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endomorphism dg-Lie algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Definition

Let Ch(𝒜)Ch(\mathcal{A}) be a category of chain complexes in a category 𝒜\mathcal{A} that is a closed monoidal category. For instance the category of chain complexes in 𝒜=\mathcal{A} = Vect k{}_k.

For VChV \in Ch any object – any chain complex – write

end(V):=[V,V]Ch end(V) := [V,V] \in Ch

for the internal hom object of morphisms from VV to VV. This is the chain complex which in degree nn is

end(V) n= iHom 𝒜(V i,V i+n) end(V)_n = \bigoplus_{i \in \mathbb{Z}} Hom_{\mathcal{A}}(V_i, V_{i+n})

and whose differential

δ end(V):end(V) nend(V) n1 \delta_{end(V)} : end(V)_n \to end(V)_{n-1}

is given by

δ end(V)(V ifV i+1)=(V ifV i+nδ VV i+n1)(1) i(V i+1δ VV ifV i+n). \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 fend(V) kf \in end(V)_k and gend(V) lg \in end(V)_l

[f,g]=fg(1) klgf. [f,g] = f \circ g - (-1)^{k \cdot l} g \circ f \,.

This

𝔤𝔩(V):=(end(V),[,]) \mathfrak{gl}(V) := (end(V), [-,-])

is the endomorphism dg-Lie algebra of VV. Or, since dg-Lie algebras are special cases of L-∞ algebras: the endomorphism L L_\infty-algebra of VV.

Representation

A representation of an L-∞ algebra 𝔤\mathfrak{g} on a chain complex VV is a morphism of L L_\infty-algebras

ρ:𝔤𝔤𝔩(V). \rho : \mathfrak{g} \to \mathfrak{gl}(V) \,.

Sometimes this is called a representation up to homotopy .

Examples

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)