---
title: Lie Algebras
description: Lie algebras study infinitesimal transformations and non-associative algebraic structures. Learners will understand the relationship between Lie groups and Lie algebras, classify semi-simple Lie algebras, and apply root systems to analyze continuous symmetries in physics and geometry.
category: mathematics
subcategory: representation-theory
difficulty: beginner, intermediate, advanced
url: /subject/lie-algebras
---

# Lie Algebras

Lie algebras study infinitesimal transformations and non-associative algebraic structures. Learners will understand the relationship between Lie groups and Lie algebras, classify semi-simple Lie algebras, and apply root systems to analyze continuous symmetries in physics and geometry.

## Available Resources

2 Books • 1 Courses • 3 Websites

## Websites

### 1. nLab: Lie Algebra

The nLab's reference entry on Lie algebras, written from a category-theoretic viewpoint. It covers definitions, the link to Lie groups, Chevalley–Eilenberg algebras, L-infinity generalizations and representations, helping readers connect classical Lie theory to higher algebra and homotopy theory.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/Lie+algebra

**Tags:** lie-algebras, lie-groups, representation-theory, category-theory

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

### 3. nLab

Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://ncatlab.org

**Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki

## Courses

### 1. Lie Algebras

Learn Lie algebras from Professor Richard Borcherds! This course covers fundamental concepts and examples.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLgMDNELGJ1Ca6RP_wDKSH7SwM5DPq1No_

**Tags:** lie-algebras, lie-groups, representation-theory, root-systems

## Books

### 1. Introduction to Lie Algebras and Representation Theory

**Author:** James E. Humphreys

Humphreys' standard graduate text on semisimple Lie algebras over the complex numbers. It develops Cartan subalgebras, root systems, Weyl groups, the classification via Dynkin diagrams and highest-weight representations, preparing readers to work with representation theory and algebraic groups.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/3540900535?tag=edmonddante07-20

**Tags:** lie-algebras, representation-theory, root-systems, semisimple-lie-algebras

### 2. Lie Algebras in Particle Physics

**Author:** Howard Georgi

A physicist's introduction to Lie groups and Lie algebras, built around SU(2), SU(3) and the unified gauge groups. Readers learn roots, weights, Young tableaux and tensor methods well enough to classify particle multiplets and analyse symmetry in particle physics.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0805331530?tag=edmonddante07-20

**Tags:** lie-algebras, group-theory, particle-physics, representation-theory

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