Theoretical / proof-based
Lie groups & Lie algebras · Graduate · Math
Learning objectives from the Lie groups & Lie algebras syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Lie Groups & Lie Algebras — scope drawn from open Lie theory notes and typical US graduate Lie groups syllabi.
Lie groups and Lie algebras
- Lie groups: definitions, examples (GL(n), SO(n), SU(n)) — Lie groups: definitions, examples (GL(n), SO(n), SU(n))
- Lie algebras: bracket, homomorphisms, and ideals — Lie algebras: bracket, homomorphisms, and ideals
- Exponential map — Exponential map
- Baker–Campbell–Hausdorff formula (introduction) — Baker–Campbell–Hausdorff formula (introduction)
- Adjoint representation and Killing form — Adjoint representation and Killing form
- Semisimple and simple Lie algebras — Semisimple and simple Lie algebras
Structure theory
- Cartan subalgebras and root systems — Cartan subalgebras and root systems
- Weyl group and root reflections — Weyl group and root reflections
- Dynkin diagrams — Dynkin diagrams
- Classification of simple Lie algebras — Classification of simple Lie algebras
- Highest weight representations (overview) — Highest weight representations (overview)
- Compact Lie groups and maximal tori — Compact Lie groups and maximal tori
Advanced topics
- Universal enveloping algebras (introduction) — Universal enveloping algebras (introduction)
- Haar measure and integration on compact groups — Haar measure and integration on compact groups
- Symmetric spaces (preview) — Symmetric spaces (preview)
- Lie cohomology (optional) — Lie cohomology (optional)
- Applications to differential geometry and physics — Applications to differential geometry and physics
Learning objectives
Click an objective for study materials.
Lie groups and Lie algebras
- Lie groups: definitions, examples (GL(n), SO(n), SU(n)) — Lie groups: definitions, examples (GL(n), SO(n), SU(n))
- Lie algebras: bracket, homomorphisms, and ideals — Lie algebras: bracket, homomorphisms, and ideals
- Exponential map — Exponential map
- Baker–Campbell–Hausdorff formula (introduction) — Baker–Campbell–Hausdorff formula (introduction)
- Adjoint representation and Killing form — Adjoint representation and Killing form
- Semisimple and simple Lie algebras — Semisimple and simple Lie algebras
Structure theory
- Cartan subalgebras and root systems — Cartan subalgebras and root systems
- Weyl group and root reflections — Weyl group and root reflections
- Dynkin diagrams — Dynkin diagrams
- Classification of simple Lie algebras — Classification of simple Lie algebras
- Highest weight representations (overview) — Highest weight representations (overview)
- Compact Lie groups and maximal tori — Compact Lie groups and maximal tori
Advanced topics
- Universal enveloping algebras (introduction) — Universal enveloping algebras (introduction)
- Haar measure and integration on compact groups — Haar measure and integration on compact groups
- Symmetric spaces (preview) — Symmetric spaces (preview)
- Lie cohomology (optional) — Lie cohomology (optional)
- Applications to differential geometry and physics — Applications to differential geometry and physics
Definitions and structure
- Lie groups: definitions, examples (GL(n), SO(n), SU(n)) — Lie groups: definitions, examples (GL(n), SO(n), SU(n))
- Lie algebras: bracket, homomorphisms, and ideals — Lie algebras: bracket, homomorphisms, and ideals
- Exponential map — Exponential map
- Baker–Campbell–Hausdorff formula (introduction) — Baker–Campbell–Hausdorff formula (introduction)
- Adjoint representation and Killing form — Adjoint representation and Killing form
Proofs and reasoning
- Semisimple and simple Lie algebras — Semisimple and simple Lie algebras
- Cartan subalgebras and root systems — Cartan subalgebras and root systems
- Weyl group and root reflections — Weyl group and root reflections
- Dynkin diagrams — Dynkin diagrams
- Classification of simple Lie algebras — Classification of simple Lie algebras
Abstraction and generalization
- Highest weight representations (overview) — Highest weight representations (overview)
- Compact Lie groups and maximal tori — Compact Lie groups and maximal tori
- Universal enveloping algebras (introduction) — Universal enveloping algebras (introduction)
- Haar measure and integration on compact groups — Haar measure and integration on compact groups
- Symmetric spaces (preview) — Symmetric spaces (preview)
Modeling and computation
- Apply lie groups: definitions, examples (gl(n), so(n), su(n)) in engi... — Apply lie groups: definitions, examples (gl(n), so(n), su(n)) in engineering contexts
- Apply lie algebras: bracket, homomorphisms, and ideals in engineering... — Apply lie algebras: bracket, homomorphisms, and ideals in engineering contexts
- Apply exponential map in engineering contexts — Apply exponential map in engineering contexts
- Apply baker–campbell–hausdorff formula (introduction) in engineering... — Apply baker–campbell–hausdorff formula (introduction) in engineering contexts
- Apply adjoint representation and killing form in engineering contexts — Apply adjoint representation and killing form in engineering contexts
Data and technology
- Use software to explore lie groups & lie algebras problems numerically — Use software to explore lie groups & lie algebras problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Semisimple and simple Lie algebras — Semisimple and simple Lie algebras
- Cartan subalgebras and root systems — Cartan subalgebras and root systems
- Weyl group and root reflections — Weyl group and root reflections
- Dynkin diagrams — Dynkin diagrams
- Classification of simple Lie algebras — Classification of simple Lie algebras
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Lie groups and Lie algebras
Lie groups: definitions, examples (GL(n), SO(n), SU(n))
Coming soon - Structure theory
Cartan subalgebras and root systems
Coming soon - Advanced topics
Universal enveloping algebras (introduction)
Coming soon
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