Killing form

Depends on

• Trace form of a representation
• Derivations form a Lie algebra and inner derivations an ideal

Used by

• Maximal compact subgroups exist and are conjugate in a connected finite center semisimple Lie group Corollary
• Semisimple Lie algebras are centerless and perfect Corollary
• Same complexification with different killing form signatures Counterexample
• Cartan involution of a real semisimple Lie algebra Definition
• Cayley transform of a theta-stable Cartan subalgebra Definition
• Compact real form of a complex semisimple Lie algebra Definition
• Dual Coxeter number for the normalized invariant form Definition
• Killing-dual vector of a root Definition
• A nonreduced bc root system from a real form Example
• A reductive algebra with degenerate Killing form Example
• Cartan involution and k plus p for sl n r Example
• Cartan subalgebra and roots of sl₂ Example
• Classical simple Lie algebras and their Killing forms Example
• Compact and split real forms of sl two c Example
• Complex simple lie algebra viewed as a real simple algebra Example
• Hyperbolic space as so zero n one mod so n Example
• Killing form of sl₂ Example
• Standard and dual representations of slₙ Example
• The Killing form identifies roots with coroot directions Example
• The root sl₂ triple inside slₙ Example
• A plain dynkin diagram classifies real forms False statement
• All real forms of a complex semisimple lie algebra are isomorphic False statement
• Dynkin diagrams classify real semisimple Lie algebras False statement
• The Killing form is nondegenerate on every reductive Lie algebra False statement
• Chevalley basis and real structure constants Lemma
• The Killing length of a root is nonzero Lemma
• Bracket relations and Killing signs in a Cartan decomposition Proposition
• Cartan decomposition gives the invariant metric and curvature of G mod K Proposition
• Complexification preserves semisimplicity Proposition
• Orthogonality of root spaces and nondegeneracy on the Cartan subalgebra Proposition
• Real Cartan subalgebras need not be conjugate Proposition
• The bracket of opposite root spaces is the root line Proposition
• The roots form a reduced crystallographic Euclidean root system Proposition
• Cartan subalgebras are exactly maximal toral subalgebras Theorem
• Cartan's solvability criterion Theorem
• Compact roots form a reduced crystallographic root system Theorem
• Conjugacy of Cartan involutions Theorem
• Conjugacy of Cartan subalgebras Theorem
• Conjugacy of compact real forms Theorem
• Every real Cartan subalgebra is conjugate to a theta-stable one Theorem

…and 9 more results.

Dependency tree · two levels

7 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

• Milne, Lie Algebras, §4, Killing form (standard reference, not scraped)