# nLab EL-∞ algebra

Contents

## Examples

### $\infty$-Lie algebras

#### Higher algebra

higher algebra

universal algebra

and

# Contents

## $h\mathcal{L}ie$- and $\mathcal{E}ilh$-algebras

The following notions have been introduced by (BKS21).

### $h\mathcal{L}ie$-algebras

###### Definition

An $h\mathcal{L}ie$-algebra $(\mathfrak{E},\varepsilon_1,\varepsilon_2^i)$ is a graded vector space $\mathfrak{E}$ together with a differential and a collection of binary products,

\begin{aligned} \varepsilon_1&: \mathfrak{E}\rightarrow \mathfrak{E}, &|\varepsilon_1|=1, \\ \varepsilon^i_2&: \mathfrak{E}\otimes \mathfrak{E}\rightarrow \mathfrak{E}, &|\varepsilon^i_2|=-i, \end{aligned}

such that

\begin{aligned} \varepsilon_1(\varepsilon_1(x_1))&=0, \\ \varepsilon_1(\varepsilon^i_2(x_1,x_2))&=(-1)^i\big(\varepsilon^i_2(\varepsilon_1(x_1),x_2)+(-1)^{|x_1|}\varepsilon^i_2(x_1,\varepsilon_1(x_2))\big) \\ &+\varepsilon^{i-1}_2(x_1,x_2)-(-1)^{i+|x_1|\,|x_2|}\varepsilon^{i-1}_2(x_2,x_1), \\ \varepsilon^i_2(\varepsilon_2^i(x_1,x_2),x_3)&=(-1)^{i(|x_1|+1)}\varepsilon^i_2(x_1,\varepsilon^i_2(x_2,x_3))-(-1)^{(|x_1|+i)|x_2|}\varepsilon^i_2(x_2,\varepsilon_2^i(x_1,x_3)) \\ &-(-1)^{(|x_2|+|x_3|)|x_1|+(i-1)|x_2|}\varepsilon^{i+1}_2(x_2,\varepsilon^{i-1}_2(x_3,x_1)), \\ \varepsilon^{j}_2(\varepsilon^{i}_2(x_1,x_2),x_3)&= (-1)^{1+j(i+1)+|x_1|(|x_2|+|x_3|)+(j-1)|x_2|}\varepsilon_2^{i+1}(x_2,\varepsilon_2^{j-1}(x_3,x_1)), \\ \varepsilon^{i}_2(\varepsilon^{j}_2(x_1,x_2),x_3)&= (-1)^{i(j+|x_1|)}\varepsilon_2^{j}(x_1,\varepsilon_2^{i}(x_2,x_3))-(-1)^{(|x_1|+j)|x_2|}\varepsilon_2^i(x_2,\varepsilon_2^j(x_1,x_3)) \\ &-(-1)^{j+|x_3|(j+|x_2|-1)+|x_1|(|x_2|+|x_3|)}\varepsilon_2^{i+1}(x_3,\varepsilon_2^{j-1}(x_2,x_1)) \end{aligned}

are satisfied for all $i,j\in \mathbb{N}$ s.t. $j$<$i$ and for all $x_1,x_2,x_3\in \mathfrak{E}$, where we regard $\varepsilon_2^{-1}=0$.

### $\mathcal{E}ilh$-algebras

###### Definition

An $\mathcal{E}ilh$-algebra $(\mathfrak{E},Q,\oslash_i)$ is a differential graded vector space $(\mathfrak{E},Q)$ equipped with binary operations $\oslash_i$ of degree $i\in \mathbb{N}$ which satisfy the quadratic identities

\begin{aligned} a\oslash_i(b\oslash_i c)&=(-1)^{i(|a|+1)}((a\oslash_i b)\oslash_i c+(-1)^{|a|\,|b|}(b\oslash_i a)\oslash_i c), \\ a\oslash_i(b\oslash_j c)&=(-1)^{ij+j|a|}(a \oslash_i b)\oslash_j c, \\ a\oslash_j(b\oslash_i c)&=\begin{cases} (-1)^{i|a|+|a|\,|b|}(b \oslash_i a)\oslash_j c &\text{if } j-i=1, \\ (-1)^{i|a|+|a|\,|b|}(b \oslash_i a)\oslash_j c &\text{if } j-i=2, \\ +(-1)^{i(|a|+j+1)+(|a|+|b|)|c|}((c\oslash_{j-1} a)\oslash_{i+1}b) \\ (-1)^{i|a|+|a|\,|b|}(b \oslash_i a)\oslash_j c &\text{if } j-i>2 \\ +(-1)^{i(|a|+j+1)+(|a|+|b|)|c|}((c\oslash_{j-1} a)\oslash_{i+1}b) \\ +(-1)^{j+|a|(|b|+i)+(|a|+|b|)|c|}((c\oslash_{i+1} b)\oslash_{j-1}a) \end{cases} \end{aligned}

for $j$>$i$, and such that differential $Q$ satisfies the property

\begin{aligned} Q(a\oslash_i b)&=(-1)^i\big((Qa)\oslash_i b+(-1)^{|a|}a\oslash_i Qb\big)\\ &+(-1)^i (a \oslash_{i+1}b)-(-1)^{|a|\,|b|} (b\oslash_{i+1} a), \end{aligned}

which is a deformed Leibniz rule.

###### Proposition

$\mathcal{E}ilh$-algebras are Koszul dual to $h\mathcal{L}ie$-algebras.

###### Definition

The Chevalley–Eilenberg algebra $\mathrm{CE}(\mathfrak{E})$ of an $h\mathcal{L}ie$-algebra $\mathfrak{E}$ whose differential and binary products are given by

\begin{aligned} \varepsilon_1&: \mathfrak{E}\rightarrow \mathfrak{E},&\tau_\alpha&\mapsto m^\beta_\alpha \tau_\beta,&&|\varepsilon_1|=1, \\ \varepsilon^i_2&: \mathfrak{E}\otimes \mathfrak{E}\rightarrow \mathfrak{E},&\tau_\alpha\otimes \tau_\beta&\mapsto m^{i,\gamma}_{\alpha\beta}\tau_\gamma,&&|\varepsilon^i_2|=i \end{aligned}

for some $m^\beta_\alpha$ and $m^{i,\gamma}_{\alpha\beta}$ taking values in the underlying ground field is the $\mathcal{E}ilh$-algebra $(\oslash_\bullet^\bullet V,Q,\oslash_i)$ with $V=\mathfrak{E}^*$ and the differential

$Q t^\alpha=-(-1)^{|\beta|}m^\alpha_\beta t^\beta-(-1)^{i(|\beta|+|\gamma|)+|\gamma|(|\beta|-1)}\,m^{i,\alpha}_{\beta\gamma}\,t^\beta \oslash_i t^\gamma,$

where $|\beta|\coloneqq|t^\beta|$.

Consider

$\oslash^\bullet_\bullet V \,\coloneqq\, \underbrace{\mathbb{R}}_{\text{deg }0}\oplus\underbrace{V}_{\text{deg }1}\oplus\underbrace{\bigoplus_{i\in \mathbb{N}}V\oslash_i V}_{\text{deg }2}\oplus\underbrace{\bigoplus_{i,j\in \mathbb{N}}(V\oslash_i V)\oslash_j V}_{\text{deg }3}\oplus\dots$

###### Definition

The Chevalley–Eilenberg algebra $\mathrm{CE}(\mathfrak{E})$ of an $h\mathcal{L}ie$-algebra $\mathfrak{E}$ whose differential and binary products are given by

\begin{aligned} \varepsilon_1&: \mathfrak{E}\rightarrow \mathfrak{E},&\tau_\alpha&\mapsto m^\beta_\alpha \tau_\beta,&&|\varepsilon_1|=1, \\ \varepsilon^i_2&: \mathfrak{E}\otimes \mathfrak{E}\rightarrow \mathfrak{E},&\tau_\alpha\otimes \tau_\beta&\mapsto m^{i,\gamma}_{\alpha\beta}\tau_\gamma,&&|\varepsilon^i_2|=i \end{aligned}

for some $m^\beta_\alpha$ and $m^{i,\gamma}_{\alpha\beta}$ is the $\mathcal{E}ilh$-algebra $(\oslash_\bullet^\bullet V,Q,\oslash_i)$ with $V=\mathfrak{E}^*$ and the differential

$Q t^\alpha=-(-1)^{|\beta|}m^\alpha_\beta t^\beta-(-1)^{i(|\beta|+|\gamma|)+|\gamma|(|\beta|-1)}\,m^{i,\alpha}_{\beta\gamma}\,t^\beta \oslash_i t^\gamma,$

where $|\beta|\coloneqq|t^\beta|$.

## $EL_\infty$-algebras

$EL_\infty$-algebras are the homotopy version of $h\mathcal{L}ie$-algebras, defined by (BKS21).

Analogously to an $L_\infty$-algebra, an $EL_\infty$-algebra structure on a graded vector space $\mathfrak{E}$ is encoded by a differential $Q$ on the $\mathcal{E}ilh$-algebra $\mathrm{CE}(\mathfrak{E})\coloneqq (\oslash_\bullet^\bullet \mathfrak{E}^\ast, Q, \oslash_i)$. The differential $Q$ is given by its action on $\mathfrak{E}^\ast$, which will be encoded by structure constants $m$ as follows:

$Q t^\alpha=\pm m^\alpha\pm m^\alpha_\beta t^\beta\pm m^{i_1,\alpha}_{\beta_1\beta_2} t^{\beta_1}\oslash_{i_1} t^{\beta_2}\pm m^{i_1i_2,\alpha}_{\beta_1\beta_2\beta_3}(t^{\beta_1}\oslash_{i_1} t^\gamma)\oslash_{i_2} t^\delta+\ldots,$

where $\{t^\alpha\}$ is a basis on $\mathfrak{E}^\ast$.

These structure constants $m$ define higher products $\varepsilon^I_n:\mathfrak{E}^{\otimes n}\rightarrow \mathfrak{E}$ with degree$-|I|$ by

\begin{aligned} \varepsilon_0=m^\alpha \tau_\alpha, \; \varepsilon_1(\tau_\alpha)=m^\beta_\alpha \tau_\beta, \; \varepsilon_2^i(\tau_\alpha,\tau_\beta)=m^{i,\gamma}_{\alpha\beta}\tau_\gamma,\ldots \\ \varepsilon^I_n(\tau_{\alpha_1},\ldots,\tau_{\alpha_n})=m_{\alpha_1\ldots\alpha_n}^{I,\beta} \tau_\beta, \end{aligned}

where $I$ is a multi-index consisting of $n-1$ indices $i_1,i_2,\ldots,\in \mathbb{N}$ and $|I|\coloneqq i_1+i_2+\ldots$.