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differential graded Lie algebra

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\infty-Lie theory

∞-Lie theory (higher geometry)

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Smooth structure

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∞-Lie groupoids

∞-Lie algebroids

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\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Rational homotopy theory

Contents

Idea

A differential graded Lie algebra, or dg-Lie algebra for short, is equivalently

Definition

Direct explicit definition

Definition

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

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

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

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

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

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

    1. it is of degree -1, :𝔤 i𝔤 i1\partial \colon \mathfrak{g}_{i} \to \mathfrak{g}_{i-1};

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

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

    [x 1,x 2]=[x 1,x 2]+(1) |x 1|[x 1,x 2], \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) |x 1||x 2|[x 2,x 1], [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,][x,-] is a graded derivation)

    [x 1,[x 2,x 3]]=[[x 1,x 2],x 3]+(1) |x 1||x 2|[x 2,[x 1,x 3]]. [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, LL, together with a bilinear map of degree zero

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

such that

[x,y]=(1) |x||y|+1[y,x][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(-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)(x,y,z) of homogeneous elements in LL.

(The first property is call antisymmetry, the second the Jacobi identity.)

A morphism f:LLf: 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)].f[x,y] = [f(x),f(y)].
Example

To any augmented pre-ga AA, one can associate a pre-gla, denoted A¯ L\bar{A}_L, with underlying gvs A¯\bar{A} and with bracket, the commutator, [x,y]=xy(1) |x||y|yx[x,y] = xy -(-1)^{|x||y|}yx for each pair (x,y)(x,y) of homogeneous elements. A¯ L\bar{A}_L is abelian (i.e. with trivial bracket) if and only if AA is graded commutative.

Example

If AA is a pre-cga and LL is a pre-gla, the tensor product ALA\otimes L has a pre-gla structure with bracket

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

for a,a,l,la,a', l, l' homogeneous.

Definition

Let LL be a pre-gla. A derivation of gla-s, of degree pp\in \mathbb{Z}, is a linear mapping θHom p(L,L)\theta \in Hom_p(L,L) such that

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

for any pair x,yx,y of homogeneous elements of LL. We denote by Der p(L)Der_p(L), the vector space of degree pp derivations of the gla, LL.

Definition

A differential \partial of a pre-gla is a Lie algebra derivation of degree -1 such that =0\partial\circ \partial = 0. The pair (L,)(L,\partial) is then called a differential pre-graded Lie algebra (pre-dgla); its homology H(L,)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 preDGLApre 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,)(A,\partial) is an augmented pre-dga, (A¯ L,)(\bar{A}_L,\partial) is a pre-dgla.

Example

If (A,)(A,\partial) is a pre-cdga and (L,)(L,\partial'), a pre-dgla, ALA \otimes L, together with the tensor product differential, is a pre-dgla.

Example

Let (V,)(V,\partial) be a pre-dgvs, then the pre-dgvs, (Hom(V,V),D)(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]=fg(1) |f||g|gf[f,g] = f\circ g - (-1)^{|f||g|}g\circ f

and

Df=[,f].Df = [\partial,f].

In particular, if (V,)=(A,d)(V,\partial) = (A,d) is a cdga (resp. (V,)=(L,)(V,\partial) = (L,\partial) is dgla), then Der(A,d)=( pDer p(A),D)Der(A,d) = (\bigoplus_p Der_p(A),D), (resp. Der(L,)=(Der p(L),D)Der(L,\partial) = (Der_p(L),D)) is a sub-pre-dgl of (Hom(V,V),D)(Hom(V,V),D).

Definition

A dgla is a pre-dgla with a lower grading; explicitly:

A differential graded Lie algebra, (L,)(L,\partial), is a graded vector space L= p0L pL = \bigoplus_{p\geq 0}L_p, together with a bilinear map of degree 0

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

and a differential \partial satisfying

L pL p1,[x,y]=(1) |x||y|+1[y,x],\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(-1)^{|x||z|}[x,[y,z]] +(-1)^{|y||x|}[y,[z,x]] +(-1)^{|z||y|}[z,[x,y]] = 0

and

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

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

Let DGLADGLA be the corresponding category.

Definition

A dgla is nn-reduced (resp. homologically nn-reduced) if L p=0L_p = 0 (resp. H p(L,)=0H_p(L,\partial) = 0) for all p<np\lt n. Denote by DGLA nDGLA_n (resp. DGLA hnDGLA_{hn}), the corresponding categories.

Definition

If (L,)(L,\partial) is a pre-dgla, a gla-filtration of LL (resp. a dgla-filtration of (L,)( L,\partial) ) is a family of subgraded vector spaces F pLF_p L, pp\in \mathbb{Z}, such that F pLF p+1LF_p L\subseteq F_{p+1}L, [F pL,F nL]F p+nL[F_p L,F_n L]\subseteq F_{p+n} L, (resp. and F pLF pL\partial F_p L\subseteq F_p L).

Definition

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

F 1L=L;F pL=[L,F p1L]ifp2.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 2LQ(L) = L/F^2L is called the space of indecomposables of LL.

If (L,)(L,\partial) is a pre-dgla, F pLF^p L is stable by \partial. Letting Q()Q(\partial) be the induced differential on Q(L)Q(L), QQ then defines a functor

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

Free Lie algebra, 𝕃(V)\mathbb{L}(V)

Let VV be a pre-gvs, T(V)T(V), the tensor algebra on VV with augmentation ideal T(V)¯\overline{T(V)} (recall T(V)= n0V nT(V) = \bigoplus_{n\geq 0} V^{\otimes n} and the augmentation sends V(=V 1V (= V^{\otimes 1} to 0).

Let T(V)¯ L\overline{T(V)}_L be T(V)¯\overline{T(V)} with the pre-gla structure given by the commutators. We denote by 𝕃(V)\mathbb{L}(V), the Lie subalgebra of T(V)¯ L\overline{T(V)}_L generated by VV.

Tim: A more explicit description may help here, cf. Quillen, Rational Homotopy theory (p.281) or MacLane, Homology.

If LL is a pre-gla, any morphism of pre-gvs f:VLf: V\to L has a unique extension to a pre-gla morphism f^:𝕃(V)L\hat{f} :\mathbb{L}(V)\to L. If (e α) αI(e_\alpha)_{\alpha\in I} is a homogeneous basis for VV, 𝕃(V)\mathbb{L}(V) may be denoted 𝕃((e α) αI)\mathbb{L}((e_\alpha)_{\alpha\in I}).

On the free Lie algebra 𝕃(V)\mathbb{L}(V), the bracket length filtration comes from a gradation 𝕃(V)= j𝕃 j(V)\mathbb{L}(V) = \bigoplus_j\mathbb{L}^j(V) , where 𝕃 j(V)\mathbb{L}^j(V) is the subspace generated by the brackets of elements of VV of length jj. The inclusion 𝕃(V)T(V)\mathbb{L}(V)\hookrightarrow T(V) identifies 𝕃 j(V)\mathbb{L}^j(V) with 𝕃(V)T j(V)\mathbb{L}(V)\cap T^j(V).

If 𝕃(V),)\mathbb{L}(V), \partial) is a dgla, free as a gla, with VV fixed, \partial is the sum of derivations k\partial_k defined by : k𝕃 k(V)\partial_k \subset \mathbb{L}^k(V). The isomorphism between VV and Q𝕃(V)Q\mathbb{L}(V) identifies 1V\partial_1V with Q()Q(\partial). 1\partial_1 (resp. 2\partial_2) is called the linear part (resp, the quadratic part) of \partial.

Definition

Let (L,)(L,\partial) and (L,)(L',\partial') be two dglas. Their product (L,)×(L,)(L,\partial)\times(L',\partial') in DGLADGLA is defined by:

  • the underlying vector space is the direct sum LLL\oplus L';

  • (L,)(L,\partial) and (L,)(L',\partial') are two sub differential graded Lie algebras of (L,)×(L,)(L,\partial)\times(L',\partial');

  • if xLx\in L and xLx' \in L', then [x,x]=0[x,x'] = 0.

Their coproduct or sum (L,)(L,)(L,\partial)\star(L',\partial') is often called their free product.

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

More generally if LL and LL' are given by generators and relations

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

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

Properties

Model category structure

Relation to L L_\infty-algebras

Every dg-Lie algebra is in an evident way an L-infinity algebra. Dg-Lie algebras are precisely those L L_\infty-algebras for which all nn-ary brackets for n>2n \gt 2 are trivial. These may be thought of as the strict L L_\infty-algebras: those for which the Jacobi identity holds on the nose and all its possible higher coherences are trivial.

Theorem

Let kk be a field of characteristic 0 and write L Alg kL_\infty Alg_k for the category of L-infinity algebras over kk.

Then every object of L Alg kL_\infty Alg_k is quasi-isomorphic to a dg-Lie algebra.

Moreover, one can find a functorial replacement: there is a functor

W:L Alg kL Alg k W : L_\infty Alg_k \to L_\infty Alg_k

such that for each 𝔤L Alg k\mathfrak{g} \in L_\infty Alg_k

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

  2. there is a quasi-isomorphism

    𝔤W(𝔤). \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 L_\infty-algebras, dg-Lie algebras are also connected by adjunction to dg-coalgebras

dgLieAlg kCEdgCoAlg k dgLieAlg_k \underoverset {\underset{CE}{\longrightarrow}} {\overset{\mathcal{L}}{\longleftarrow}} {\bot} dgCoAlg_k

Here

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

    CE(𝔤,,[,])( 𝔤[1],D=+[,]), 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)(X,D) a dg-coalgebra, then

    (X,D)(F(X¯[1]),D+(Δ1idid1)) \mathcal{L}(X,D) \coloneqq \left( F(\overline{X}[-1]),\; \partial \coloneqq D + (\Delta - 1 \otimes id - id \otimes 1) \right)

    where

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

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

    3. on the right we are extending (Δ1idid1):X¯X¯X¯(\Delta - 1 \otimes id - id \otimes 1) \colon \overline{X} \to \overline{X} \otimes \overline{X} as a Lie algebra derivation

Moreover

Hom((X),𝔤)Hom(X,CE(𝔤))MC(Hom(X¯,𝔤)) 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 X¯\overline{X} to 𝔤\mathfrak{g}.

For dg-Lie algebras concentrated in degrees n1 \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

There is an adjunction

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

between simplicial Lie algebras and dg-Lie algebras, where NN 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

(LN *N˜):Ho(LieAlg Δ) 1N˜LN *Ho(dgLieAlg) 1 (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 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:

Revised on December 7, 2017 16:01:26 by Urs Schreiber (178.24.246.134)