$$\newcommand{\deg}{\mathrm{deg}} \newcommand{\Lie}{\mathrm{Lie}} \newcommand{\Ab}{\mathrm{Ab}} \newcommand{\Univ}{\mathcal{E}} \newcommand{\Chain}{\mathrm{Chain}} \newcommand{\Fun}{\mathrm{Fun}}$$

Lie algebras and homotopy theory

These are notes from a lecture by Jacob Lurie.

Definition. A Lie algebra is an abelian group $L$ with a bilinear operation $$[\cdot, \cdot] : L\times L\to L$$ satisfying skew-symmetry $$[x, y] + [y, x] = 0$$ and the Jacobi identity $$[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.$$

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

Recall. If $G$ is a group, the group commutator is given by $$\begin{align} G\times G &\to G \\ (x, y) &\mapsto xyx^{-1}y^{-1}. \end{align}$$ If $G$ is a Lie group, we can differentiate the commutator at the identity: $$[\cdot, \cdot] : T_{G,e}\times T_{G,e}\to T_{G,e}$$ to get a bilinear operator which endows the tangent space $T_{G,e}$ the structure of a Lie algebra.

The fundamental group of a space $X$ is $$\begin{align} \pi_1(X) &= \pi_1(X, x) \\ &= \pi_0(\Omega X). \end{align}$$ where $\Omega X$ is the based loop space of $X$: $$\Omega X := \{ p:[0,1]\to X | p(0) = p(1) = x\}.$$ This space has an operation, concatenation: $$\begin{align} \star : \Omega X \times \Omega X &\to \Omega X \\ (p, q) &\mapsto (p\star q)(t) \end{align}$$ $$(p\star q)(t) = \begin{cases} q(2t) & 0\le t\le \frac{1}{2}, \\ p(2t-1) & \frac{1}{2}\le t\le 1. \end{cases}$$ which is not associative, but induces an associative operation on $\pi_0(\Omega X).$ There is also an inverse operation on $\Omega X$ $$p \mapsto {\bar p}, \ \ {\bar p}(t) = p(1-t).$$

This $\Omega X$ is not a group, but we can still form a "commutator": $$\begin{align} \Omega X \times \Omega X &\to \Omega X \\ (p, q) &\mapsto (p\star q)\star({\bar p}\star{\bar q}) \end{align}$$ This induces maps $$\begin{align} \pi_a(\Omega X)\times \pi_b(\Omega X) \to \pi_{a+b}(\Omega X) \end{align}$$ for $a, b\ge 0.$ Note that $\pi_a(\Omega X) \simeq \pi_{a+1}(X)$. This gives the Whitehead bracket: $$\begin{align} [\cdot,\cdot] : \pi_{a+1}(X)\times \pi_{b+1}(X) \to \pi_{a+b+1}(X) \end{align}$$

The higher homotopy groups of a space $X$ are $$\pi_a(X) = \{ \text{homotopy classes of maps}\ ([0,1]^a, \partial [0,1]^a) \to (X, x). \}$$ These groups $\pi_a(X)$ are abelian for $a>1.$

Fact. The Whitehead bracket endows $$L_{\star} = \bigoplus_{n>0} L_n = \bigoplus_{n\ge 0} \pi_{n+1}(X)$$ with the structure of a graded Lie algebra.

Definition. A graded Lie algebra is a graded abelian group $L_{\star}$ with a homogenous bilinear operation: $$[\cdot, \cdot] : L_a\times L_b\to L_{a+b}$$ satisfying skew-symmetry: $$[x, y] + (-1)^{\deg(x)\deg(y)} [y, x] = 0$$ and the Jacobi identity: $$(-1)^{\deg(x)\deg(z)} [x, [y, z]] + (-1)^{\deg(x)\deg(y)} [y, [z, x]] + (-1)^{\deg(y)\deg(z)} [z, [x, y]] = 0.$$

A homotopy operation of $n$ variables and arity $(a_1,...,a_n)\to b$ is a map $$\pi_{a_1}(X)\times \pi_{a_2}(X) \times ... \times \pi_{a_n}(X)\to \pi_b(X)$$ natural in $X$.

Example. The Whitehead bracket gives a 2-variable homotopy operation of arity $(a+1, b+1)\to(a+b+1)$ for all $a,b\ge 0.$

Example. Let $f:S^n\to S^m$ be a pointed map. Then $f$ defines a homotopy operation of 1 variable: $$\begin{align} \pi_m(X) &\to \pi_n(X) \\ \eta &\mapsto \eta\circ f \end{align}$$ 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. $$\begin{align} \pi_{\star}(S^n)\otimes \Rational = \begin{cases} \Rational \text{id} & \text{if}\ \star = n, \\ \Rational [\text{id},\text{id}] & \text{if}\ \star = 2n-1, n\ \text{even}, \\ 0 & \text{otherwise.} \end{cases} \end{align}$$

Definition. A differential graded Lie algebra, or d.g. Lie algebra, is a pair $(L_{\star}, d)$ with $L_{\star}$ a graded Lie algebra, and $d$ a linear map $d : L_{\star} \to L_{(\star-1)}$ such that $d^2=0$ and satisfying the compatibility condition $$d[x, y] = [dx, y] + (-1)^{\deg(x)} [x, dy].$$ This compatibility condition means that the homology $H_{\star}(L_{\star}, d)$ inherits the structure of a graded Lie algebra.

Construction. (Quillen) There is a construction that takes a simply connected ponted space $(X,x)$ to a differential graded Lie algebra $L_{\star}(X)$ over $\Rational.$ such that $$H_{\star}(L_{\star}(X)) \simeq \pi_{\star+1}(X)\otimes \Rational$$ and respecting the Lie bracket (on the right hand side the bracket is the Whitehead bracket).

Definition. A map $f:(X,x)\to (Y,y)$ between simply connected pointed spaces is a rational homotopy equivalence if it induces an isomorphism $$\pi_{\star}(X)\otimes\Rational \xrightarrow{\simeq} \pi_{\star}(Y)\otimes\Rational.$$ Equivalently, $f$ induces an isomorphism $$H_{\star}(X;\Rational) \to H_{\star}(Y;\Rational)$$ Equivalently, $f$ induces an isomorphism $$H^{\star}(X;\Rational) \to H^{\star}(Y;\Rational)$$ Equivalently, $f$ induces a quasi-isomorphism $$L_{\star}(X) \to L_{\star}(Y).$$ 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 $L_{\star}$ with $L_{\star}=0$ for $\star\le 0.$

Theorem. (Quillen) The construction $(X,x) \mapsto L_{\star}(X)$ induces an equivalence: $$\begin{align} \{\text{simply connected pointed spaces} \} &/ \text{rational homotopy equivalence} \\ &\simeq \\ \{\text{connected d.g. Lie algebras}\} &/ \text{quasi-isomorphism}. \end{align}$$

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

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

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

Definition. A Lie algebra object internal to $A$ is an object $L\in A$ and a morphism $\mathrm{br}:L\otimes L\to L$ satisfying skew-symmetry (an equation in $\Hom_A(L\otimes L, L)$) and the Jacobi identity (an equation in $\Hom_A(L\otimes L\otimes L, L)$)

The category of Lie algebra objects in $A$ we denote $\Lie(A).$

Examples. $$\begin{array}{ll} A = \{\mathrm{abelian\ groups}\}, &\Lie(A) = \{\mathrm{Lie\ algebras}\} \\ A = \{\mathrm{graded\ abelian\ groups}\}, &\Lie(A) = \{\mathrm{graded\ Lie\ algebras}\} \\ A = \{\mathrm{chain\ complexes}\}, &\Lie(A) = \{\mathrm{d.g.\ Lie\ algebras}\} \end{array}$$

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 $\Univ$ with a Lie algebra object $$L_u \in \Lie(\Univ).$$ By universality we mean that $$\Lie(A) \simeq \mathrm{Fun}_{\otimes}(\Univ, A)$$ where functors $F\in \mathrm{Fun}_{\otimes}(\cdot, \cdot)$ preserve tensor product and colimits. This isomorphism sends $F$ to $F(L_u).$ By abstract nonsense, if $\Univ$ exists, then it is unique up to isomorphism. (Question: is $\Univ$ the walking Lie algebra?)

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

So now we have: $$\begin{align} \mathrm{Fun}_{\otimes}(\Univ, Ab) &= \{\mathrm{Lie\ algebras}\} \\ \mathrm{Fun}_{\otimes}(\Univ, \Chain(Ab)) &= \{\mathrm{differential\ graded\ Lie\ algebras}\} \end{align}$$

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

Recall. If $A$ is an abelian category, we can consider $\Chain(A)$, the category of chain complexes with values in $A$. The derived category of $A$ is written $D(A)$ is $\Chain(A)$ with quasi-isomorphisms formally inverted: $$D(A) := \Chain(A) [\mathrm{quasi-isos}^{-1}].$$

Example. There is a construction $$\{\mathrm{d.g.\ Lie\ algebras} \} \to \Fun_{\otimes}(D(\Univ), D(\Ab))$$ that sends quasi-isomorphisms to $\otimes$-functor isomorphisms.

Warning. There can be inequivalent abelian categories $A, B$ such that $D(A)\simeq D(B).$

Definition. The category ${\mathcal E}'$ is a functor category: $${\mathcal E}' := \mathrm{Fun}( \{ \mathrm{finite\ sets\ and\ surjections} \}, \mathrm{Ab} ).$$ This is a tensor category with tensor product is given by $$(F\otimes G)(I) = \bigoplus_{J\subseteq I} F(J)\otimes G(I-J).$$

Claim. There is an equivalence of derived categories $$D({\mathcal E}) \simeq D({\mathcal E'}).$$

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