nLab endomorphism dg-Lie algebra

Definition

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

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

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

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

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

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 $f \in end(V)_k$ and $g \in end(V)_l$

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

This

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

is the endomorphism dg-Lie algebra of $V$. 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 $\mathfrak{g}$ on a chain complex $V$ is a morphism of $L_\infty$-algebras

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

Sometimes this is called a representation up to homotopy .

Examples

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

References

See for instance the paragraph above theorem 5.4 in

• Tom Lada, Martin Markl, Strongly homotopy Lie algebras (arXiv:9406095)