∞-Lie theory (higher geometry)
and
A differential graded Lie algebra, or dg-Lie algebra for short, is equivalently
a graded Lie algebra equipped with a differential that acts as a graded derivation with respect to the Lie bracket;
a strict L-∞-algebra, i.e. an $L_\infty$-algebra in which only the unary and the binary brackets may be nontrivial.
A dg-Lie algebra $(\mathfrak{g},\partial,[-,-])$ is
a $\mathbb{Z}$-graded vector space $\mathfrak{g} = \bigoplus_{i} \mathfrak{g}_i$;
a linear map $\partial \colon \mathfrak{g} \longrightarrow \mathfrak{g}$;
a bilinear map $[-,-] \colon \mathfrak{g}\otimes\mathfrak{g} \longrightarrow \mathfrak{g}$, the bracket;
such that (all conditions are expressed for homogeneously graded elements $x_i \in \mathfrak{g}_{\vert x_i \vert}$):
$\partial$ is a differential that makes $(\mathfrak{g},\partial)$ into a chain complex, i.e.
it is of degree -1, $\partial \colon \mathfrak{g}_{i} \to \mathfrak{g}_{i-1}$;
it squares to zero, $\partial \circ \partial = 0$;
$\partial$ is a graded derivation of the bilinear pairing, i.e.
the bilinear pairing is graded skew-symmetric, i.e.
the bilinear pairing satisfies the graded Jacobi identity (saying that $[x,-]$ is a graded derivation)
A pre-graded Lie algebra (pre-gla) is a pre-gvs, $L$, together with a bilinear map of degree zero
such that
and
for every triple $(x,y,z)$ of homogeneous elements in $L$.
(The first property is call antisymmetry, the second the Jacobi identity.)
A morphism $f: L \to L'$ of pre-glas is a linear map of degree zero, that is compatible with the brackets,
To any augmented pre-ga $A$, one can associate a pre-gla, denoted $\bar{A}_L$, with underlying gvs $\bar{A}$ and with bracket, the commutator, $[x,y] = xy -(-1)^{|x||y|}yx$ for each pair $(x,y)$ of homogeneous elements. $\bar{A}_L$ is abelian (i.e. with trivial bracket) if and only if $A$ is graded commutative.
If $A$ is a pre-cga and $L$ is a pre-gla, the tensor product $A\otimes L$ has a pre-gla structure with bracket
for $a,a', l, l'$ homogeneous.
Let $L$ be a pre-gla. A derivation of gla-s, of degree $p\in \mathbb{Z}$, is a linear mapping $\theta \in Hom_p(L,L)$ such that
for any pair $x,y$ of homogeneous elements of $L$. We denote by $Der_p(L)$, the vector space of degree $p$ derivations of the gla, $L$.
A differential $\partial$ of a pre-gla is a Lie algebra derivation of degree -1 such that $\partial\circ \partial = 0$. The pair $(L,\partial)$ is then called a differential pre-graded Lie algebra (pre-dgla); its homology $H(L,\partial)$, is a pre-gla.
A morphism of pre-dglas is a morphism for both the underlying pre-gla and the pre-dgvs. We denote the corresponding category by $pre DGLA$.
This means that a differential graded Lie algebra is an internal Lie algebra in the symmetric monoidal category of chain complexes with tensor product given as in differential graded vector spaces.
If $(A,\partial)$ is an augmented pre-dga, $(\bar{A}_L,\partial)$ is a pre-dgla.
If $(A,\partial)$ is a pre-cdga and $(L,\partial')$, a pre-dgla, $A \otimes L$, together with the tensor product differential, is a pre-dgla.
Let $(V,\partial)$ be a pre-dgvs, then the pre-dgvs, $(Hom(V,V),D)$, constructed earlier is a pre-dga for the multiplication law given by composition of mappings. Its associated pre-dgla has
and
In particular, if $(V,\partial) = (A,d)$ is a cdga (resp. $(V,\partial) = (L,\partial)$ is dgla), then $Der(A,d) = (\bigoplus_p Der_p(A),D)$, (resp. $Der(L,\partial) = (Der_p(L),D)$) is a sub-pre-dgl of $(Hom(V,V),D)$.
A dgla is a pre-dgla with a lower grading; explicitly:
A differential graded Lie algebra, $(L,\partial)$, is a graded vector space $L = \bigoplus_{p\geq 0}L_p$, together with a bilinear map of degree 0
and a differential $\partial$ satisfying
and
for every triple $(x,y,z)$ of homogeneous elements in $L$.
Let $DGLA$ be the corresponding category.
A dgla is $n$-reduced (resp. homologically $n$-reduced) if $L_p = 0$ (resp. $H_p(L,\partial) = 0$) for all $p\lt n$. Denote by $DGLA_n$ (resp. $DGLA_{hn}$), the corresponding categories.
If $(L,\partial)$ is a pre-dgla, a gla-filtration of $L$ (resp. a dgla-filtration of $( L,\partial)$ ) is a family of subgraded vector spaces $F_p L$, $p\in \mathbb{Z}$, such that $F_p L\subseteq F_{p+1}L$, $[F_p L,F_n L]\subseteq F_{p+n} L$, (resp. and $\partial F_p L\subseteq F_p L$).
Let $L$ be a pre-gla. Its bracket length filtration is obtained from the descending central series:
It is a gla-filtration.
$Q(L) = L/F^2L$ is called the space of indecomposables of $L$.
If $(L,\partial)$ is a pre-dgla, $F^p L$ is stable by $\partial$. Letting $Q(\partial)$ be the induced differential on $Q(L)$, $Q$ then defines a functor
Free Lie algebra, $\mathbb{L}(V)$
Let $V$ be a pre-gvs, $T(V)$, the tensor algebra on $V$ with augmentation ideal $\overline{T(V)}$ (recall $T(V) = \bigoplus_{n\geq 0} V^{\otimes n}$ and the augmentation sends $V (= V^{\otimes 1}$ to 0).
Let $\overline{T(V)}_L$ be $\overline{T(V)}$ with the pre-gla structure given by the commutators. We denote by $\mathbb{L}(V)$, the Lie subalgebra of $\overline{T(V)}_L$ generated by $V$.
Tim: A more explicit description may help here, cf. Quillen, Rational Homotopy theory (p.281) or MacLane, Homology.
If $L$ is a pre-gla, any morphism of pre-gvs $f: V\to L$ has a unique extension to a pre-gla morphism $\hat{f} :\mathbb{L}(V)\to L$. If $(e_\alpha)_{\alpha\in I}$ is a homogeneous basis for $V$, $\mathbb{L}(V)$ may be denoted $\mathbb{L}((e_\alpha)_{\alpha\in I})$.
On the free Lie algebra $\mathbb{L}(V)$, the bracket length filtration comes from a gradation $\mathbb{L}(V) = \bigoplus_j\mathbb{L}^j(V)$, where $\mathbb{L}^j(V)$ is the subspace generated by the brackets of elements of $V$ of length $j$. The inclusion $\mathbb{L}(V)\hookrightarrow T(V)$ identifies $\mathbb{L}^j(V)$ with $\mathbb{L}(V)\cap T^j(V)$.
If $\mathbb{L}(V), \partial)$ is a dgla, free as a gla, with $V$ fixed, $\partial$ is the sum of derivations $\partial_k$ defined by : $\partial_k \subset \mathbb{L}^k(V)$. The isomorphism between $V$ and $Q\mathbb{L}(V)$ identifies $\partial_1V$ with $Q(\partial)$. $\partial_1$ (resp. $\partial_2$) is called the linear part (resp, the quadratic part) of $\partial$.
Let $(L,\partial)$ and $(L',\partial')$ be two dglas. Their product $(L,\partial)\times(L',\partial')$ in $DGLA$ is defined by:
the underlying vector space is the direct sum $L\oplus L'$;
$(L,\partial)$ and $(L',\partial')$ are two sub differential graded Lie algebras of $(L,\partial)\times(L',\partial')$;
if $x\in L$ and $x' \in L'$, then $[x,x'] = 0$.
Their coproduct or sum $(L,\partial)\star(L',\partial')$ is often called their free product.
More generally if $L$ and $L'$ are given by generators and relations
The differential on $L\star L'$ is the unique Lie algebra derivation extending $\partial$ and $\partial'$.
Every dg-Lie algebra is in an evident way an L-infinity algebra. Dg-Lie algebras are precisely those $L_\infty$-algebras for which all $n$-ary brackets for $n \gt 2$ are trivial. These may be thought of as the strict $L_\infty$-algebras: those for which the Jacobi identity holds on the nose and all its possible higher coherences are trivial.
Let $k$ be a field of characteristic 0 and write $L_\infty Alg_k$ for the category of L-infinity algebras over $k$.
Then every object of $L_\infty Alg_k$ is quasi-isomorphic to a dg-Lie algebra.
Moreover, one can find a functorial replacement: there is a functor
such that for each $\mathfrak{g} \in L_\infty Alg_k$
$W(\mathfrak{k})$ is a dg-Lie algebra;
there is a quasi-isomorphism
This appears for instance as (KrizMay, cor. 1.6).
Via the above relation to $L_\infty$-algebras, dg-Lie algebras are also connected by adjunction to dg-coalgebras
Here
$CE$ is the Chevalley-Eilenberg algebra functor. It sends a dg-Lie algebra $(\mathfrak{g}, \partial, [-,-])$ to
where on the right the extension of $\partial$ and $[-,-]$ to graded derivations is understood.
For $(X,D)$ a dg-coalgebra, then
where
$\overline{X} \coloneqq ker(\epsilon)$ is the kernel of the counit, regarded as a chain complex;
$F$ is the free Lie algebra functor (as graded Lie algebras);
on the right we are extending $(\Delta - 1 \otimes id - id \otimes 1) \colon \overline{X} \to \overline{X} \otimes \overline{X}$ as a Lie algebra derivation
Moreover
is the Maurer-Cartan elements in the Hom-dgLie algebra from $\overline{X}$ to $\mathfrak{g}$.
For dg-Lie algebras concentrated in degrees $\geq n \geq 1$ this is due to (Quillen 69, appendix B, prop 6.1, 6.2). For unbounded dg-algebras, this is due to (Hinich 98, 2.2).
For more see at model structure on dg-Lie algebras.
There is an adjunction
between simplicial Lie algebras and dg-Lie algebras, where $N$ acts on the underlying simplicial vector spaces as the Moore complex functor.
This is (Quillen, prop. 4.4). For more see at simplicial Lie algebra.
This adjunction is a Quillen adjunction with respect to the projective model structure on dg-Lie algebras and the projective model structure on simplicial Lie algebras (this prop.).
The corresponding derived functors constitute an equivalence of categories between the corresponding homotopy categories
of 1-connected objects on both sides.
This is in the proof of (Quillen, theorem. 4.4).
crossed complex $\Leftarrow$ crossed module $\Rightarrow$ 2-crossed module
dg-Lie algebra $\Leftarrow$ differential crossed module $\Rightarrow$ differential 2-crossed module
A standard reference in the context of rational homotopy theory is
For the unbounded case there is general discussion in
The relation to $L_\infty$-algebras is discussed for instance in
See also the regerences at model structure on dg-Lie algebras.
A discussion of how formal neighbourhoods of points in infinity-stacks are governed by dg-Lie algebras: