Representations of Lie algebras

Depends on

• Lie algebras over a field
• Homomorphisms of possibly infinite-dimensional Lie algebras
• The space $\mathcal L(V,W)$ of linear maps with pointwise addition and scalar multiplication

Used by

• A Hecke action alone does not imply a double centralizer Counterexample
• A two-dimensional irreducible real representation of an abelian Lie algebra Counterexample
• Positive-characteristic failure of Lie's theorem Counterexample
• The full weight lattice need not integrate through a central quotient Counterexample
• Arakawa-Suzuki Casimir operators on tensor functors Definition
• Chevalley–Eilenberg cochains Definition
• Classical Lie-action BRST cochain complex Definition
• Highest-weight vector and module for Virasoro Definition
• Nilpotent transformations and nil representations Definition
• Restricted affine-current module Definition
• Subrepresentations, quotient representations, and intertwiners Definition
• Symplectic comoment maps and Hamiltonian observable actions Definition
• The special linear Lie algebra sl₂ Definition
• The type-A finite W-algebra and the Whittaker category Definition
• The type-A nilpotent pyramid, good grading and Whittaker character Definition
• The Whittaker functor from the relevant parabolic category-O summand Definition
• Trace form of a representation Definition
• Weight and weight space Definition
• Adjoint and trivial representations Example
• All irreducible finite-dimensional sl2 modules Example
• Quadratic sl2 comoment action on the symplectic plane Example
• Standard representations of classical matrix Lie algebras Example
• The standard two-dimensional sl₂ module extends to the enveloping algebra Example
• Vust's double centralizer in a small stable range Example
• Every irreducible real representation of a solvable Lie algebra is one-dimensional False statement
• Lie's theorem is field- and characteristic-free False statement
• Chevalley–Eilenberg cohomology computes Ext of the trivial module Proposition
• Comoment maps act on observables by Poisson derivations Proposition
• Degree-zero Lie algebra cohomology is the invariant subspace Proposition
• Direct-sum, dual, Hom, and tensor representations Proposition
• Finite-dimensional modules decompose into weight spaces Proposition
• Lie representations as actions before enveloping Proposition
• Representation kernels are ideals Proposition
• Root vectors shift weights Proposition
• The normalizer acts on Lie algebra cohomology Proposition
• Ado's theorem with nilpotent nilradical action Theorem
• Finite-dimensional representations of sl₂ Theorem
• Lie representations are U(g)-modules Theorem
• Lie's theorem Theorem
• The Casimir and place-permutation operators satisfy the degenerate affine Hecke relations Theorem

…and 2 more results.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Etingof, MIT 18.745 notes, §11.1, printed pp. 61–62 (standard reference, not scraped)
• Kirillov, An Introduction to Lie Groups and Lie Algebras, §4.1, printed pp. 49–50 (standard reference, not scraped)