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EL-∞ algebra

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\infty-Lie theory

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Contents

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 EL_\infty-algebras

EL EL_\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

Created on September 13, 2021 at 07:13:48. See the history of this page for a list of all contributions to it.