REU Meeting - 2026-07-23

This following is a brief summary of our research meeting on 2026-07-23.

Meeting summary

We went over the constructions and properties of both the free Lie algebra functor (from ${\mathbf{V}\mathbf{e}\mathbf{c}}_k\to {\mathbf{L}\mathbf{i}\mathbf{e}\mathbf{A}\mathbf{l}\mathbf{g}}_k$) and universal enveloping algebra functor (from ${\mathbf{L}\mathbf{i}\mathbf{e}\mathbf{A}\mathbf{l}\mathbf{g}}_k\to {\mathbf{A}\mathbf{s}\mathbf{s}\mathbf{o}\mathbf{c}\mathbf{A}\mathbf{l}\mathbf{g}}_k$). There are/were way too many details to check, but the main constructions involved using the tensor product (and basically the tensor algebra), and taking quotients by appropriate ideals.

Tasks for next meeting

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)

References

nLab