# Ideals and Lie subalgebras

Ideals and Lie subalgebras are represented by the types LieAlgebraIdeal and LieSubalgebra respectively. They are used similarly in most cases.

## Functions

### Ideals

dimMethod
dim(I::LieAlgebraIdeal) -> Int

Return the dimension of the ideal I.

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basisMethod
basis(I::LieAlgebraIdeal) -> Vector{LieAlgebraElem}

Return the basis of the ideal I.

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basisMethod
basis(I::LieAlgebraIdeal, i::Int) -> LieAlgebraElem

Return the i-th basis element of the ideal I.

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inMethod
in(x::LieAlgebraElem, I::LieAlgebraIdeal) -> Bool

Return true if x is in the ideal I, false otherwise.

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bracketMethod
bracket(I1::LieAlgebraIdeal, I2::LieAlgebraIdeal) -> LieAlgebraIdeal

Return $[I_1,I_2]$.

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normalizerMethod
normalizer(L::LieAlgebra, I::LieAlgebraIdeal) -> LieSubalgebra

Return the normalizer of I in L, i.e. $\{x \in L \mid [x, I] \subseteq I\} = L$. As I is an ideal in L, this is just L.

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centralizerMethod
centralizer(L::LieAlgebra, I::LieAlgebraIdeal) -> LieSubalgebra

Return the centralizer of I in L, i.e. $\{x \in L \mid [x, I] = 0\}$.

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### Lie subalgebras

dimMethod

dim(S::LieSubalgebra) -> Int

Return the dimension of the Lie subalgebra S.

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basisMethod
basis(S::LieSubalgebra{C}) -> Vector{LieAlgebraElem{C}}

Return a basis of the Lie subalgebra S.

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basisMethod
basis(S::LieSubalgebra{C}, i::Int) -> LieAlgebraElem{C}

Return the i-th basis element of the Lie subalgebra S.

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inMethod
in(x::LieAlgebraElem, S::LieSubalgebra) -> Bool

Return true if x is in the Lie subalgebra S, false otherwise.

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bracketMethod
bracket(S1::LieSubalgebra, S2::LieSubalgebra) -> LieAlgebraIdeal

Return $[S_1, S_2]$.

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normalizerMethod
normalizer(L::LieAlgebra, S::LieSubalgebra) -> LieSubalgebra

Return the normalizer of S in L, i.e. $\{x \in L \mid [x, S] \subseteq S\}$.

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centralizerMethod
centralizer(L::LieAlgebra, S::LieSubalgebra) -> LieSubalgebra

Return the centralizer of S in L, i.e. $\{x \in L \mid [x, S] = 0\}$.

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is_self_normalizingMethod
is_self_normalizing(S::LieSubalgebra) -> Bool

Return true if S is self-normalizing, i.e. if its normalizer is S.

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## Constructors

### Ideals

idealMethod
ideal(L::LieAlgebra, gens::Vector{LieAlgebraElem}; is_basis::Bool=false) -> LieAlgebraIdeal

Return the smallest ideal of L containing gens. If is_basis is true, then gens is assumed to be a basis of the ideal.

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idealMethod
ideal(L::LieAlgebra, gen::LieAlgebraElem) -> LieAlgebraIdeal

Return the smallest ideal of L containing gen.

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idealMethod
ideal(L::LieAlgebra) -> LieAlgebraIdeal

Return L as an ideal of itself.

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### Lie subalgebras

subMethod
sub(L::LieAlgebra, gens::Vector{LieAlgebraElem}; is_basis::Bool=false) -> LieSubalgebra

Return the smallest Lie subalgebra of L containing gens. If is_basis is true, then gens is assumed to be a basis of the subalgebra.

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subMethod
sub(L::LieAlgebra, gen::LieAlgebraElem) -> LieSubalgebra

Return the smallest Lie subalgebra of L containing gen.

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subMethod
sub(L::LieAlgebra) -> LieSubalgebra

Return L as a Lie subalgebra of itself.

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## Conversions

lie_algebraMethod
lie_algebra(S::LieSubalgebra) -> LieAlgebra

Return S as a Lie algebra LS, together with an embedding LS -> L, where L is the Lie algebra where S lives in.

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lie_algebraMethod
lie_algebra(I::LieAlgebraIdeal) -> LieAlgebra

Return I as a Lie algebra LI, together with an embedding LI -> L, where L is the Lie algebra where I lives in.

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subMethod
sub(L::LieAlgebra, I::LieAlgebraIdeal) -> LieSubalgebra

Return I as a subalgebra of L.

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