## Lie algebras and homotopy theory

These are notes from a lecture by Jacob Lurie.

*Definition.*
A Lie algebra is an abelian group
with a bilinear operation
satisfying skew-symmetry
and the Jacobi identity

(Question: these equations look like a complete 2-cycle and a complete 3-cycle?)

*Recall.*
If is a group, the group commutator is given by
If is a Lie group, we can differentiate the commutator
at the identity:
to get a bilinear operator which endows the tangent space the
structure of a Lie algebra.

The fundamental group of a space is
where is the *based loop space* of :
This space has an operation, concatenation:
which is not associative, but induces an associative operation on
There is also an inverse operation on

This is not a group, but we can still form a "commutator":
This induces maps
for
Note that .
This gives the *Whitehead bracket*:

The higher homotopy groups of a space are These groups are abelian for

*Fact.* The Whitehead bracket endows
with the structure of a graded Lie algebra.

*Definition.*
A *graded Lie algebra*
is a graded abelian group
with a homogenous bilinear operation:
satisfying skew-symmetry:
and the Jacobi identity:

A *homotopy operation*
of variables and arity
is a map
natural in .

*Example.* The Whitehead bracket gives a 2-variable homotopy operation
of arity for all

*Example.* Let be a pointed map.
Then defines a homotopy operation of 1 variable:
By general nonsense these are all of the homotopy operations of 1-variable.
Note: homotopy groups of spheres are very complicated!

*Theorem (Hilton-Milnor).*
Paraphrase: all homotopy operations can be built from 1-variable
operations using the Whitehead bracket.
All relations among homotopy operations can be explained using the
fact that the Whitehead bracket satisfies the graded Lie algebra identities.

Try to simplify the story by studying rational homotopy groups of spheres.

*Definition.*
A *differential graded Lie algebra*, or d.g. Lie algebra,
is a pair with
a graded Lie algebra, and a linear map
such that and satisfying the compatibility condition
This compatibility condition means that the homology
inherits the structure of a graded Lie algebra.

*Construction. (Quillen)*
There is a construction
that takes a simply connected ponted space to a differential graded Lie algebra
over
such that
and
respecting the Lie bracket (on the right hand side the bracket is the Whitehead bracket).

*Definition.*
A map between simply connected pointed spaces
is a rational homotopy equivalence if it induces an isomorphism
Equivalently, induces an isomorphism
Equivalently, induces an isomorphism
Equivalently, induces a quasi-isomorphism
Recall that a map between chain complexes is a quasi-isomorphism
if it induces an isomorphism on the homology.

Quillen's theorem is a converse of this.

A *connected* d.g. Lie algebra is a d.g. Lie algebra
with for

*Theorem. (Quillen)*
The construction
induces an equivalence:

This theorem has a "torsion-sensitive" generalization, see Heuts reference below.

Now we want to consider Lie algebras in a variety of contexts.

Let be a category such that
(1) has a colimites
(2) is additive
(3) is symmetric monoidal with preserving colimits.
(Call these *abelian tensor categories*.)

*Definition.*
A *Lie algebra* object internal to is an object
and a morphism
satisfying skew-symmetry (an equation in )
and the Jacobi identity (an equation in )

The category of Lie algebra objects in we denote

*Examples.*

In words: a d.g. Lie algebra is a Lie algebra object internal to the (a?) category of chain complexes.

*Claim.*
There exists a universal abelian tensor category with
a Lie algebra object
By universality we mean that
where functors
preserve tensor product and colimits.
This isomorphism sends to
By abstract nonsense, if exists, then it is unique up to isomorphism.
(Question: is the walking Lie algebra?)

What does look like? It has an object and tensor power objects , colimit objects and cokernel objects $ \mathrm{coker}( \bigoplus_{\alpha} L_u^{\otimes n_{\alpha}} \to \bigoplus_{\beta} L_u^{\otimes n_{\beta}}). $

So now we have:

Problem: we want to classify d.g. Lie algebras up to *quasi-isomorphism*.

*Recall.*
If is an abelian category, we can consider , the
category of chain complexes with values in .
The *derived category* of is written is with
quasi-isomorphisms formally inverted:

*Example.*
There is a construction
that sends quasi-isomorphisms to -functor isomorphisms.

*Warning.*
There can be inequivalent abelian categories
such that

*Definition.*
The category is a functor category:
This is a tensor category with tensor product is given by

*Claim.*
There is an equivalence of derived categories

## References

From a lecture "Lie algebras and homotopy theory", by Jacob Lurie. https://www.youtube.com/watch?v=LeaiPHAh0X0

"Lie algebra models for unstable homotopy theory" Gijs Heuts https://arxiv.org/abs/1907.13055

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