Contents

Contents

Idea

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.

Definition

Direct explicit definition

Definition

A dg-Lie algebra $(\mathfrak{g},\partial,[-,-])$ is

1. a $\mathbb{Z}$-graded vector space $\mathfrak{g} = \bigoplus_{i} \mathfrak{g}_i$;

2. a linear map $\partial \colon \mathfrak{g} \longrightarrow \mathfrak{g}$;

3. a bilinear map $[-,-] \colon \mathfrak{g}\otimes\mathfrak{g} \longrightarrow \mathfrak{g}$, the bracket;

such that (all conditions are expressed for homogeneously graded elements $x_i \in \mathfrak{g}_{\vert x_i \vert}$):

1. $\partial$ is a differential that makes $(\mathfrak{g},\partial)$ into a chain complex, i.e.

1. it is of degree -1, $\partial \colon \mathfrak{g}_{i} \to \mathfrak{g}_{i-1}$;

2. it squares to zero, $\partial \circ \partial = 0$;

2. $\partial$ is a graded derivation of the bilinear pairing, i.e.

$\partial [x_1,x_2] = [\partial x_1, x_2] + (-1)^{\vert x_1 \vert} [x_1, \partial x_2] \,,$
3. the bilinear pairing is graded skew-symmetric, i.e.

$[x_1, x_2] = -(-1)^{\vert x_1\vert \vert x_2 \vert} [x_2,x_1] \,,$
4. the bilinear pairing satisfies the graded Jacobi identity (saying that $[x,-]$ is a graded derivation)

$[x_1,[x_2,x_3]] = [[x_1,x_2], x_3] + (-1)^{\vert x_1 \vert \vert x_2 \vert} [x_2, [x_1, x_3]] \,.$

As graded Lie algebras with nilpotent derivations

Definition

A pre-graded Lie algebra (pre-gla) is a pre-gvs, $L$, together with a bilinear map of degree zero

$[\quad,\quad ] : L\otimes L \to L,$

such that

$[x,y] = (-1)^{|x||y|+1}[y,x]$

and

$(-1)^{|x||z|}[x,[y,z]] +(-1)^{|y||x|}[y,[z,x]] +(-1)^{|z||y|}[z,[x,y]] = 0$

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,

$f[x,y] = [f(x),f(y)].$
Example

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.

Example

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

$[a\otimes l,a'\otimes l'] = (-1)^{|a'||l|}aa' \otimes [l,l']$

for $a,a', l, l'$ homogeneous.

Definition

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

$\theta[x,y] = [\theta{x},y] + (-1)^{p|x|}[x,\theta(y)]$

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, $L$.

Definition

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.

Example

If $(A,\partial)$ is an augmented pre-dga, $(\bar{A}_L,\partial)$ is a pre-dgla.

Example

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.

Example

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

$[f,g] = f\circ g - (-1)^{|f||g|}g\circ f$

and

$Df = [\partial,f].$

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 $(Hom(V,V),D)$.

Definition

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

$[\quad ,\quad] : L\otimes L \to L,$

and a differential $\partial$ satisfying

$\partial L_p \subseteq L_{p-1}, \quad [x,y] = (-1)^{|x||y|+1}[y,x],$
$(-1)^{|x||z|}[x,[y,z]] +(-1)^{|y||x|}[y,[z,x]] +(-1)^{|z||y|}[z,[x,y]] = 0$

and

$\partial[x,y] = [\partial x,y] + (-1)^{|x|}[x,\partial y]$

for every triple $(x,y,z)$ of homogeneous elements in $L$.

Let $DGLA$ be the corresponding category.

Definition

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.

Definition

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 $\partial F_p L\subseteq F_p L$).

Definition

Let $L$ be a pre-gla. Its bracket length filtration is obtained from the descending central series:

$F^1 L = L; \quad F^p L = [L,F^{p-1} L] \quad if \quad p\geq 2.$

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

$Q : pre DGLA \to pre DGVS.$
Example

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 $\partial$.

Definition

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.

Example
$\mathbb{L}(V)\star \mathbb{L}(V') \cong \mathbb{L}(V\oplus V').$

More generally if $L$ and $L'$ are given by generators and relations

$L = \mathbb{L}(V)/I , \quad L' = \mathbb{L}(V')/I' ,$
$L\star L' = \mathbb{L}(V\oplus V')/{I,I'}.$

The differential on $L\star L'$ is the unique Lie algebra derivation extending $\partial$ and $\partial'$.

Properties

Relation to $L_\infty$-algebras

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.

Theorem

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

$W : L_\infty Alg_k \to L_\infty Alg_k$

such that for each $\mathfrak{g} \in L_\infty Alg_k$

1. $W(\mathfrak{k})$ is a dg-Lie algebra;

2. there is a quasi-isomorphism

$\mathfrak{g} \stackrel{\simeq}{\to} W(\mathfrak{g}) \,.$

This appears for instance as (KrizMay, cor. 1.6).

For more see at

Relation to dg-coAlgebras

Via the above relation to $L_\infty$-algebras, dg-Lie algebras are also connected by adjunction to dg-coalgebras

$dgLieAlg_k \underoverset {\underset{CE}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {\bot} dgCoAlg_k$

Here

• $CE$ is the Chevalley-Eilenberg algebra functor. It sends a dg-Lie algebra $(\mathfrak{g}, \partial, [-,-])$ to

$CE(\mathfrak{g},\partial,[-,-]) \;\coloneqq\; \left( \vee^\bullet \mathfrak{g}[1] ,\; D = \partial + [-,-] \right) \,,$

where on the right the extension of $\partial$ and $[-,-]$ to graded derivations is understood.

• For $(X,D)$ a dg-coalgebra, then

$\mathcal{L}(X,D) \coloneqq \left( F(\overline{X}[-1]),\; \partial \coloneqq D + (\Delta - 1 \otimes id - id \otimes 1) \right)$

where

1. $\overline{X} \coloneqq ker(\epsilon)$ is the kernel of the counit, regarded as a chain complex;

2. $F$ is the free Lie algebra functor (as graded Lie algebras);

3. 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

$Hom(\mathcal{L}(X), \mathfrak{g}) \simeq Hom(X, CE(\mathfrak{g})) \simeq MC(Hom(\overline{X},\mathfrak{g}))$

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.

Relation to simplicial Lie algebras

Theorem

$(N^* \dashv N) : LieAlg^\Delta \stackrel{\overset{N^*}{\leftarrow}}{\underset{N}{\to}} dgLieAlg$

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.

Theorem

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

$(L N^* \dashv \tilde N) : Ho(LieAlg^\Delta)_1 \stackrel{\overset{L N^*}{\leftarrow}}{\underset{\tilde N}{\to}} Ho(dgLieAlg)_1$

of 1-connected objects on both sides.

This is in the proof of (Quillen, theorem. 4.4).

References

A standard reference in the context of rational homotopy theory is

• Dan Quillen, Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (JSTOR)

For the unbounded case there is general discussion in

The relation to $L_\infty$-algebras is discussed for instance in