\big)\in \frg_v^*~,
\end{aligned}
\end{equation}

as one verifies by direct computation. This $\ophLie$-algebra is obtained from a differential graded Lie algebra $\frG$ by <ref>, and we have

\begin{equation}\label{eq:ghsk-complex2}
\frG=\left(
\begin{tikzcd}[row sep=0cm,column sep=1.6cm]
\frg^*_v\arrow[r]{}{\mu_1=\sfid} & \frg^*_u & \IR^*_s \arrow[r]{}{\mu_1=\sfid} & \IR_p^*
\\
& \oplus & \oplus & \oplus
\\
\underbrace{\phantom{\frg^*_v}}_{\frG_{-4}} & \underbrace{\IR_q}_{\frG_{-3}} \arrow[r]{}{\mu_1=\sfid} & \underbrace{\IR_r}_{\frG_{-2}} & \underbrace{\frg_t}_{\frG_{-1}} \arrow[r]{}{\mu_1=\sfid} & \underbrace{\frg_{\hat t}}_{\frG_0}
\end{tikzcd}\right)
\end{equation}

with the non-trivial Lie brackets $[-,-]_\frG$ fixed by

\begin{equation}
\begin{aligned}
[\hat t_1,\hat t_2]_\frG&\coloneqq [\hat t_1,\hat t_2]\in\frg_{\hat t}~,
~~~&[\hat t_1,t_2]_\frG&\coloneqq [\hat t_1,t_2]\in\frg_{t}~,
\\
[\hat t_1,u]_\frG&\coloneqq u([-,t])\in\frg_u^*~,
~~~&[\hat t_1,v]_\frG&\coloneqq v([-,t])\in\frg_u^*~,
\\
[t_1,t_2]_\frG&\coloneqq (t_1,t_2)\in\IR_t~,
~~~&[t_1,s]_\frG&\coloneqq \alpha_2 s(-,t_1)\in\frg_u^*~.
\end{aligned}
\end{equation}

This is an extension of the example presented at the end of <ref>.

We thus see that we have the following sequence that leads to a construction of $\hat\frg^\omega$:

\begin{equation}
\mbox{dg Lie algebra}~\frG~\xrightarrow{\text{\Cref{thm:ophLie_from_dgLA}}}~~\mbox{$\ophLie$-algebra}~\frE~\xrightarrow{\text{\Cref{thm:antisym_hLie}}}~~\mbox{$L_\infty$-algebra}~\hat\frg^\omega,
\end{equation}

specializing the picture (<ref>). The additional information (i.e. structure constants) contained in the $E_2L_\infty$-algebra are vital for constructing the adjusted form of the curvatures.

A corresponding adjusted Weil algebra was found in <cit.>, and it agrees with the one obtained from our construction of a firmly adjusted Weil algebra from <ref>:

\begin{equation}
\def\arraystretch{1.5}
\begin{aligned}
Q_{\rm fadj}~&:~&t^\alpha &\mapsto  -\tfrac12 f^\alpha_{\beta\gamma} t^\beta t^\gamma+\hat t^\alpha~,~~~& p&\mapsto -s+\hat p~,\\
&&\hat t^\alpha &\mapsto  -f^\alpha_{\beta\gamma} t^\beta \hat t^\gamma~,~~~& \hat p&\mapsto \hat s~,\\
&&r&\mapsto \tfrac{1}{3!} f_{\alpha\beta\gamma} t^\alpha t^\beta t^\gamma-\kappa_{\alpha\beta}t^\alpha \hat t^\beta+q+\hat r~,~~~& s&\mapsto \hat s~,\\
&&\hat r&\mapsto \kappa_{\alpha\beta}\hat t^\alpha \hat t^\beta-\hat q~,~~~& \hat s&\mapsto 0~,\\
&&u_\alpha &\mapsto -f^\gamma_{\alpha\beta}t^\beta u_\gamma-\tfrac12 f_{\alpha\beta\gamma}t^\beta t^\gamma s-v_\alpha+\hat u_\alpha~,~~~&
q &\mapsto \hat q~,\\
&&\hat u_\alpha &\mapsto -f^\gamma_{\alpha\beta}t^\beta \hat u_\gamma+\hat v_\alpha~,~~~&
\hat q &\mapsto 0~,\\
&&v_\alpha&\mapsto -f^\gamma_{\alpha\beta}t^\beta v_\gamma - f^\gamma_{\alpha\beta}\hat t^\beta u_\gamma +f_{\alpha\beta\gamma}t^\beta\hat t^\gamma s-\tfrac12 f_{\alpha\beta\gamma} t^\beta t^\gamma \hat s+\hat v_\alpha~,\\
&&\hat v_\alpha&\mapsto -f^\gamma_{\alpha\beta}t^\beta \hat v_\gamma+f^\gamma_{\alpha\beta}\hat t^\beta \hat u_\gamma~.
\end{aligned}
\end{equation}

§ TENSOR HIERARCHIES IN TERMS OF \(E_2L_\INFTY\)-ALGEBRAS

Let us ignore the link between tensor hierarchies and gauged supergravity for a moment; clearly, the resulting kinematical data is potentially of interest in higher gauge theory in a much wider context.

The construction prescription is rather straightforward. We consider a Lie algebra $\frg$, which we enlarge to a differential graded Lie algebra

\begin{equation}
\begin{aligned}
V=\Big(~\dotsb \xrightarrow{~\rmd~} V_{-2} \xrightarrow{~\rmd~} V_{-1} \xrightarrow{~\rmd~} V_0=\frg \xrightarrow{~\rmd~} V_1 \xrightarrow{~\rmd~} \dotsb~\Big)~,
\end{aligned}
\end{equation}

where we allowed for additional vector spaces $V_i$ with $i>0$. All vector spaces $V_i$ are $\frg$-modules, and the Lie bracket on $V_0$ as well as the Lie brackets on $V_0\otimes V_i$ are given. Further Lie brackets $[-,-]:V_i\otimes V_j\rightarrow V_{i+j}$ can be introduced, but due to the Jacobi identity, the underlying structure constants have to be invariant tensors of $\frg$ (as we shall also see below in an example). The differentials do not have to satisfy this restriction. As an additional constraint, we can also impose the condition that $V_{-p}^*=V_{p+2-d}$ as required by the U-duality condition from supergravity. This can be useful in the construction of action principles.

To illustrate the above, let us construct a generic example in $d=5$. Let $\frg$ be a Lie algebra and $V_{-1}$ any representation. Imposing the duality constraint and allowing for an extension in one degree on either side leads to the differential complex

\begin{equation}
\begin{aligned}
V=\Big(~V_{-4}\cong{\rm coker}(\Theta)^* \xrightarrow{~\rmd~} V_{-3}\cong\frg^* &\xrightarrow{~\rmd~} V_{-2}\cong V_{-1}^*
\\
&\xrightarrow{~\rmd~} V_{-1} \xrightarrow{~\Theta~} V_0=\frg \xrightarrow{~\rmd~} V_1\cong{\rm coker}(\Theta)~\Big)~.
\end{aligned}
\end{equation}

Let us now switch to the Chevalley–Eilenberg description $\sfCE(V)$ of the differential graded Lie algebra $V$ we want to construct, which is generated by coordinates $r^\mu$, $r^\alpha$, $r^a$, $r_a$, $r_\alpha$, $r_\mu$ of degrees $0,1,2,3,4,5$, respectively. We note that we have a natural symplectic form on $V[1]^*$ of degree 5,

\begin{equation}
\omega=\rmd r^\alpha \wedge \rmd r_\alpha+\rmd r^a\wedge \rmd r_a+\rmd r^\mu \wedge \rmd r_\mu~.
\end{equation}

Compatibility of the Lie algebra action with the duality pairing amounts to the fact that the Chevalley–Eilenberg differential $Q$ is Hamiltonian for the Poisson bracket of degree $-5$,

\begin{equation}\label{eq:Poisson_5d}
\begin{aligned}
\{f,g\}\coloneqq &-\parder[f]{r_\alpha}~\parder[g]{r^\alpha}+(-1)^{|f|+1}\parder[f]{r^\alpha}~\parder[g]{r_\alpha}-\parder[f]{r_a}~\parder[g]{r^a}+(-1)^{|g|+1}\parder[f]{r^a}~\parder[g]{r_a}
\\
\end{aligned}
\end{equation}

induced by $\omega$. That is,

\begin{equation}
\end{equation}

The most generic Hamiltonian $\caQ$ of degree $6$ that is at most cubic in the generators[This restriction is required to obtain a differential graded Lie algebra, as opposed to an $L_\infty$-algebra] is

\begin{equation}
\begin{aligned}
\caQ&=\tfrac12 f_{\beta\gamma}{}^\alpha r_\alpha r^\beta r^\gamma+t_{\alpha a}{}^b r^\alpha r^a r_b+\tfrac1{3!}d_{abc} r^ar^br^c+\tfrac12 Z^{ab}r_ar_b+\Theta_{a}{}^{\alpha}r^ar_\alpha
\\
&~~~~+g^{\mu}_{1\alpha}r_\mu r^\alpha+g^{\mu}_{2\alpha\nu}r_\mu r^\alpha r^\nu+g_{3\mu a}^\alpha r^\mu r^ar_\alpha+g_{4\mu}^{ab} r^\mu r_ar_b~,
\end{aligned}
\end{equation}

where besides the structure constants $f_{\beta\gamma}{}^\alpha$ and the embedding tensor $\Theta_{a}{}^{\alpha}$ we have the deformation tensors $d_{abc}$ and $Z^{ab}$, which are totally symmetric and antisymmetric, respectively, due to the grading of the generators. The remaining structure constants will be called auxiliary. For $\caQ$ to give rise to a Chevalley–Eilenberg differential, we have to impose

\begin{equation}
\end{equation}

This equation imposes conditions on the structure constants. For example, we have

\begin{equation}\label{eq:cond_1}
\Theta_{a}{}^{\gamma} f_{\beta\gamma}{}^{\alpha}+t_{\beta a}{}^b\Theta_{b}{}^{\alpha}-g^\mu_{1\beta}g_{3\mu a}^\alpha=0~.
\end{equation}

For $g_{1}=g_{3}=0$, this implies that the embedding tensor is an invariant tensor, which is clearly too strong a condition. We can make a non-canonical choice of an embedding

\begin{equation}
i:{\rm coker}(\Theta)\hookrightarrow\frg~,
\end{equation}

which is given by structure constants $i^\alpha_\mu$ such that

\begin{equation}
i^\alpha_\mu g_{1\alpha}^\nu=\delta_\mu^\nu~.
\end{equation}

With this choice, we can split the condition (<ref>) into

\begin{equation}
\begin{aligned}
\Theta_c{}^\beta\Theta_{a}{}^{\gamma} f_{\beta\gamma}{}^{\alpha}+X_{ca}^b\Theta_{b}{}^{\alpha}&=0~,
\\
i^\beta_\mu(\Theta_{a}{}^{\gamma} f_{\beta\gamma}{}^{\alpha}+t_{\beta a}{}^b\Theta_{b}{}^{\alpha})&=g_{3\mu a}^\alpha~,
\end{aligned}
\end{equation}

and the first condition is the usual one encountered in the $d=5$ tensor hierarchy, while the second condition fixes one of the auxiliary structure constants. Besides the above condition and the fact that $f_{\beta\gamma}{}^{\alpha}$ and $t_{\alpha a}{}^b$ are the structure constants of the Lie algebra $\frg$ and a representation of $\frg$, we also have

\begin{equation}
\begin{aligned}
Z^{ab}\Theta_{b}{}^{\alpha} &=0
\Theta_{a}{}^{\alpha} g_{1\alpha}^\mu&=0~,
\\
Z^{a[b} t_{\alpha a}{}^{c]}+2g_{1\alpha}^\mu g_{4\mu}^{bc}&=0~,
\\
t_{\alpha (a}{}^d d_{bc)d}&=0~,
\end{aligned}
\end{equation}

as well as a number of conditions for the auxiliary structure constants. As expected, the tensor $d_{abc}$ capturing the Lie bracket $V_{-1}\otimes V_{-1}\rightarrow V_{-2}$ has to be an invariant tensor.

The kinematical data of a generic tensor hierarchy can then be constructed from the firmly adjusted Weil algebra of the corresponding $L_\infty$-algebra as described in detail in <ref>.

We note that the condition that $d_{abc}$ be an invariant tensor is too strong a constraint, e.g. for the non-maximally supersymmetric case. From the formulas of the curvatures, it is clear that there is no differential graded Lie algebra underlying this case, if the higher gauge algebra is constructed using the formulas of <ref>. This observation strongly suggests that there are  generalisations of these derived bracket constructions, but this is beyond the scope of this thesis.

§.§ Example: d=5 maximal supergravity

Let us give a concrete and complete picture of the interpretation of a tensor hierarchy using $\ophLie$-algebras, including the construction of curvatures. We choose the case $d=5$, which allows us to recycle observations made in <ref>. For a detailed discussion of this theory, see <cit.>.

Maximal supergravity in $d=5$ dimensions has the non-compact global symmetry group $\sfE_{6(6)}(\IR)$ <cit.>. When dimensionally reducing from $d=11$, in order to make manifest the $\fre_{6(6)}$ structure of the scalar sector in $d=5$, one must first dualise the 3-form potential, as described in detail in <cit.>. This gives a total of 42 scalars parameterizing

The fully dualised bosonic Lagrangian with manifest $\sfE_{6(6)}(\IR)$-invariance can be written as

\begin{equation}\label{D5L}
\mathcal{L}_5 = R\star 1 +\tfrac{1}{2} g_{xy} \rmd \varphi^x \wedge \star \rmd \varphi^y  -\tfrac{1}{2} a_{ab} F^{a} \wedge \star F^{b} -\tfrac{1}{6} d_{abc}F^{a}_{\2} \wedge F^{b}_{\2} \wedge A^{c}_{\1}~.
\end{equation}

The 1-form potentials transform linearly in the ${\rep{27 } }_c$ of  $\mathfrak{e}_{6(6)}$, and $a,b,c \in \{ 1,\ldots, 27\}$. In addition to the singlet $\rep{1}\in{\rep{27 } }_c\otimes \rep{27}$, used to construct the 1-form kinetic term, there is a singlet in the totally symmetric 3-fold tensor product $\rep{1}\in\bigodot^3(\mathbf{{27} }_c)$, which is used to construct the topological cubic term.

For the construction of the tensor hierarchy, we shall need the following $\sfE_6$-invariant tensors:

\begin{equation}
\begin{aligned}
f_{\alpha\beta\gamma} & \in \bigwedge\nolimits^3~\mathbf{78}
t_{\alpha}{}_{a}{}^{b}& \in \mathbf{78}\otimes \mathbf{{27}}\otimes\mathbf{{27} }_c~,
\\
d^{abc}& \in \bigodot\nolimits^3~\mathbf{27}
d_{abc}& \in \bigodot\nolimits^3~\mathbf{{27} }_c~.
\end{aligned}
\end{equation}

To optimise our notation, we also introduce the following tensors:

\begin{equation}
\begin{aligned}
X_{ab}{}^{c}& = \Theta_{a}{}^{\alpha} t_{\alpha}{}_{b}{}^{c}~,~~~&Y_{a}{}_{\alpha}{}^{\beta}& =  \Theta_{a}{}^{\gamma} f_{\gamma\alpha}{}^{\beta}+ t_{\alpha}{}_{a}{}^{b}\Theta_{b}{}^{\beta}\equiv\delta_\alpha \Theta_{b}{}^{\beta}~,
\\
X_{a\alpha}{}^{\beta}& =\Theta_{a}{}^{\gamma} f_{\gamma\alpha}{}^{\beta}~,~~~
&Z^{ab}& = \Theta_{c}{}^{\alpha} t_{\alpha}{}_{d}{}^{a}d^{bcd}= X_{cd}{}^{a}d^{bcd}=Z^{[ab]}~,
\end{aligned}
\end{equation}

The above tensors satisfy the following identities <cit.>:

\begin{equation}
d_{a cd}d^{b cd} = \delta_{a}{}^{b}~,~~~
X_{(ab)}{}^{c} = d_{ab d}Z^{cd}~,~~~X_{[ab]}{}^{c} = 10 d_{adf}d_{beg}d^{c de}Z^{fg}~,
\end{equation}

and in addition, we have the following three equivalent forms of the closure constraints:

\begin{equation}
\end{equation}

Using these, we can now apply the formalism of <ref> and construct the differential graded Lie algebra. It helps to broaden the perspective a bit and derive the latter from a graded Lie algebra $V$, with underlying vector space consisting of $\mathfrak{e}_{6(6)}$-modules:

\begin{equation}
\begin{array}{ccccccccccccccccccccc}
V_{\mathfrak{e}_{6(6)}}& =& V_{-5}&\oplus& V_{-4} &\oplus& V_{-3}&\oplus& V_{-2}&\oplus& V_{-1}&\oplus& V_{0} &\oplus& V_{1}\\[5pt]
\rho_{\sst{(k)}}&& \mathbf{27\oplus 1728}&& \mathbf{351}_c && \mathbf{78} && \mathbf{27} && \mathbf{27}_c && \mathbf{78} && \mathbf{351} \\[5pt]
e_{\sst{(k)}}&& (e^{a}, e_{ab}{}^{\alpha}) && e_{a}{}^{\alpha} && e^{\alpha} &&e^{a} && e_a &&  e_\alpha  &&  e_{\alpha}{}^{a}
\end{array}
\end{equation}

We have indicated the $\mathfrak{e}_{6(6)}$-representations $\rho_{\sst{(k)}}$ carried by each $V_{\mathfrak{e}_{6(6)}}$-degree $k$ summand, $V_k$, and their corresponding basis elements $e_{\sst{(k)}}$, e.g. $(e_\0)_\alpha=e_\alpha$, where $(e_\alpha)$ is some basis for the exceptional Lie algebra $\mathfrak{e}_{6(6)}$. Note that the embedding tensor $\Theta=\Theta_{a}{}^{\alpha}e_{\alpha}{}^{a}$ is an element of $V_1$ and $e_{\alpha}{}^{a}= P_{\mathbf{351}}e_{\alpha}\otimes e^{a}$.

The graded Lie bracket on $V$ is now given mostly by the obvious projections of the graded tensor products,

\begin{equation}\label{D5_gla_com}
[e_\alpha,e_\beta]=f_{\alpha\beta}{}^\gamma e_\gamma~,~~~
[e_\alpha, e_{\sst{(k)}}] = \rho_{\sst{(k)}}(e_\alpha) e_{\sst{(k)}}~,~~~
[e_{\sst{(k)}}, e_{\sst{(l)}}] = T_{k,l}e_{\sst{(k+l)}}~.
\end{equation}

Here, $T_{k,l}$ are the intertwiners dual to the projectors  $P_{k,l}:V_k\wedge V_l\rightarrow V_{k+l}$. For example,

\begin{equation}\label{eq:e6_gla_comm_def}
[e_a, e_b] = 2d_{abc}e^c~,~~~[e_a, e^b] = (t_{\alpha})_{a}{}^{b} e^\alpha~,
\end{equation}

\begin{equation}%\label{eq:e6_gla_comm_def}
e^a \coloneqq \tfrac12 d^{abc}[e_b, e_c], \qquad  e^\alpha \coloneqq (t_{\alpha})_{b}{}^{a} [e_a, e^b]~.
\end{equation}

The adjoint indices are raised/lowered with $\eta_{\alpha\beta}= \tr (t_\alpha t_\beta)$, which is proportional to the Cartan–Killing form.

Selecting an element $\Theta=\Theta_{a}{}^{\alpha}e_{\alpha}{}^{a}\in V_1$ now defines a differential

\begin{equation}
\rmd v \coloneqq [\Theta, v]
\end{equation}

for $v\in V$, and we note that $[\Theta,\Theta]=0$ for degree reasons. The explicit action of $\rmd e_{\sst{(k)}} \coloneqq [\Theta, e_{\sst{(k)}} ] $ can be determined using the graded Jacobi identity  from the initial condition

\begin{equation}
[ e_{\alpha}{}^{a} , e_b] = P_{\rep{351}}\delta_{b}{}^{a}e_\alpha~,
\end{equation}

where $P_{\rep{351}}$ is the projector $P_{\mathbf{351}}: \rep{78}\otimes \rep{27}_c \rightarrow \rep{351}$,

\begin{equation}
(P_{\rep{351}})_{\alpha}{}^{a}{}_{b}{}^{\beta} = -\tfrac65 t_{\alpha}{}_{b}{}^{c}t^{\beta}{}_{c}{}^{a}+\tfrac{3}{10} t_{\alpha}{}_{c}{}^{a}t^{\beta}{}_{b}{}^{c}+\tfrac15 \delta_{\alpha}{}^{\beta}\delta^{a}{}_{b}~.
\end{equation}

We thus obtain a differential graded Lie algebra, and this is a special case of the algebra called dgLie (THA$'$) in <ref>.

Let us now construct the $\ophLie$-algebra $\frE$ of this differential graded Lie algebra using <ref>. We arrive at the graded vector space

\begin{equation}
\begin{array}{ccccccccccccccccccccc}
\frE_{\mathfrak{e}_{6(6)}}=&  \frE_{-4} &\oplus& \frE_{-3}&\oplus& \frE_{-2}&\oplus& \frE_{-1}&\oplus& \frE_{0} \\
& \mathbf{27\oplus 1728} && \mathbf{351}_c && \mathbf{78} && \mathbf{27} && \mathbf{27}_c \\
& (e^{a}, e_{ab}{}^{\alpha}) && e_{a}{}^{\alpha} &&e^{\alpha} &&  e^{a}&&  e_a
\end{array}
\end{equation}

with non-trivial products

\begin{equation}
\eps_1(x)\coloneqq [\Theta, x]
\eps^0_2(x,y)\coloneqq     [[\Theta, x] , y]
\eps^{1}_{2}(x,y)\coloneqq     (-1)^{|x|}[x , y]~.
\end{equation}

Explicitly, we have the differentials

\begin{equation}
\begin{split}
\eps_1(e^a)&=  \Theta_{b}{}^{\alpha} t_{\alpha}{}_{c}{}^{d}d^{bca}e_d = X_{bc}{}^{d}d^{bca}e_d=-Z^{ad}e_d~,
\\
\eps_1(e^\alpha)&=   \Theta_{b}{}^{\alpha}e^{b}~,\\
\eps_1(e_{a}{}^{\alpha})&=  - \delta_{\beta}\Theta_{a}{}^{\alpha}e^{\beta}=-Y_{a}{}_{\beta}{}^{\alpha}e^{\beta}~,
\end{split}
\end{equation}

the Leibniz-like products

\begin{equation}
\begin{aligned}
\eps^0_2(e_a, e_b)&=    [\Theta_{a}{}^{\alpha} e_\alpha  , e_b] =\Theta_{a}{}^{\alpha} t_{\alpha}{}_{b}{}^{c}e_{c}  =X_{ab}{}^{c}e_{c}~,
\\
\eps^0_2(e_a, e^b)& =    [\Theta_{a}{}^{\alpha} e_\alpha  , e^b] =-\Theta_{a}{}^{\alpha} t_{\alpha}{}_{c}{}^{b}e^{c}  =-X_{ac}{}^{b}e^{c}~,
\\
\eps^0_2(e_a, e^\beta)& =    [\Theta_{a}{}^{\alpha} e_\alpha  , e^\beta] =-\Theta_{a}{}^{\alpha}f_{\alpha\gamma}{}^\beta e^\gamma =-X_{a\gamma}{}^\beta e^\gamma~,
\\
\eps^0_2(e_a, e_{b}{}^{\beta})& =    [\Theta_{a}{}^{\alpha} e_\alpha  , e_{b}{}^{\beta}] =-\Theta_{a}{}^{\alpha}f_{\alpha\gamma}{}^\beta e_{b}{}^{\gamma}+\Theta_{a}{}^{\alpha}t_{\alpha b}{}^{c} e_{c}{}^{\beta} =-X_{a\gamma}{}^\alpha   e_{b}{}^{\gamma}+X_{ab}{}^{c}e_{c}{}^{\beta}~,
\end{aligned}
\end{equation}

as well as the alternator-type products

\begin{equation}
\begin{aligned}
\eps^{1}_{2}(e_a, e_b)&=  2 d_{abc}e^{c}~,
\\
\eps^{1}_{2}(e_a, e^b)&=  t_{\alpha a}{}^{b}e^{\alpha}   =\eps^{1}_{2}(e^b, e_a)~,
\\
\eps^{1}_{2}(e_a, e^\alpha)&= e_{a}{}^{\alpha}  = P_{\rep{351}_c}e_{b}{}^{\beta} =-\eps^{1}_{2}(e^\alpha, e_a)~,
\\
\eps^{1}_{2}(e^a, e^b)&= -e^{[ab]}  = t_{\alpha c}{}^{[a}d^{b]cd}e_{d}{}^{\alpha}~,
\end{aligned}
\end{equation}

where we used that  $t_{\alpha c}{}^{[a}d^{b]cd}$ is the intertwiner between the $\mathbf{351 }_c\in \mathbf{27 }_c \otimes \mathbf{78}$ and $\mathbf{351 }_c\cong  \bigwedge^2\,\mathbf{27}$.

We can now construct the corresponding curvatures. We start from the Chevalley–Eilenberg algebra of $\frE_{\mathfrak{e}_{6(6)}}$ with the following generators $(r^A)$ spanning $\frE_{\mathfrak{e}_{6(6)}}[1]^*$:

\begin{equation}
\begin{array}{ccccccccccccccccccccc}
\text{degree} & 1&& 2&&3&&4&&5\\[5pt]
\frE_{\mathfrak{e}_{6(6)}}[1]^*= & \mathbf{27} &\oplus& \mathbf{27 }_c &\oplus& \mathbf{78} &\oplus& \mathbf{351} &\oplus& \mathbf{27 }_c\oplus\rep{1728 }_c \\[5pt]
& r^a &&r_{a} &&r_{\alpha} &&  r_{\alpha}{}^{a} &&  (r_{a}, r_{\alpha}{}^{ab})
\end{array}
\end{equation}

Consider now the Weil algebra $\sfW(\frE_{\mathfrak{e}_{6(6)}})$, cf. <ref>. Here, we introduce a second copy of shifted generators $(\hat r^A)$ spanning $\frE_{\mathfrak{e}_{6(6)}}[2]^*$ with $|\hat r^A|=|r^A|+1$. The usual Weil differential up to degree $3$ elements, dual to scalars in $d=5$, then reads as

\begin{equation}\label{N=8D5QW}
\begin{aligned}
Q_{\sf W} r^{a} &=-Z^{ab}r_{b} - X_{bc}{}^{a}    r^{b} \oslash_0 r^{c}+\hat r^a~,
\\
Q_{\sf W} r_{a} &=
\Theta_{a}{}^{\alpha}r_{\alpha}
+X_{ba}{}^{c}r^b \oslash_0 r_c
- d_{abc} r^{b} \mathbin{\hat \oslash_1}   r^{c}+ \hat r_a~,
\\
Q_{\sf W}r_{\alpha} &=
+X_{a\alpha}{}^{\beta}r^{a} \oslash_0 r_{\beta}
+ t_{\alpha}{}_{a}{}^{b}  r^{a} \mathbin{\hat \oslash_1} r_{b}
+  \hat r_\alpha~,
\\
Q_{\sf W} \hat r^{a} &=
Z^{ab} \hat r_{b} + X_{bc}{}^{a}  \hat  r^{b} \oslash_0 r^{c}- X_{bc}{}^{a}    r^{b} \oslash_0 \hat r^{c}~,
\\
Q_{\sf W} \hat r_{a} &=
- \Theta_{a}{}^{\alpha}\hat r_{\alpha}
- X_{ba}{}^{c}\hat r^b \oslash_0 r_c
+ X_{ba}{}^{c} r^b \oslash_0 \hat r_c
+2 d_{abc} \hat r    ^{b} \mathbin{\hat \oslash_1} r    ^{c}~,
\\
Q_{\sf W} \hat r_{\alpha} &=-Y_{a\alpha}{}^{\beta}\hat r_{\beta}{}^{a}
-X_{a\alpha}{}^{\beta} \hat r^{a} \oslash_0 r_{\beta}
+ X_{a\alpha}{}^{\beta}  r^{a} \oslash_0 \hat r_{\beta}
- t_{\alpha}{}_{a}{}^{b} \hat r^{a} \mathbin{\hat \oslash_1} r_{b}
+ t_{\alpha}{}_{a}{}^{b}  \hat r_{b} \mathbin{\hat \oslash_1} r^{a}~,
\end{aligned}
\end{equation}

where we have introduced the notation

\begin{equation}
\begin{split}
a \mathbin{\hat \oslash_i} b& =  a  \oslash_i b +(-1)^{i+|a|\,|b|} b  \oslash_i a~, \\
a \mathbin{\check \oslash_i} b& = a  \oslash_i b -(-1)^{i+|a|\,|b|} b  \oslash_i a~.\\
\end{split}
\end{equation}

The deformed Leibniz rule (<ref>), together with the remaining $\opEilh$-relations (<ref>) and the identities (<ref>), then imply $Q_\sfW^2=0$ as one can check by direct computation.

In order to define the curvatures of the tensor hierarchy, we symmetrise to an $L_\infty$-algebra using <ref>. We can then use the formalism of <ref> to construct an adjusted Weil algebra in the sense of <cit.>, ensuring closure of the gauge algebra without any further constraints on the field strengths.

To illustrate in more detail the procedure and what it achieves, we can perform the coordinate change already at the level of the Weil algebra of the $\ophLie$-algebra. This coordinate change yields a symmetrised and firmly adjusted Weil algebra through an  evident  coordinate change, $r^A\mapsto \tilde r^A$, which removes all appearances of $\oslash_1$ in $Q_{\sf W}\tilde r^A$ via the deformed Leibniz rule (<ref>). Hence,  by <ref> we are left with an $L_\infty$-algebra. Explicitly, the following  coordinate change manifestly removes all appearances of $\oslash_1$:

\begin{equation}
\begin{aligned}\label{eq:can_rotation}
r^a&\mapsto a^a \coloneqq  r^a~,
\\
r_a&\mapsto b_a \coloneqq  r_a  +\tfrac12  d_{abc} r^b \mathbin{\check \oslash_0} r^c~,
\\
r_{\alpha}&\mapsto c_{\alpha}\coloneqq  r_{\alpha}- \tfrac12   t_{\alpha}{}_{a}{}^{b}  r^{a} \mathbin{\check  \oslash_0} r_{b}~,
\\
r_{\alpha}{}^{a}&\mapsto d_{\alpha}{}^{a} \coloneqq  r_{\alpha}{}^{a}+\tfrac12 P_{\rep{351}_c} r^{b} \mathbin{\check  \oslash_0} r_{\beta}+\tfrac12 t_{\alpha c}{}^{[b}d^{c]ad}r_{b} \mathbin{\check  \oslash_0} r_{c}~,
\end{aligned}
\end{equation}

where $ d_{\alpha}{}^{a}$ is included as it is needed for $Q_{\sf W}\tilde r_{\alpha}$. The corresponding coordinate change on $\hat r^A$ is firmly adjusted by simply first ordering the occurrences of  $\hat r^B$ in  $\hat{\tilde{r}}^A$ to the left (which is permitted by the appearance of only $\check \oslash_0$ in $\hat{\tilde{r}}^A$) and then  sending $\check \oslash_i$ to $\check \oslash_i+\hat \oslash_i=2\oslash_i$. The choice of left ordering follows from the choice of left Leibniz rule, which is a matter of convention. Applied to (<ref>) this yields

\begin{equation}
\begin{aligned}
\hat   r^a&\mapsto f^a \coloneqq  \hat r^a~,
\\
\hat    r_a&\mapsto h_a \coloneqq \hat  r_a  + 2d_{abc}\hat r^b  \oslash_0  r^c~,
\\
\hat    r_{\alpha}&\mapsto g_{\alpha} \coloneqq  \hat r_{\alpha}-  t_{\alpha}{}_{a}{}^{b} (\hat  r^{a}   \oslash_0  r_{b} -  \hat r_{b}  \oslash_0  r^{a})~,
\\
\hat  r_{\alpha}{}^{a}&\mapsto k_{\alpha}{}^{a} \coloneqq  \hat r_{\alpha}{}^{a}+ P_{\rep{351}_c} (\hat r^{b}   \oslash_0  r_{\beta} +\hat r_{\beta}  \oslash_0  r^{b})+2 t_{\alpha c}{}^{[b}d^{c]ad} \hat r_{b}   \oslash_0  r_{c}~.
\end{aligned}
\end{equation}

Note, this is a special case of the transformation (<ref>) for a firm adjustment.

The result of this coordinate change is the differential graded commutative algebra $\sfW_{\rm adj}(\frE_{\mathfrak{e}_{6(6)}})$ generated by $\frE_{\mathfrak{e}_{6(6)}}[1]^*\oplus\frE_{\mathfrak{e}_{6(6)}}[2]^*$ and differential

\begin{equation}\label{symN=8D5QW}
\begin{aligned}
Q_{\sfW_{\rm adj}} a^{a} &=-Z^{ab}b_{b} - \tfrac12 X_{bc}{}^{a}    a^{b}   a^{c}+f^a~,
\\
Q_{\sfW_{\rm adj}} b_{a} &=
\Theta_{a}{}^{\alpha}c_{\alpha}
+ \tfrac12 X_{ba}{}^{c}a^b   b_c + \tfrac16 d_{abc} X_{de}{}^{b}a^c   a^d   a^e
- d_{abc} f^{b}   a^{c}+ h_a~,
\\
Q_{\sfW_{\rm adj}}c_{\alpha} &=
+\tfrac12 X_{a\alpha}{}^{\beta}a^{a}   c_{\beta}
+(\tfrac14 X_{a\alpha}{}^{\beta}t_{\beta b}{}^{c}+\tfrac13t_{\alpha a}{}^{d} X_{(db)}{}^{c})a^a  a^b b_c
\\
&~~~~~+\tfrac12 t_{\alpha}{}_{a}{}^{b} f^a   b_b -\tfrac12 t_{\alpha}{}_{a}{}^{b} h_b a^a    -\tfrac{1}{6} t_{\alpha}{}_{a}{}^{b} d_{bcd}a^{a}  a^{c}  f^{d}+ g_\alpha~,\\
Q_{\sfW_{\rm adj}} f^{a} &=
Z^{ab} h_{b} + X_{bc}{}^{a} a^{b}    f^{c} ~,
\\
Q_{\sfW_{\rm adj}} h_{a} &=
+ X_{ab}{}^{c} a^b   h_c
+ d_{abc}f^b   f^c~,
\\
Q_{\sfW_{\rm adj}} g_{\alpha} &=-Y_{a\alpha}{}^{\beta}k_{\beta}{}^{a}
+X_{a\alpha}{}^{\beta}  a^{a}   g_{\beta}
- t_{\alpha}{}_{a}{}^{b}   h_{b}    f^{a}~.
\end{aligned}
\end{equation}

We can now define the corresponding curvatures in the adjusted higher gauge theory as usual as a morphism of differential graded algebras

\begin{equation}\label{eq:dga_morphism_5d}
(\caA,\caF): \sfW_{\rm adj}(\frE_{\mathfrak{e}_{6(6)}})~\longrightarrow~\Omega^\bullet(M)~,
\end{equation}

where[The additional signs here follow from the choice of sign convention in (<ref>).]

\begin{equation}
\begin{split}
(a^a,b_a,c_\alpha,  d_{\alpha}{}^{a}) &\mapsto  (A^a, B_a, -C_\alpha,  -D_{\alpha}{}^{a})~,\\
(f^a,h_a,g_\alpha,  k_{\alpha}{}^{a}) &\mapsto  (F^a, H_a, -G_\alpha, -K_{\alpha}{}^{a})~.
\end{split}
\end{equation}

This indeed yields  the gauge potentials and curvatures of the $d=5$ tensor hierarchy:

\begin{align}
F^a &= \rmd A^a +\tfrac{1}{2} X_{bc}{}^{a} A^b\wedge A^c  + Z^{ab}B_b~,
\\
&= \rmd B_a -\tfrac 12 X_{ba}{}^{c} A^b\wedge B_c   - \tfrac{1}{6} d_{abc} X_{de}{}^{b} A^c\wedge A^d \wedge A^e  +d_{abc} A^b\wedge F^c + \Theta_{a}{}^{\alpha}  C_\alpha~,
\\
G_\alpha  &= \rmd  C_\alpha-\tfrac12 X_{a\alpha}{}^{\beta}A^a\wedge  C_\gamma+(\tfrac14 X_{a\alpha}{}^{\beta}t_{\beta b}{}^{c}+\tfrac13t_{\alpha a}{}^{d} X_{(db)}{}^{c})A^a\wedge  A^b \wedge B_c \\
&\phantom{=} +\tfrac12 t_{\alpha}{}_{a}{}^{b} F^a \wedge B_b -\tfrac12 t_{\alpha}{}_{a}{}^{b}H_b  \wedge A^a  -\tfrac{1}{6} t_{\alpha}{}_{a}{}^{b} d_{bcd}A^{a}\wedge A^{c}\wedge F^{d}- Y_{a\alpha}{}^{\beta} D_{\beta}{}^{a}~,
\end{align}

along with the corresponding Bianchi identities,

\begin{align}
0&=\rmd F^{a}
- X_{bc}{}^{a} A^{b} \wedge   F^{c} -Z^{ab} H_{b}  ~,
\\
0 &= \rmd H_{a}
- X_{ab}{}^{c} A^b  \wedge H_c
- d_{abc}F^b \wedge  F^c-\Theta_{a}{}^{\alpha}G_{\alpha}~,
\\
0&=  \rmd G_{\alpha}- X_{a\alpha}{}^{\beta}  A^{a} \wedge  G_{\beta}- t_{\alpha}{}_{a}{}^{b}   H_{b} \wedge   F^{a} +Y_{a\alpha}{}^{\beta}K_{\beta}{}^{a}~.
\end{align}

We note that the full kinematical data is determined in this way: the Bianchi identities are implied by compatibility of the morphism (<ref>) with the differential, and the gauge transformations are constructed as infinitesimal partially flat homotopies, cf. e.g. <cit.> for details.

To make contact with the expressions in the supergravity literature, cf. <cit.>, one must make the field redefinitions

\begin{equation}
\begin{aligned}
C_{\alpha}&\mapsto C_\alpha+\tfrac12t_{\alpha a}{}^b A^a\wedge B_b~,
\\
D_{\alpha}{}^{a} &\mapsto D_{\alpha}{}^{a} - \tfrac12 P_{\rep{351 }_c}  A^a\wedge  C_\alpha~.
\end{aligned}
\end{equation}

Similar field redefinitions were also used in <cit.> to link another elegant derivation of the curvature forms (in which, however, the link to higher gauge algebras also is somewhat obscured) to the supergravity literature. We stress that from the higher gauge algebra point of view, the form (<ref>) is special in the sense that all exterior derivatives of gauge potentials in non-linear terms have been absorbed in field strengths. This makes (<ref>) particularly useful, as it exposes cleanly the separation of unadjusted curvature and adjustment. From the former, one can straightforwardly identify the higher Lie algebra of the structure group of the underlying higher principal bundle. Moreover, gauge transformations are readily derived from partially flat homotopies, as mentioned above. As a side effect, it is interesting to note that the arising higher products are at most ternary.

An interesting aspect of (<ref>) is the fact that the covariantisations of the differentials $\rmd B$ and $\rmd C$ contain a perhaps unexpected factor of $\tfrac12$. This factor is a clear indication that
the origin of the gauge $L_\infty$-algebra is indeed an $\ophLie$-algebra: the action $\acton$ of $A$ on $B$ and $C$ is encoded in an $\ophLie$-algebra with

\begin{equation}
\eps_2^0(A,B)\coloneqq A\acton B\eand \eps_2^0(A,C)\coloneqq A\acton C~,
\end{equation}

which is then antisymmetrised by <ref> to

\begin{equation}
\mu_2(A,B)\coloneqq \tfrac12 \eps^0_2(A,B)\eand \mu_2(A,C)\coloneqq \tfrac12 \eps^0_2(A,C)~,
\end{equation}

at the cost of introducing non-trivial higher products $\mu_3$, cf. (<ref>).

§ COMPARISON TO THE LITERATURE

We compare our results with algebraic structures previously introduced in the literature to capture the gauge structure underlying the higher gauge theories obtained in the tensor hierarchies of gauged supergravity. We shall focus on the particularities of the gauge algebraic structures of the tensor hierarchies; for other work linking the tensor hierarchy to ordinary $L_\infty$-algebras, see also <cit.>.

§.§ Enhanced Leibniz algebras

A notion of enhanced Leibniz algebras was introduced in <cit.> to capture the parts of the higher gauge algebraic structures appearing in the tensor hierarchy. See also <cit.> for a discussion of the higher gauge theory employing these enhanced Leibniz algebras and the link to the tensor hierarchy.

An enhanced Leibniz algebra is a Leibniz algebra $(\sfV,[-,-])$ together with a vector space $\sfW$ and a linear map $t:\sfW\rightarrow \sfV$ as well as a binary operation $\circ:\sfV\otimes\sfV\to\sfW$ such that

\begin{equation}
\begin{aligned}
\\
t(w)\circ t(w)&=0~,~~~&[v,v]&=t(v\circ v)
\end{aligned}
\end{equation}

for all $u,v\in \sfV$ and $w\in \sfW$, where $u\stackrel{s}{\circ}v$ denotes the symmetric part of $u\circ v$.

A symmetric enhanced Leibniz algebra additionally satisfies the condition that

\begin{equation}
u\circ v=v\circ u
\end{equation}

for all $u,v\in \sfV$.

A symmetric enhanced Leibniz algebra is an $\ophLie$-algebra concentrated in degrees $-1$ and $0$ with a few axioms missing. We can identify the structure maps as follows.

\begin{equation}
\begin{gathered}
\frE=(\frE_{-1}\xrightarrow{~\eps_1~}\frE_0)~=~(\sfW\xrightarrow{~\sft~}\sfV)~,
\\
\eps_2(v_1,v_2)=[v_1,v_2]~,~~~\eps_2(v,w)=0~,~~~\sfalt(v_1,v_2)=v_1\circ v_2~,
\end{gathered}
\end{equation}

for $v,v_1,v_2\in \sfV$ and $w\in\sfW$. The $\ophLie$-algebra relations (<ref>) are trivially satisfied since $\eps_2$ is a Leibniz bracket. Moreover, $\eps_1$ is trivially a differential and a derivation of $\eps_2$. The relation $\eps_2(v_1,v_2)+\eps_2(v_2,v_1)=\eps_1(\sfalt(v_1,v_2))$ is the polarisation of $[v,v]=t(v\circ v)$. The relation $u\stackrel{s}{\circ}[v,v]=v\stackrel{s}{\circ}[u,v]$ fails to accurately reproduce the relation between $\eps_2$ and the alternator, $\sfalt(v_1,\eps_2(v_2,v_3))=\sfalt(\eps_2(v_2,v_3),v_1)$. Moreover, the relation $t(w)\circ t(w)=0$ fails to reproduce the appropriate relation for the alternator, $\sfalt(v_1,\sft(w_1))=\sfalt(\sft(w_1),v_1)=0$.

The original definition in <cit.> of a (not necessarily symmetric) “enhanced Leibniz algebra” is slightly more general, allowing for the operation $\circ$ to be not symmetric. However, this is not very natural, as discussed in <ref>. Moreover, the algebraic structure underlying the tensor hierarchy is an $\ophLie$-algebra, so enhanced Leibniz algebras require axiomatic completion.

§.§ Infinity-Enhanced Leibniz algebras

A similar notion of extended Leibniz algebras was formulated in <cit.>, see also <cit.> as well as the previous work on Leibniz algebra gauge theories <cit.>.

An \(\infty\)-enhanced Leibniz algebra is an $\IN$-graded differential complex $(X=\oplus_{i\in\IN} X_i,\rmd)$ with differential of degree $-1$, endowed with two binary operations

\begin{equation}
\begin{aligned}
\circ&:X_0\otimes X_0\rightarrow X_0~,
\\
\bullet&:X_i\otimes X_j\rightarrow X_{i+j+1}~,
\end{aligned}
\end{equation}

satisfying the following relations:

\begin{align}
(x\circ y)\circ z&=x \circ(y\circ z)-y\circ (x\circ z)~,\label{eq:eL_Leibniz}
\\
a\bullet b&=(-1)^{|a|\,|b|}(b\bullet a)~,\label{eq:eL_sym_bullet}
\\
(\rmd w)\circ x &= 0~, \label{eq:ielax1}
\\ % ax1
\rmd(x\bullet y)&= x\circ y+y\circ x~,\label{eq:ielax2}
\\
\rmd(u\bullet v) &= - (\rmd u)\bullet v + (-1)^{|u|+1}u\bullet\rmd v~, \label{eq:ielax5} % ax5
\\
(a\bullet b)\bullet c&=
(-1)^{|a|+1}a\bullet(b\bullet c)-(-1)^{(|a|+1)|b|} b\bullet(a\bullet c)~, \label{eq:ielax6} % ax6, slightly rearranged by changing argument order for the last term by using graded symmetry of the outer alt
\\ % ax2
\rmd(x\bullet(y\bullet z)) &= (x\circ y)\bullet z + (x\circ z)\bullet y - (y\circ z+z\circ y)\bullet x~, \label{eq:ielax3} % ax3, but with the last term order swapped using symmetry of alternator
\\
\left[\rmd(x\bullet(y\bullet z))\right]_{x\leftrightarrow y}&=\left[(x\circ y)\bullet u-2x\bullet\rmd(y\bullet u)-x\bullet(y\bullet \rmd u)
\right]_{x\leftrightarrow y}~, \label{eq:ielax4} % ax4
\end{align}

where \(x,y,z\) range over degree $0$ elements, \(w\) ranges over degree \(1\) elements, \(u,v\) range over positive degree elements, and \(a,b,c\) over arbitrary elements of homogeneous degrees, and where \(
[\dotsb]_{x\leftrightarrow y}
\)
signifies that the enclosed expression is antisymmetrised with respect to the permutation between \(x\) and \(y\).

An $\infty$-enhanced Leibniz algebra is a particular type of $\ophLie$-algebra with some axioms missing. Clearly, to compare the axioms, we have to invert the sign of the degree. We thus consider an $\ophLie$-algebra $\frE$ concentrated in non-positive degrees with $\eps_2^0=\circ$ non-trivial only on elements of degree $0$. Moreover, we are led to identify $\eps_2^1$ with $\bullet$; all other $\eps_2^i$ are trivial. Then we have the following relations between the axioms of an $\infty$-enhanced Leibniz algebra and an $\ophLie$-algebra:

(<ref>) is simply the Leibniz identity and follows from the quadratic relation for $\eps_2^0$.

(<ref>) amounts to $\eps_2^1$ being graded symmetric and follows from the modified Leibniz rule, as do (<ref>)–(<ref>).

(<ref>) follows from the $\ophLie$-axiom for $\eps_2^1\circ \eps_2^1$.

(<ref>) follows from the modified Leibniz rule together with the $\ophLie$-axioms for $\eps_2^1\circ \eps_2^0$ and $\eps_2^0\circ \eps_2^1$:

\begin{equation}
\begin{aligned}
\eps_1(\eps_2^1(x,\eps_2^1(y,z))) &=
\eps_2^0(x,\eps_2^1(y,z))
+  \eps_2^0(\eps_2^1(y,z),x)
- \eps_2^1(x,\eps_2^0(y,z)+\eps_2^0(z,y)) \\
&= \eps_2^1(\eps_2^0(x,y),z) + \eps_2^1(y,\eps_2^0(x,z))
- \eps_2^1(x,\eps_2^0(y,z)+\eps_2^0(z,y))~,
\end{aligned}
\end{equation}

as does (<ref>): we have:

\begin{equation}
\begin{aligned}
\eps_2^0(\eps_2^1(-,-),-)&=0~,
\\
\eps_1(\eps_2^1(x,\eps_2^1(y,u))) &= \eps_2^1(x,\eps_2^1(y,\eps_1(u)))
+ \eps_2^0(x,\eps_2^1(y,u)) - \eps_2^1(x,\eps_2^0(y,u)+\eps_2^0(u,y))
\\
&= \eps_2^1(x,\eps_2^1(y,\eps_1(u)))
+\eps_2^1(\eps_2^0(x,y),u) + \eps_2^1(y,\eps_2^0(x,u))
\\
&\qquad - \eps_2^1(x,\eps_2^0(y,u)) -\eps_2^1(x,\eps_2^0(u,y))~,
\\
\eps_2^1(x,\eps_1(\eps_2^1(y,u))) &= -\eps_2^1(x,\eps_2^1(y,\eps_1(u))) + \eps_2^1(x,\eps_2^0(y,u)+\eps_2^0(u,y))~,
\end{aligned}
\end{equation}

and putting this together, we obtain

\begin{equation}
\begin{aligned}
\right]_{x\leftrightarrow y}
\\
-\eps_2^1(x,\eps_2^1(y,\rmd u))
]_{x\leftrightarrow y}~.
\end{aligned}
\end{equation}

Note, however, that while the $\ophLie$-algebra axioms imply the axioms of an $\infty$-enhanced Leibniz algebra, the reverse statement is not true, even for $\infty$-enhanced Leibniz algebras concentrated in degrees $0$ and $1$. The latter essentially implies that $\infty$-enhanced Leibniz algebras are an incomplete abstraction of homotopy Lie algebras and thus do not give the full picture. Altogether, we arrive at the same conclusion as for enhanced Leibniz algebras.

As a side remark, we note that in the outlook of <cit.>, the authors mentioned the desire for the interpretation of $\infty$-enhanced Leibniz algebras as the homotopy algebras of some simpler algebraic structure. Our discussion suggests that this is not possible; instead, the axiomatic completion of $\infty$-enhanced Leibniz algebras yields $\ophLie$-algebras whose homotopy algebras form $E_2L_\infty$-algebras, a much weaker version of $L_\infty$-algebras.

§.§ Algebras producing the tensor hierarchies

We now come to larger picture of algebras that lead to the gauge structures visible in the tensor hierarchies, see <ref>. Note that this picture has only been applied in the context of the tensor hierarchy for maximal supersymmetry. We shall be less detailed in the following.

[row sep=2.8cm,column sep=2.3cm]

[swap,""]      r [swap]

r[swap]     u[sloped] [swap,sloped]

The relation between the various algebraic structures in the literature and how $\ophLie$-algebras fit into the picture.

In <cit.>, Palmkvist constructs an infinite-dimensional $\IZ/2$-graded Lie algebra, which he calls the tensor hierarchy algebra, “gLie (THA)” in <ref>. For further work on the tensor hierarchy algebra, see also <cit.>. As observed in <cit.>, see also <cit.>, this $\IZ/2$-grading can be naturally refined into a $\IZ$-grading, and picking an element of degree 1 and subsequent restriction induces the structure of a differential graded Lie algebra, “dgLie (THA\('\))” in <ref>. In <cit.>, Lavau called this differential graded Lie algebra the “tensor hierarchy algebra” (not to be confused with Palmkvist's larger graded Lie algebra), and derived it from a further algebraic structure called Lie–Leibniz triples, “LieLeibTriple” in <ref>. This differential graded Lie algebra then naturally gives rise to $\infty$-enhanced Leibniz algebras, as described in <cit.>. As explained above, the $\infty$-enhanced Leibniz algebra were an incomplete “guess” of the axioms of an $\ophLie$-algebra with $\eps_2^i=0$ for $i\geq 2$. Thus, from our perspective, $\infty$-enhanced Leibniz algebras are appropriately replaced by these, and we then have the construction of the gauge $L_\infty$-algebra via the picture (<ref>), which is refined in <ref>. We note that the composition of the arrows “complete axioms” and “antisym.”, which produces an $L_\infty$-algebra from an $\infty$-enhanced Leibniz algebra, is found in <cit.>. As indicated in <ref>, the direct construction of an $L_\infty$-algebra from a differential graded Lie algebra is the Fiorenza–Manetti–Getzler construction <Ref>, as pointed out in <cit.>, where Getzler's formulas were specialised to the tensor hierarchy differential graded Lie algebra.

For prior relations amongst tensor hierarchies, the embedding tensor formalism and (homotopy) algebras see also <cit.>. We again stress that from our point of view, it is not natural to consider gauge theories with infinitesimal symmetries that are not (weaker forms of) Lie algebras. Axiomatically completing the various forms of Leibniz algebras to $\ophLie$-algebras solves this issue.

As a side remark, let us note that the fact that Leibniz algebras naturally produce $L_\infty$-algebras has been pointed out in <cit.>. This is <ref> stating that any Leibniz algebra naturally extends to an $\ophLie$-algebra combined with <ref> antisymmetrising this $\ophLie$-algebra to an $L_\infty$-algebra.