∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
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 -algebra in which only the unary and the binary brackets may be nontrivial.
A dg-Lie algebra is
a -graded vector space ;
a linear map ;
a bilinear map , the bracket;
such that (all conditions are expressed for homogeneously graded elements ):
is a differential that makes into a chain complex, i.e.
it is of degree -1, ;
it squares to zero, ;
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 is a graded derivation)
A pre-graded Lie algebra (pre-gla) is a pre-gvs, , together with a bilinear map of degree zero
such that
and
for every triple of homogeneous elements in .
(The first property is call antisymmetry, the second the Jacobi identity.)
A morphism of pre-glas is a linear map of degree zero, that is compatible with the brackets,
To any augmented pre-ga , one can associate a pre-gla, denoted , with underlying gvs and with bracket, the commutator, for each pair of homogeneous elements. is abelian (i.e. with trivial bracket) if and only if is graded commutative.
If is a pre-cga and is a pre-gla, the tensor product has a pre-gla structure with bracket
for homogeneous.
Let be a pre-gla. A derivation of gla-s, of degree , is a linear mapping such that
for any pair of homogeneous elements of . We denote by , the vector space of degree derivations of the gla, .
A differential of a pre-gla is a Lie algebra derivation of degree -1 such that . The pair is then called a differential pre-graded Lie algebra (pre-dgla); its homology , 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 .
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 is an augmented pre-dga, is a pre-dgla.
If is a pre-cdga and , a pre-dgla, , together with the tensor product differential, is a pre-dgla.
Let be a pre-dgvs, then the pre-dgvs, , constructed earlier is a pre-dga for the multiplication law given by composition of mappings. Its associated pre-dgla has
and
In particular, if is a cdga (resp. is dgla), then , (resp. ) is a sub-pre-dgl of .
A dgla is a pre-dgla with a lower grading; explicitly:
A differential graded Lie algebra, , is a graded vector space , together with a bilinear map of degree 0
and a differential satisfying
and
for every triple of homogeneous elements in .
Let be the corresponding category.
A dgla is -reduced (resp. homologically -reduced) if (resp. ) for all . Denote by (resp. ), the corresponding categories.
If is a pre-dgla, a gla-filtration of (resp. a dgla-filtration of ) is a family of subgraded vector spaces , , such that , , (resp. and ).
Let be a pre-gla. Its bracket length filtration is obtained from the descending central series:
It is a gla-filtration.
is called the space of indecomposables of .
If is a pre-dgla, is stable by . Letting be the induced differential on , then defines a functor
Free Lie algebra,
Let be a pre-gvs, , the tensor algebra on with augmentation ideal (recall and the augmentation sends to 0).
Let be with the pre-gla structure given by the commutators. We denote by , the Lie subalgebra of generated by .
Tim: A more explicit description may help here, cf. Quillen, Rational Homotopy theory (p.281) or MacLane, Homology.
If is a pre-gla, any morphism of pre-gvs has a unique extension to a pre-gla morphism . If is a homogeneous basis for , may be denoted .
On the free Lie algebra , the bracket length filtration comes from a gradation , where is the subspace generated by the brackets of elements of of length . The inclusion identifies with .
If is a dgla, free as a gla, with fixed, is the sum of derivations defined by : . The isomorphism between and identifies with . (resp. ) is called the linear part (resp, the quadratic part) of .
Let and be two dglas. Their product in is defined by:
the underlying vector space is the direct sum ;
and are two sub differential graded Lie algebras of ;
if and , then .
Their coproduct or sum is often called their free product.
More generally if and are given by generators and relations
The differential on is the unique Lie algebra derivation extending and .
Every dg-Lie algebra is in an evident way an L-infinity algebra. Dg-Lie algebras are precisely those -algebras for which all -ary brackets for are trivial. These may be thought of as the strict -algebras: those for which the Jacobi identity holds on the nose and all its possible higher coherences are trivial.
Let be a field of characteristic 0 and write for the category of L-infinity algebras over .
Then every object of is quasi-isomorphic to a dg-Lie algebra.
Moreover, one can find a functorial replacement: there is a functor
such that for each
is a dg-Lie algebra;
there is a quasi-isomorphism
This appears for instance as (KrizMay, cor. 1.6).
For more see at
model structure on dg-Lie algebras the section Relation to L-infinity algebras.
model structure for L-infinity algebras, the section on dg-Lie algebras.
Via the above relation to -algebras, dg-Lie algebras are also connected by adjunction to dg-coalgebras
Here
is the Chevalley-Eilenberg algebra functor. It sends a dg-Lie algebra to
where on the right the extension of and to graded derivations is understood.
For a dg-coalgebra, then
where
is the kernel of the counit, regarded as a chain complex;
is the free Lie algebra functor (as graded Lie algebras);
on the right we are extending as a Lie algebra derivation
Moreover
is the Maurer-Cartan elements in the Hom-dgLie algebra from to .
For dg-Lie algebras concentrated in degrees 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 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).
dg-Lie algebra differential crossed module 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 -algebras is discussed for instance in
See also the references at model structure on dg-Lie algebras.
A discussion of how formal neighbourhoods of points in infinity-stacks are governed by dg-Lie algebras:
Last revised on July 23, 2021 at 14:37:47. See the history of this page for a list of all contributions to it.