# Lie Groups & Lie Algebras Representations Intro

Canonical: https://manyspace.aporia.systems/topics/lie-groups-lie-algebras-representations-intro-mth-repth-361
Updated: 2026-10-13

Mathematics · Representation Theory · MTH-REPTH-361

- Difficulty: 5 / 5
- Importance: 4 / 5
- Study time: 9.5–16.5 h

weights, highest weight theorem, SU(2) and spin

## Why it matters

Studies groups through their actions on vector spaces; the framework for symmetries in particle physics and quantum mechanics.

## Where it is used

Particle classification (standard model), quantum chemistry, crystallography, signal symmetries, machine learning equivariance.

## How you know you have learned it

Explain/derive key result from memory + solve 2 non-routine checks.

## What you need first

- [Groups & Subgroups](https://manyspace.aporia.systems/topics/groups-subgroups-mth-absalg-256) (required)
- [Gaussian Elimination](https://manyspace.aporia.systems/topics/gaussian-elimination-mth-linalg-155) (helpful)

## Resources

- Representation Theory (Fulton & Harris) — Fulton & Harris