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Cartan subalgebra
Definition
Let be a finite-dimensional Lie algebra over a field. A Cartan subalgebra of is a Lie subalgebra which is nilpotent (Lower central series and nilpotent Lie algebras) and satisfies for the normalizer of Normalizer of a Lie subalgebra.
This definition is stated for arbitrary finite-dimensional Lie algebras and carries no semisimplicity hypothesis. In particular the zero subalgebra of the zero Lie algebra is a Cartan subalgebra, since the zero algebra is nilpotent and its normalizer is again zero; in a nonzero Lie algebra the zero subalgebra is not a Cartan subalgebra, because its normalizer is the whole algebra.
Depends on
Used by
- A maximal abelian subalgebra that is not a Cartan subalgebra Counterexample
- Two nonconjugate real cartan subalgebras Counterexample
- Root and root space Definition
- Split real form Definition
- Theta-stable Cartan subalgebras and their compact and split parts Definition
- Weight and weight space Definition
- Cartan subalgebra and roots of sl₂ Example
- Cartan subalgebras of a direct sum Example
- Compact and split cartan subalgebras of sl two r Example
- Compact and split real forms of sl two c Example
- Diagonal Cartan subalgebra and roots of slₙ Example
- Root systems B₂ and C₂ from matrix Lie algebras Example
- A Cartan subalgebra of an arbitrary Lie algebra means a maximal abelian subalgebra False statement
- All cartan subalgebras of a real semisimple lie algebra are conjugate False statement
- Real Cartan subalgebras need not be conjugate Proposition
- Root systems of the classical complex Lie algebras Proposition
- Split Cartan subalgebras of classical matrix Lie algebras Proposition
- Cartan subalgebras are exactly maximal toral subalgebras Theorem
- Cayley transforms connect theta-stable Cartans in the classification Theorem
- Centralizer of a regular semisimple element is Cartan Theorem
- Classification of real forms by Vogan diagrams Theorem
- Compact roots form a reduced crystallographic root system Theorem
- Complexification dichotomy for a real simple lie algebra Theorem
- Conjugacy of Cartan subalgebras Theorem
- Every real Cartan subalgebra is conjugate to a theta-stable one Theorem
- Existence and uniqueness up to isomorphism of the split real form Theorem
- Existence of Cartan subalgebras Theorem
- Root-space decomposition Theorem
- Serre presentation theorem Theorem
- Vogan and Satake diagrams give equivalent real form classifications Theorem
Dependency tree · two levels
4 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
- Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I, Lectures 19–24 (standard reference, not scraped)