Summer REU 2026 - Categorical Lie theory

Project summary

Lie theory, which is the study of Lie groups and their associated Lie algebras, is a field in an interesting position at the moment. There are tons of concrete examples, mostly matrix Lie groups, many arising from applications in physics. The category of Lie groups is well understood, and there are explicit functors between the category of Lie groups and the category of Lie algebras. However, the latter category still seems somewhat mysterious, even to the experts. So our goal is get to the point where we can ask (and possibly answer!) categorical questions about the category of Lie algebras.

Our project will break down into roughly three phases:

• Phase I: Get familiar with classical Lie theory, by learning about matrix Lie groups and their associated Lie algebras
• Phase II: Learn the more general definitions of Lie groups and algebras, and general facts about the categories of each
• Phase III: Learn and investigate the structures of the categories of Lie groups and Lie algebras in relation to other categories; e.g., what is special about the category of Lie groups? how is it intrinsically defined as a category in its own right?

Meeting notes

Task list

Tasks will be added after each meeting.

Classical Lie theory

• Learn about classical Lie groups by reading Chapter 1 ("Matrix Lie Groups") in Hall and then check out the following exercises in Section 1.6:
  • Exercise 1
  • Exercise 2
  • Exercise 3
  • Exercise 7
  • Exercise 11
  • Exercise 16
  • Exercise 17
• Learn about the classical matrix exponential function by reading Chapter 2 ("The Matrix Exponential") in Hall and then check out the following exercises in Section 2.6:
  • Exercise 2
  • Exercise 4
  • Exercise 8
  • Exercise 9
• Learn about classical Lie algebras by reading Chapter 3 ("Lie Algebras") in Hall and then check out the following exercises in Section 3.9:
  • Exercise 1
  • Exercise 2
  • Exercise 3
  • [?] Exercise 6
  • Exercise 8
  • Exercise 13
  • Exercise 16
  • Exercise 17
• Take our first steps beyond matrix Lie groups by reading the following in Lie groups beyond an introduction by Knapp:
  • Chapter 1: Sections 3, 10, 14, and 17
  • Chapter 3 (skim)

Monoidal and Cartesian categories

• Learn about monoidal categories by reading Sections 1 of Chapters VII and XI in Mac Lane's Categories for the Working Mathematician. Then try the following exercises at the end Section 1 in Chapter VII:
  • Exercise 4
  • Exercise 5
• Find two monoidal structures on the category of sets
• Suppose a category $C$ has a all finite products and a terminal object. Construct a monoid structure on $C$
• Look up the definition of a Cartesian monoidal category
• For each of the following familiar categories, determine if it is Cartesian monoidal:
  • $\mathbf{\text{Group}}$
  • $\mathbf{\text{Ab}}$
  • $\mathbf{\text{Ring}}$
  • $\mathbf{\text{CRing}}$
  • $R\mathbf{\text{-Mod}}$, using the tensor product
  • $R\mathbf{\text{-Mod}}$, using the Cartesian product
  • $\mathbf{\text{Field}}$
  • $\mathbf{\text{Diff}}$, the category of smooth manifolds

A categorical definition of a Lie group

• Look up the definition of a group object in a Cartesian monoidal category
• Show groups (as classically defined) are precisely group objects in the category of sets
• What is a group object in the category of abelian groups? (An abelian group)
• What is a group object in the category of groups? (Again, an abelian group)
• What is a group object in the category of sets if we use the disjoint union for our monoidal product? (The diagrams required for the unit, inversion, etc. don't make sense if the monoidal product isn't the usual (Cartesian) product)
• Show that Lie groups are precisely groups objects in the category of smooth manifolds

Properties of the category of Lie groups

• Investigate whether the category of Lie groups has (or does not have) each of the following categorical properties:
  • Has a terminal object? (Yes, the trivial Lie group.)
  • Has an initial object? (Yes, the trivial Lie group.)
  • Has a null object? (Yes, the trivial Lie group.)
  • Has finite products? (Yes.)
  • Has all products? (No.)
  • Has pullbacks? (Yes.)
  • Has finite coproducts? (No.)
  • Has all coproducts? (No.)
  • Has pushouts? (No.)
  • Has equalizers? (Yes.)
  • Has kernels? (Yes.)
  • Has quotients? (Yes.)
  • Has an analogue of the First Isomorphism Theorem? (Yes.)
  • Is an additive category? (No.)
  • Is an abelian category? (No.)

Properties of the category of Lie algebras

• Investigate whether the category of Lie algebras has (or does not have) each of the following categorical properties:
  • Has a terminal object? (Yes, the trivial Lie algebra.)
  • Has an initial object? (Yes, the trivial Lie algebra.)
  • Has a null object? (Yes, the trivial Lie algebra.)
  • Has finite products? (Yes, with component-wise brackets.)
  • Has all products? (Yep!)
  • Has pullbacks? (Yes.)
  • Has equalizers? (Yes.)
  • Has kernels? (Yes.)
  • Has finite coproducts? (Yes, but we need some stuff below to make sense of this.)
  • Has all coproducts? (Yes, but we need some stuff below to make sense of this.)
  • Has pushouts? (Yes, but we need some stuff below to make sense of this.)
  • Has quotients? (Yes. Needed the concept of a Lie ideal. Check kernels are ideals.)
  • Has an analogue of the First Isomorphism Theorem? (Yep!)
  • Is an additive category? (No.)
  • Is an abelian category? (No, but close. It's semi-abelian.)
  • Is an enriched category?

Important functors: Part I

Free Lie algebras

Let $U:{\mathbf{L}\mathbf{i}\mathbf{e}\mathbf{A}\mathbf{l}\mathbf{g}}_k\to {\mathbf{V}\mathbf{e}\mathbf{c}}_k$ be the forgetful functor that sends each Lie algebra over $k$ to its underlying $k$-vector space, i.e., forgets the bracket operation.

• It turns out that the functor $U$ has a left adjoint, $L:{\mathbf{V}\mathbf{e}\mathbf{c}}_k\to {\mathbf{L}\mathbf{i}\mathbf{e}\mathbf{A}\mathbf{l}\mathbf{g}}_k$. What does this mean?
• Given a $k$-vector space $V$, describe the Lie algebra $L(V)$. (This is called the free Lie algebra on the vector space $V$.)

Universal enveloping algebras

Let ${\mathbf{A}\mathbf{s}\mathbf{s}\mathbf{o}\mathbf{c}\mathbf{A}\mathbf{l}\mathbf{g}}_k$ be the category of (unital) associative $k$-algebras, and let $G:{\mathbf{A}\mathbf{s}\mathbf{s}\mathbf{o}\mathbf{c}\mathbf{A}\mathbf{l}\mathbf{g}}_k\to {\mathbf{L}\mathbf{i}\mathbf{e}\mathbf{A}\mathbf{l}\mathbf{g}}_k$ be the functor that sends each associative $k$-algebra $A$ to the Lie algebra whose vector space is the underlying vector space of $A$ (i.e., forget the multiplication in general, but remember the addition and multiplication by elements of the base field, $k$) and with bracket defined by $[x,y]:=xy-yx$.

• Verify that $G(A)$ is indeed a Lie algebra over $k$.
• What is the arrow map of the functor $G$?
• It turns out the functor $G$ has a left adjoint, $U:{\mathbf{L}\mathbf{i}\mathbf{e}\mathbf{A}\mathbf{l}\mathbf{g}}_k\to {\mathbf{A}\mathbf{s}\mathbf{s}\mathbf{o}\mathbf{c}\mathbf{A}\mathbf{l}\mathbf{g}}_k$. What does this mean?
• Give a Lie algebra $\mathfrak{g}$, describe the Lie algebra $U(\mathfrak{g})$. (This is called the universal enveloping algebra of $\mathfrak{g}$.)
• Look up some properties of the universal enveloping algebra, e.g., how is $\mathfrak{g}$ related to $U(\mathfrak{g})$?

Aside: Applications of Lie theory

A popular question you'll be asked often is "What are some applications of Lie groups?" So let's make sure to have some answers ready for that! Find some specific applications of Lie theory. Some possible places to look are:

• Particle physics
• Relativity
• Quantum computing
• Machine learning
• Robotics (including robot vision)

Important functors: Part II

The Lie functor

• Describe a functor $\mathbf{L}\mathbf{i}\mathbf{e}\mathbf{G}\mathbf{r}\mathbf{p}\to \mathbf{L}\mathbf{i}\mathbf{e}\mathbf{A}\mathbf{l}\mathbf{g}$. Make sure to describe both the object function and the arrow function.
• Does this functor have any nice properties; e.g., does it preserve products?
• What does Lie's Third Theorem say about this functor?
• What is the adjoint representation of a group, $G$?
• If $G\to \mathrm{Gl}(\mathfrak{g})$ is the adjoint representation of $G$, how do we get a Lie algebra morphism $\mathfrak{g}\to \mathfrak{gl}\mathfrak{(}\mathfrak{g}\mathfrak{)}$?
• Read a bit about Tannaka-Krein duality, at least for (Lie) groups.

Other important results and questions

• What are Lie’s three theorems?
• Can a given group G always be equipped with a Lie group structure? Can it be equipped with different Lie group structures?
• Which topological spaces can be equipped with a Lie group structure?
• Can non-isomorphic Lie groups have isomorphic Lie algebras?
• What is Lie integration?
• Is there a classification of all Lie groups?

Wrap-up and future research

• Briefly investigate some of the ideas below, any of which could be an avenue for future research:
  • Lie algebra objects (analogous to group objects internal to a Cartesian closed category)
  • Lie groupoids and Lie algebroids (Maybe even just starting with groupoids and the idea of “oidification”)
  • Higher “dimensional” versions of Lie’s three theorems, e.g., analogues for Lie groupoids, Lie 2-groups, etc.
  • The Lie operad (which also involves learning what an operad is)
• Write up what we've learned and done this summer
• Start planning the poster

The team

Matt Richards

mricha49@calpoly.edu

Zoey Pieper

zpieper@calpoly.edu

References

• Lie groups, Lie algebras, and representations, by Brian C. Hall
• Structure and geometry of Lie groups, by Hilgert and Neeb