[[Lie theory MOC]]
# Lie algebras MOC

While the [[Lie algebra]] was first invoked to study the [[Lie group]],
they are an interesting object in their own right.
See also the more general [[Algebra theory MOC]].

## Objects

- [[Lie algebra]]

### Additional structure

- [[Graded Lie algebra]]
- [[Invariant bilinear form on a Lie algebra]]
- [[Quadratic Lie algebra]]

### Special constructions of Lie algebras

- [[Associated Lie algebra of a positive definite even lattice]]
- [[Affine Lie algebra]]
- [[Twisted affine Lie algebra]]

### Types of Lie algebra

- [[Simple Lie algebra]]
- [[Abelian Lie algebra]]
- [[Solvable Lie algebra]]
- [[Nilpotent Lie algebra]]
- [[Heisenberg algebra]]
- [[Triangular Lie algebra]]

### Special Lie algebras

- [[Witt algebra]]
- [[Virasoro algebra]]

### Subalgebras

- [[Cartan subalgebra]]

## Morphisms

- [[Lie algebra homomorphism]], [[Lie algebra isomorphism]], [[Lie algebra automorphism]]
- [[Kernel of a Lie algebra homomorphism]]

## Internally

- [[Lie algebra ideal]]
- [[Centralizer in a Lie algebra]]
- [[Normalizer in a Lie algebra]]
- [[Centre of a Lie algebra]]
- [[Radical of a Lie algebra]]


## Externally

- [[Quotient Lie algebra]]
- [[Lie algebra extension]]
- [[Direct product of Lie algebras]]
- [[Semidirect product of Lie algebras]]



## Representations

- [[Lie algebra representation]], [[Module over a Lie algebra]]
- [[Adjoint Lie algebra representation]]
- [[Sum of commuting Lie algebra representations]]
- [[Tensor product of Lie algebra representations]]


## Other things

- [[Killing form]]

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