nLab EL-∞ algebra

Idea

The notion of $E L_\infty$-algebras is meant to be the fully homotopy theoretic (i.e. $(\infty,0)$-categorified) higher structure enhancing the mathematical structure of Lie algebras: For $E L_\infty$-algebras both the Jacobi identity and the skew symmetry of the Lie bracket are relaxed up to potentially infinite coherent higher homotopy.

This is in contrast to the more widely considered notion of $L_\infty$-algebras, which relax the Jacobi identity but retain strict skew symmetry. (Whence the terminology “$E L_\infty$”: the “$E$” is for “everything homotopy”, a whimsical but time-honored terminology, enshrined in the now classical terminology of $E_\infty$-algebras).

The homotopy theory of $E L_\infty$-algebra is in fact equivalent to that of $L_\infty$-algebras (and thus both are equivalent even to that of dg-Lie algebras, which are further rectified $L_\infty$-algebras): $L_\infty$-algebras are a special case of $E L_\infty$-algebras and every $E L_\infty$-algebra is weakly equivalent to one that is an $L_\infty$-algebra (i.e. the homotopy-skew-symmetry may always be rectified to strict skew symmetry).

Nevertheless, in some circumstances it is practically useful to work with instances of $E L_\infty$-algebras up to isomorphism without passing to a weakly equivalent $L_\infty$-algebra. In particular, Borsten, Kim & Saemann 2021 argue that the notion of $E L_\infty$-algebra serves to give a transparent way to understand adjusted Weil algebras for $L_\infty$-algebras, and then to understand tensor hierarchies (in gauged supergravity-theory) in terms of the resulting $\infty$-connections/higher gauge theory.

$h\mathcal{L}ie$- 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 $V=\mathfrak{E}[1]^*$ 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 $V=\mathfrak{E}[1]^*$ 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 $|\beta|\coloneqq|t^\beta|$.

$E L_\infty$-algebras

$E L_\infty$-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 $\mathrm{CE}(\mathfrak{E})\coloneqq (\oslash_\bullet^\bullet \mathfrak{E}[1]^\ast, Q, \oslash_i)$. The differential $Q$ is given by its action on $\mathfrak{E}[1]^\ast$, 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 $\mathfrak{E}[1]^\ast$.

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 $|I|\coloneqq i_1+i_2+\ldots$.

Related concepts

• A-∞ algebra

• E-∞ algebra

• L-∞ algebra

  • homomorphism of L-∞ algebras

  • infinity-Lie algebra cohomology

  • universal enveloping E-n algebra

  • nilpotent L-∞ algebra

  • L-∞ algebroid

  • derived L-∞ algebroid

  • sheaf of L-∞ algebras

• relation between L-∞ algebras and dg-Lie algebras

• adjusted Weil algebra

• tensor hierarchy

• Leibniz algebra

References

The general notion is discussed in:

• Leron Borsten, Hyungrok Kim, Christian Saemann: $EL_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies (arXiv:2106.00108)

The special case of weak Lie 2-algebras was originally considered in:

• Dmitry Roytenberg: On weak Lie 2-algebras, in: XXVI Workshop on Geometrical Methods in Physics, AIP Conference Proceedings 956, American Institute of Physics, (2007) [arXiv:0712.3461, hdl:10993/10465]

and the more general special case of weak Lie 3-algebras in:

• Malte Dehling, On weak Lie 3-algebras [arXiv:1710.11104]