Context
-Lie theory
∞-Lie theory
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Cohomology
Homotopy
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
Contents
Definition
Let be a category of chain complexes in a category that is a closed monoidal category. For instance the category of chain complexes in Vect.
For any object – any chain complex – write
end(V) := [V,V] \in Ch
for the internal hom object of morphisms from to . This is the chain complex which in degree is
end(V)_n = \bigoplus_{i \in \mathbb{Z}} Hom_{\mathcal{A}}(V_i, V_{i+n})
and whose differential
\delta_{end(V)} : end(V)_n \to end(V)_{n-1}
is given by
\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 and
[f,g] = f \circ g - (-1)^{k \cdot l} g \circ f
\,.
This
\mathfrak{gl}(V) := (end(V), [-,-])
is the endomorphism dg-Lie algebra of . Or, since dg-Lie algebras are special cases of L-∞ algebras: the endomorphism -algebra of .
Representation
A representation of an L-∞ algebra on a chain complex is a morphism of -algebras
\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