Mathematics / advanced
Lie groups and continuous symmetry
Study continuous groups through tangent algebras, exponential maps, homogeneous spaces, integration, and compact-group structure.
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Inside this module
Module 01 advanced
Lie groups and continuous symmetry
Study continuous groups through tangent algebras, exponential maps, homogeneous spaces, integration, and compact-group structure.
- 01
Matrix Lie groups and smooth subgroups
Combine matrix symmetries with smooth structure and distinguish a closed subgroup from an arbitrary parametrized subgroup.
30 min - 02
Tangent Lie algebras and left-invariant fields
Identify tangent vectors with invariant vector fields and fix the bracket sign through concrete matrix calculations.
32 min - 03
Exponential maps and one-parameter subgroups
Integrate a constant identity generator into a global one-parameter motion while keeping the exponential's coordinate chart local.
32 min - 04
The Baker–Campbell–Hausdorff formula
Recover a local product from sums and nested brackets, with separate formal, convergent, and nilpotent interpretations.
34 min - 05
Lie subgroups and Lie correspondence
Integrate subalgebras and homomorphisms while tracking immersed topology, covering groups, and period obstructions.
34 min - 06
Adjoint and coadjoint actions
Differentiate conjugation and its dual action with a fixed inverse and sign convention.
34 min - 07
Homogeneous spaces and stabilizers
Build smooth quotients from closed stabilizers and distinguish orbit manifolds from quotient groups and badly behaved images.
34 min - 08
Haar measure and integration
Construct translation-invariant volume and compute the left/right discrepancy through an explicitly defined modular function.
34 min - 09
Compact Lie groups and maximal tori
Reduce compact connected symmetry to conjugate torus subgroups while distinguishing rank, dimension, and singular centralizers.
34 min - 10
Roots, Weyl groups, and compact group structure
Extract finite reflection data from a compact connected group while retaining the lattice and central quotient that distinguish global groups.
36 min