nLab EL-∞ algebra

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Higher algebra

Rational homotopy theory

Contents

Idea

The notion of EL E L_\infty-algebras is meant to be the fully homotopy theoretic (i.e. ( , 0 ) (\infty,0) -categorified) higher structure enhancing the mathematical structure of Lie algebras: For EL 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 L_\infty -algebras, which relax the Jacobi identity but retain strict skew symmetry. (Whence the terminology “EL E L_\infty”: the “EE” is for “everything homotopy”, a whimsical but time-honored terminology, enshrined in the now classical terminology of E E_\infty -algebras).

The homotopy theory of EL E L_\infty-algebra is in fact equivalent to that of L L_\infty -algebras (and thus both are equivalent even to that of dg-Lie algebras, which are further rectified L L_\infty-algebras): L L_\infty-algebras are a special case of EL E L_\infty-algebras and every EL E L_\infty-algebra is weakly equivalent to one that is an L 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 EL E L_\infty-algebras up to isomorphism without passing to a weakly equivalent L L_\infty -algebra. In particular, Borsten, Kim & Saemann 2021 argue that the notion of EL E L_\infty-algebra serves to give a transparent way to understand adjusted Weil algebras for L L_\infty -algebras, and then to understand tensor hierarchies (in gauged supergravity-theory) in terms of the resulting \infty -connections/higher gauge theory.

hieh\mathcal{L}ie- and ilh\mathcal{E}ilh-algebras

The following notions have been introduced by (BKS21).

hieh\mathcal{L}ie-algebras

Definition

An hieh\mathcal{L}ie-algebra (𝔈,ε 1,ε 2 i)(\mathfrak{E},\varepsilon_1,\varepsilon_2^i) is a graded vector space 𝔈\mathfrak{E} together with a differential and a collection of binary products,

ε 1 :𝔈𝔈, |ε 1|=1, ε 2 i :𝔈𝔈𝔈, |ε 2 i|=i, \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

ε 1(ε 1(x 1)) =0, ε 1(ε 2 i(x 1,x 2)) =(1) i(ε 2 i(ε 1(x 1),x 2)+(1) |x 1|ε 2 i(x 1,ε 1(x 2))) +ε 2 i1(x 1,x 2)(1) i+|x 1||x 2|ε 2 i1(x 2,x 1), ε 2 i(ε 2 i(x 1,x 2),x 3) =(1) i(|x 1|+1)ε 2 i(x 1,ε 2 i(x 2,x 3))(1) (|x 1|+i)|x 2|ε 2 i(x 2,ε 2 i(x 1,x 3)) (1) (|x 2|+|x 3|)|x 1|+(i1)|x 2|ε 2 i+1(x 2,ε 2 i1(x 3,x 1)), ε 2 j(ε 2 i(x 1,x 2),x 3) =(1) 1+j(i+1)+|x 1|(|x 2|+|x 3|)+(j1)|x 2|ε 2 i+1(x 2,ε 2 j1(x 3,x 1)), ε 2 i(ε 2 j(x 1,x 2),x 3) =(1) i(j+|x 1|)ε 2 j(x 1,ε 2 i(x 2,x 3))(1) (|x 1|+j)|x 2|ε 2 i(x 2,ε 2 j(x 1,x 3)) (1) j+|x 3|(j+|x 2|1)+|x 1|(|x 2|+|x 3|)ε 2 i+1(x 3,ε 2 j1(x 2,x 1)) \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,ji,j\in \mathbb{N} s.t. jj<ii and for all x 1,x 2,x 3𝔈x_1,x_2,x_3\in \mathfrak{E}, where we regard ε 2 1=0\varepsilon_2^{-1}=0.

ilh\mathcal{E}ilh-algebras

Definition

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

a i(b ic) =(1) i(|a|+1)((a ib) ic+(1) |a||b|(b ia) ic), a i(b jc) =(1) ij+j|a|(a ib) jc, a j(b ic) ={(1) i|a|+|a||b|(b ia) jc if ji=1, (1) i|a|+|a||b|(b ia) jc if ji=2, +(1) i(|a|+j+1)+(|a|+|b|)|c|((c j1a) i+1b) (1) i|a|+|a||b|(b ia) jc if ji>2 +(1) i(|a|+j+1)+(|a|+|b|)|c|((c j1a) i+1b) +(1) j+|a|(|b|+i)+(|a|+|b|)|c|((c i+1b) j1a) \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 jj>ii, and such that differential QQ satisfies the property

Q(a ib) =(1) i((Qa) ib+(1) |a|a iQb) +(1) i(a i+1b)(1) |a||b|(b i+1a), \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

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

Definition

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

ε 1 :𝔈𝔈, τ α m α βτ β, |ε 1|=1, ε 2 i :𝔈𝔈𝔈, τ ατ β m αβ i,γτ γ, |ε 2 i|=i \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 α βm^\beta_\alpha and m αβ i,γm^{i,\gamma}_{\alpha\beta} taking values in the underlying ground field is the ilh\mathcal{E}ilh-algebra ( V,Q, i)(\oslash_\bullet^\bullet V,Q,\oslash_i) with V=𝔈[1] *V=\mathfrak{E}[1]^* and the differential

Qt α=(1) |β|m β αt β(1) i(|β|+|γ|)+|γ|(|β|1)m βγ i,αt β it γ, 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 |β||t β||\beta|\coloneqq|t^\beta|.

Consider

V deg 0V deg 1 iV iV deg 2 i,j(V iV) jV deg 3 \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 CE(𝔈)\mathrm{CE}(\mathfrak{E}) of an hieh\mathcal{L}ie-algebra 𝔈\mathfrak{E} whose differential and binary products are given by

ε 1 :𝔈𝔈, τ α m α βτ β, |ε 1|=1, ε 2 i :𝔈𝔈𝔈, τ ατ β m αβ i,γτ γ, |ε 2 i|=i \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 α βm^\beta_\alpha and m αβ i,γm^{i,\gamma}_{\alpha\beta} is the ilh\mathcal{E}ilh-algebra ( V,Q, i)(\oslash_\bullet^\bullet V,Q,\oslash_i) with V=𝔈[1] *V=\mathfrak{E}[1]^* and the differential

Qt α=(1) |β|m β αt β(1) i(|β|+|γ|)+|γ|(|β|1)m βγ i,αt β it γ, 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 |β||t β||\beta|\coloneqq|t^\beta|.

EL E L_\infty-algebras

EL E L_\infty-algebras are the homotopy version of hieh\mathcal{L}ie-algebras, defined by (BKS21).

Analogously to an L L_\infty-algebra, an EL EL_\infty-algebra structure on a graded vector space 𝔈\mathfrak{E} is encoded by a differential QQ on the ilh\mathcal{E}ilh-algebra CE(𝔈)( 𝔈[1] *,Q, i)\mathrm{CE}(\mathfrak{E})\coloneqq (\oslash_\bullet^\bullet \mathfrak{E}[1]^\ast, Q, \oslash_i). The differential QQ is given by its action on 𝔈[1] *\mathfrak{E}[1]^\ast, which will be encoded by structure constants mm as follows:

Qt α=±m α±m β αt β±m β 1β 2 i 1,αt β 1 i 1t β 2±m β 1β 2β 3 i 1i 2,α(t β 1 i 1t γ) i 2t δ+, 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 α}\{t^\alpha\} is a basis on 𝔈[1] *\mathfrak{E}[1]^\ast.

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

ε 0=m ατ α,ε 1(τ α)=m α βτ β,ε 2 i(τ α,τ β)=m αβ i,γτ γ, ε n I(τ α 1,,τ α n)=m α 1α n I,βτ β, \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 II is a multi-index consisting of n1n-1 indices i 1,i 2,,i_1,i_2,\ldots,\in \mathbb{N} and |I|i 1+i 2+|I|\coloneqq i_1+i_2+\ldots.

References

The general notion is discussed in:

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

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

Last revised on July 18, 2022 at 01:12:00. See the history of this page for a list of all contributions to it.