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

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