Lie algebras over a field

Depends on

• Vector space over a field
• Linear map between vector spaces over the same field

Used by

• A maximal abelian subalgebra that is not a Cartan subalgebra Counterexample
• Abstract Jordan decomposition Definition
• Classical complex matrix Lie algebras Definition
• Classical Lie-action BRST cochain complex Definition
• Complexification of a real Lie algebra Definition
• Derivations of Lie algebras Definition
• Derived series and solvable Lie algebras Definition
• Direct products and direct sums of Lie algebras Definition
• Finite-dimensional Heisenberg Lie algebra Definition
• Homomorphisms of possibly infinite-dimensional Lie algebras Definition
• Lie bialgebras, degreewise duality, and root-graded Manin triples Definition
• Lie subalgebras, ideals, and center Definition
• Lower central series and nilpotent Lie algebras Definition
• Regular element and rank Definition
• Representations of Lie algebras Definition
• Symplectic comoment maps and Hamiltonian observable actions Definition
• The special linear Lie algebra sl₂ Definition
• Witt algebra of Laurent polynomial vector fields Definition
• Lie algebras of the additive, infinitesimal and general linear groups Example
• Quadratic sl2 comoment action on the symplectic plane Example
• Restricted roots of sl n r Example
• The standard two-dimensional sl₂ module extends to the enveloping algebra Example
• A Cartan subalgebra of an arbitrary Lie algebra means a maximal abelian subalgebra False statement
• An augmented coideal ideal of an enveloping algebra is generated by its primitive part Lemma
• The Heisenberg bracket satisfies the Lie algebra axioms Lemma
• The nonequivariance defect of an infinitesimal moment map is a constant Lie-algebra two-cocycle Lemma
• The opposite Borels of a symmetrizable Kac–Moody algebra are root-degreewise dual Lie bialgebras Lemma
• Complexification has a canonical conjugation with fixed algebra g zero Proposition
• Derivations form a Lie algebra and inner derivations an ideal Proposition
• Real Cartan subalgebras need not be conjugate Proposition
• The normalizer acts on Lie algebra cohomology Proposition
• A root-graded Manin triple gives dual Lie bialgebras Theorem
• Coadjoint orbits are symplectic manifolds Theorem
• The Chevalley–Eilenberg differential squares to zero Theorem
• The Lie bracket from infinitesimals and the adjoint action Theorem
• Universal property of the free Lie algebra Theorem

Dependency tree · two levels

9 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, §§3.2 and 11.1 (standard reference, not scraped)
• Kirillov, An Introduction to Lie Groups and Lie Algebras, §§3.4 and 4.1 (standard reference, not scraped)