2.3 Ideals and quotient Lie algebras

Ideals and quotient Lie algebras are crucial for understanding Lie algebra structure. Ideals are special subspaces closed under the Lie bracket with any element, allowing us to create smaller Lie algebras from larger ones.

Quotient Lie algebras, formed by dividing a Lie algebra by an ideal, help simplify complex structures. This process reveals key properties and relationships, aiding in the classification and analysis of Lie algebras.

Ideals in Lie Algebras

Definition and Properties

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• An ideal of a Lie algebra is a vector subspace closed under the Lie bracket with any element of the Lie algebra
  • If $I$ is an ideal of a Lie algebra $L$, then for any $x \in L$ and $y \in I$, $[x,y] \in I$
• The kernel of a Lie algebra homomorphism is always an ideal
• The center of a Lie algebra, consisting of elements that commute with all other elements under the Lie bracket, is always an ideal (abelian Lie algebras)
• The derived algebra (or commutator subalgebra) of a Lie algebra, generated by all Lie brackets, is an ideal (solvable Lie algebras)

Examples and Applications

• In the Lie algebra of $n \times n$ matrices $\mathfrak{gl}(n, \mathbb{R})$, the subspace of trace-zero matrices $\mathfrak{sl}(n, \mathbb{R})$ is an ideal
• In any Lie algebra $L$, the trivial subspaces $\{0\}$ and $L$ itself are always ideals
• Ideals play a crucial role in the structure theory of Lie algebras, allowing for the construction of quotient Lie algebras and the classification of Lie algebras up to isomorphism

Identifying Ideals

Checking the Ideal Property

• To check if a subspace $I$ is an ideal, one must verify that for any $x \in L$ and $y \in I$, the Lie bracket $[x,y]$ is in $I$
• For finite-dimensional Lie algebras, it is sufficient to check the closure property for a basis of the Lie algebra and a spanning set of the subspace
• Ideals are always Lie subalgebras, but not all Lie subalgebras are ideals (non-normal subalgebras)

Intersection, Union, and Preimage of Ideals

• The intersection of ideals is an ideal, but the union of ideals may not be an ideal
  • If $I$ and $J$ are ideals of $L$, then $I \cap J$ is an ideal of $L$, but $I \cup J$ may not be an ideal
• The preimage of an ideal under a Lie algebra homomorphism is an ideal
  • If $\phi: L \to L'$ is a Lie algebra homomorphism and $I'$ is an ideal of $L'$, then $\phi^{-1}(I')$ is an ideal of $L$

Quotient Lie Algebras

Construction and Properties

• Given a Lie algebra $L$ and an ideal $I$, the quotient space $L/I$ is a Lie algebra under the induced bracket operation $[x+I, y+I] = [x,y] + I$
• The natural projection map $\pi: L \to L/I$ defined by $\pi(x) = x+I$ is a surjective Lie algebra homomorphism with kernel $I$
• The dimension of the quotient Lie algebra $L/I$ is equal to the dimension of $L$ minus the dimension of $I$ (rank-nullity theorem)

Applications and Examples

• Quotient Lie algebras are useful for studying the structure of Lie algebras and classifying them up to isomorphism
• The quotient of a Lie algebra by its center is always centerless (has trivial center)
  • If $Z(L)$ is the center of $L$, then $L/Z(L)$ has trivial center
• The quotient of a Lie algebra by its derived algebra is always abelian (has trivial derived algebra)
  • If $[L,L]$ is the derived algebra of $L$, then $L/[L,L]$ is abelian

Isomorphism Theorems for Lie Algebras

First Isomorphism Theorem

• If $\phi: L \to L'$ is a Lie algebra homomorphism, then $L/\ker(\phi) \cong \operatorname{im}(\phi)$
  • The induced map $\overline{\phi}: L/\ker(\phi) \to \operatorname{im}(\phi)$ defined by $\overline{\phi}(x+\ker(\phi)) = \phi(x)$ is an isomorphism of Lie algebras

Second Isomorphism Theorem

• If $I$ is an ideal and $K$ is a Lie subalgebra of $L$, then $K/(I \cap K) \cong (K+I)/I$
  • The map $\psi: K \to (K+I)/I$ defined by $\psi(x) = x+I$ is a surjective Lie algebra homomorphism with kernel $I \cap K$

Third Isomorphism Theorem

• If $I$ and $J$ are ideals of $L$ with $I \subseteq J$, then $(L/I)/(J/I) \cong L/J$
  • The map $\theta: L/I \to L/J$ defined by $\theta(x+I) = x+J$ is a surjective Lie algebra homomorphism with kernel $J/I$
• The isomorphism theorems help in understanding the relationship between Lie subalgebras, ideals, and quotient Lie algebras (lattice of subalgebras)