A vector space 𝔤 with a bilinear bracket [·,·] that is anti-symmetric and satisfies the Jacobi identity. Tangent structure of Lie groups; classification of semisimple complex Lie algebras by Dynkin diagrams.
Lie algebra
Related concepts
- Vector space
- Jacobi identity
- Cartan-Killing classification
- Ring (R, +, ·)
- Moment map
- Poisson manifold
- SU(2) Lie algebra: T_a = σ_a/2, [T_a, T_b] = i·ε_abc·T_c
- Casimir invariant C_2 = Σ T^a T_a in centre of U(g)
- Jordan algebra (commutative nonassociative with Jordan identity)
- Alternative algebra (weakening of associativity: left/right alt)
- Flexible algebra and Malcev algebras (Jacobi generalisation)
- Cartan-Killing classification (simple Lie algebras)
- Baker-Campbell-Hausdorff formula
- Kac-Moody algebra (affine extension)
- Verma module / highest-weight
- Freudenthal-Tits magic square
- Malcev algebra (Lie-admissible)
- Octonions (Cayley-Graves 1843-1845)
- Lie bracket (Jacobi 1862)
- Killing-Cartan classification