Semisimple Lie Algebras — Mathematics Knowledge System

Overview

A semisimple Lie algebra is a Lie algebra that has no nonzero abelian ideals — equivalently, one that decomposes as a direct sum of simple Lie algebras. The structure theory of semisimple Lie algebras, developed by Killing and Cartan, reveals an extraordinarily rigid structure: every semisimple Lie algebra over ℂ is completely determined by its root system, which in turn is encoded by its Dynkin diagram. This structure theory underpins both the classification of the algebras themselves and the classification of their representations.

Intuition

A semisimple Lie algebra is one with no 'flat' directions — no sub-algebra that commutes with everything else. It splits into irreducible blocks (simple algebras), each of which has a distinguished maximal abelian sub-algebra (Cartan subalgebra) whose simultaneous eigenvectors (root spaces) organise the entire structure. The roots encode how all elements of the algebra interact.

Formal Definition

Definition

A Lie algebra 𝔤 over a field k is semisimple if its radical (largest solvable ideal) is zero. Equivalently, the Killing form is non-degenerate.

\text{rad}(\mathfrak{g}) = 0 \iff \mathfrak{g} \text{ is semisimple}

Semisimplicity via radical

\kappa(X,Y) = \operatorname{tr}(\operatorname{ad} X \circ \operatorname{ad} Y)

Killing form

\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha,\quad \mathfrak{g}_\alpha = \{X \in \mathfrak{g} : [H, X] = \alpha(H)X \; \forall H \in \mathfrak{h}\}

Root space decomposition

[\mathfrak{g}_\alpha, \mathfrak{g}_\beta] \subseteq \mathfrak{g}_{\alpha+\beta}

Bracket of root spaces

Notation

Notation	Meaning
$\mathfrak{g}$	Semisimple Lie algebra
$\mathfrak{h}$	Cartan subalgebra (maximal abelian, ad-diagonalisable)
$\Phi$	Root system of 𝔤 relative to 𝔥
$\mathfrak{g}_\alpha$	Root space for root α
$\kappa$	Killing form: κ(X,Y) = tr(ad X ∘ ad Y)

Properties

Cartan's criterion

\mathfrak{g} \text{ is semisimple} \iff \kappa \text{ is non-degenerate}

Complete reducibility (Weyl)

\text{Every finite-dimensional representation of a semisimple Lie algebra over } \mathbb{C} \text{ is completely reducible}

Dimension of root spaces

\dim \mathfrak{g}_\alpha = 1 \quad \forall\, \alpha \in \Phi

Simple decomposition

\mathfrak{g} \cong \mathfrak{g}_1 \oplus \cdots \oplus \mathfrak{g}_k \text{ where each } \mathfrak{g}_i \text{ is simple}

Worked Examples

1
sl₂(ℂ) has basis e = [[0,1],[0,0]], f = [[0,0],[1,0]], h = [[1,0],[0,-1]].
$\mathfrak{sl}_2 = \operatorname{span}\{e, f, h\}$
2
Bracket relations: [h,e] = 2e, [h,f] = -2f, [e,f] = h.
$[h,e]=2e,\quad [h,f]=-2f,\quad [e,f]=h$
3
Any nonzero ideal I must contain a linear combination aₑe + aff + ahh. Applying ad h and ad e forces I = 𝔤, so no proper nonzero ideal exists.
$\text{No proper nonzero ideal} \implies \mathfrak{sl}_2 \text{ is simple}$

✓ Answer

sl₂(ℂ) is simple (and hence semisimple) of rank 1.

Practice Problems

Hardproof writing

Prove that a Lie algebra 𝔤 is semisimple if and only if it is a direct sum of simple Lie algebras.

Common Mistakes

Common Mistake

Confusing semisimple and reductive Lie algebras

A reductive Lie algebra decomposes as centre ⊕ semisimple part (e.g. gl_n = scalars ⊕ sl_n). Semisimple Lie algebras have trivial centre.

Common Mistake

Thinking the Cartan subalgebra is unique

Cartan subalgebras are unique up to conjugation by inner automorphisms, but not literally unique as subsets. Any maximal toral subalgebra qualifies.

Quiz

Cartan's criterion states that 𝔤 is semisimple if and only if:

The Killing form is positive definiteThe Killing form is non-degenerateThe centre Z(𝔤) = 0The adjoint representation is faithful

Weyl's complete reducibility theorem fails in:

Characteristic 0 over ℂCharacteristic 0 over ℝPrime characteristic p > 0Both characteristic 0 cases

Historical Background

Élie Cartan's 1894 doctoral thesis completed Killing's classification of simple Lie algebras, establishing the Cartan decomposition and the role of the Cartan subalgebra. The landmark structure theorems — Engel's theorem, Lie's theorem, Cartan's criterion for solvability, and Weyl's complete reducibility theorem — were assembled between 1893 and 1925, creating one of the most complete theories in algebra.

1893
Engel's theorem: nilpotent Lie algebras have faithful representations by strictly upper-triangular matrices
Friedrich Engel
1894
Cartan completes classification of simple Lie algebras over ℂ
Élie Cartan
1925
Weyl proves complete reducibility of representations of semisimple Lie algebras
Hermann Weyl

Summary

A semisimple Lie algebra has zero radical; equivalently, its Killing form is non-degenerate.
Over ℂ, every semisimple Lie algebra is a direct sum of simple Lie algebras, classified by Dynkin diagrams.
The root space decomposition 𝔤 = 𝔥 ⊕ ⊕_{α∈Φ} 𝔤_α encodes the full structure relative to a Cartan subalgebra.
Weyl's theorem ensures complete reducibility of finite-dimensional representations in characteristic 0.

References

BookHumphreys, J.E. — Introduction to Lie Algebras and Representation Theory (1972), Chapters II–IV
WebsiteWikipedia — Semisimple Lie algebra
WebsitenLab — semisimple Lie algebra