Mathematics / research
Semisimple Lie algebras and representations
Develop the algebraic structure of continuous symmetry using roots, weights, highest-weight modules, and classification.
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Module 01 research
Semisimple Lie algebras and representations
Develop the algebraic structure of continuous symmetry using roots, weights, highest-weight modules, and classification.
- 01
Ideals, solvability, and Levi decomposition
Separate the solvable radical from a semisimple complement without confusing a vector-space splitting with a direct product.
31 min - 02
The Killing form and Cartan's criteria
Use the adjoint trace pairing to distinguish nondegeneracy, solvability, and positivity.
32 min - 03
sl(2) triples and weight strings
Fix raising and lowering conventions, derive the finite string coefficients, and recognize where finite dimensionality enters.
32 min - 04
Cartan subalgebras and root decompositions
Simultaneously diagonalize a maximal toral adjoint action and distinguish roots, root vectors, and coroots.
33 min - 05
Root systems and Dynkin diagrams
Recover integral reflection data from root geometry while fixing the Cartan-matrix orientation.
33 min - 06
Weyl groups and reflection geometry
Compute finite reflection actions, chamber representatives, and stabilizers while distinguishing ordinary and shifted actions.
32 min - 07
Highest-weight modules and Verma modules
Construct an induced highest-weight module before taking the simple quotient, and keep finite-dimensional truncation explicit.
34 min - 08
Finite-dimensional representations and dominant weights
Classify finite simple modules using integral coroot labels and retain the distinction between a Lie algebra and a quotient group.
33 min - 09
Weyl character and dimension formulas
Recover finite weight multiplicities from an alternating quotient and take the identity limit correctly.
34 min - 10
Enveloping algebras and Casimir elements
Use PBW and invariant forms to construct central operators while declaring the normalization of their eigenvalues.
35 min