∞-Lie theory

# Contents

## Definition

Let $\mathrm{Ch}\left(𝒜\right)$ be a category of chain complexes in a category $𝒜$ that is a closed monoidal category. For instance the category of chain complexes in $𝒜=$ Vect${}_{k}$.

For $V\in \mathrm{Ch}$ any object – any chain complex – write

$\mathrm{end}\left(V\right):=\left[V,V\right]\in \mathrm{Ch}$end(V) := [V,V] \in Ch

for the internal hom object of morphisms from $V$ to $V$. This is the chain complex which in degree $n$ is

$\mathrm{end}\left(V{\right)}_{n}=\underset{i\in ℤ}{⨁}{\mathrm{Hom}}_{𝒜}\left({V}_{i},{V}_{i+n}\right)$end(V)_n = \bigoplus_{i \in \mathbb{Z}} Hom_{\mathcal{A}}(V_i, V_{i+n})

and whose differential

${\delta }_{\mathrm{end}\left(V\right)}:\mathrm{end}\left(V{\right)}_{n}\to \mathrm{end}\left(V{\right)}_{n-1}$\delta_{end(V)} : end(V)_n \to end(V)_{n-1}

is given by

${\delta }_{\mathrm{end}\left(V\right)}\left({V}_{i}\stackrel{f}{\to }{V}_{i+1}\right)=\left({V}_{i}\stackrel{f}{\to }{V}_{i+n}\stackrel{{\delta }_{V}}{\to }{V}_{i+n-1}\right)-\left(-1{\right)}^{i}\left({V}_{i+1}\stackrel{{\delta }_{V}}{\to }{V}_{i}\stackrel{f}{\to }{V}_{i+n}\right)\phantom{\rule{thinmathspace}{0ex}}.$\delta_{end(V)} (V_i \stackrel{f}{\to} V_{i+1}) = (V_i \stackrel{f}{\to} V_{i+n} \stackrel{\delta_V}{\to} V_{i+n-1}) - (-1)^i (V_{i+1} \stackrel{\delta_V}{\to} V_i \stackrel{f}{\to} V_{i+n}) \,.

This complex naturally carries the structure of an internal Lie algebra, hence of a dg-Lie algebra with Lie bracket given by the graded commutator for $f\in \mathrm{end}\left(V{\right)}_{k}$ and $g\in \mathrm{end}\left(V{\right)}_{l}$

$\left[f,g\right]=f\circ g-\left(-1{\right)}^{k\cdot l}g\circ f\phantom{\rule{thinmathspace}{0ex}}.$[f,g] = f \circ g - (-1)^{k \cdot l} g \circ f \,.

This

$\mathrm{𝔤𝔩}\left(V\right):=\left(\mathrm{end}\left(V\right),\left[-,-\right]\right)$\mathfrak{gl}(V) := (end(V), [-,-])

is the endomorphism dg-Lie algebra of $V$. Or, since dg-Lie algebras are special cases of L-∞ algebras: the endomorphism ${L}_{\infty }$-algebra of $V$.

## Representation

A representation of an L-∞ algebra $𝔤$ on a chain complex $V$ is a morphism of ${L}_{\infty }$-algebras

$\rho :𝔤\to \mathrm{𝔤𝔩}\left(V\right)\phantom{\rule{thinmathspace}{0ex}}.$\rho : \mathfrak{g} \to \mathfrak{gl}(V) \,.

Sometimes this is called a representation up to homotopy .

## Examples

• For $V$ a vector space regarded as a chain complex concentrated in degree-0, $\mathrm{𝔤𝔩}\left(V\right)$ is the ordinary general linear Lie algebra of $V$.

## References

See for instance the paragraph above theorem 5.4 in

Revised on April 4, 2011 15:02:04 by Urs Schreiber (131.211.42.205)