PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Classical types correspond to sl, so and sp

Statement

Assume the Axiom of Choice. The simple Lie algebras of classical type are $A_{n} : {s l}_{n + 1} (C)$ for $n \geq 1$ , $B_{n} : {s o}_{2 n + 1} (C)$ for $n \geq 2$ , $C_{n} : {s p}_{2 n} (C)$ for $n \geq 3$ , and $D_{n} : {s o}_{2 n} (C)$ for $n \geq 4$ . The low-rank coincidences are ${s o}_{3} ≅ {s l}_{2}$ and ${s p}_{2} ≅ {s l}_{2}$ , ${s p}_{4} ≅ {s o}_{5}$ , ${s o}_{4} ≅ {s l}_{2} \oplus {s l}_{2}$ , and ${s o}_{6} ≅ {s l}_{4}$ .

Facts & Assumptions

Given: The classical matrix Lie algebras and their diagonal Cartan subalgebras, with the root systems computed in Root systems of the classical complex Lie algebras.

[A1]

AC is assumed and is used through the isomorphism theorem (The Axiom of Choice).

[L1]

The root systems of ${s l}_{n}, {s p}_{2 n}, {s o}_{2 n}, {s o}_{2 n + 1}$ with respect to the diagonal Cartan subalgebra are the standard coordinate models of types $A_{n - 1}, C_{n}, D_{n}, B_{n}$ , with one-dimensional root spaces (Root systems of the classical complex Lie algebras).

[L2]

In the simple ranges, the Killing forms of ${s l}_{m}$ , ${s o}_{m}$ and ${s p}_{2 m}$ are respectively the nonzero multiples $2 m tr (X Y)$ , $(m - 2) tr (X Y)$ and $2 (m + 1) tr (X Y)$ , and are nondegenerate (Classical simple Lie algebras and their Killing forms).

[L3]

Two finite-dimensional complex semisimple Lie algebras with isomorphic based root systems are isomorphic, and the connected classical diagrams occur in the ranges $A_{n}$ for $n \geq 1$ , $B_{n}$ for $n \geq 2$ , $C_{n}$ for $n \geq 3$ , and $D_{n}$ for $n \geq 4$ (Isomorphism theorem for complex semisimple Lie algebras, Cartan-Killing classification of complex simple Lie algebras).

[L4]

Remark 23.18 of the cited Etingof notes records the root-system coincidences $D_{2} ≅ A_{1} ⊔ A_{1}$ , $D_{3} ≅ A_{3}$ , and $B_{2} ≅ C_{2}$ ; the rank-one coordinate models give $B_{1} = C_{1} = A_{1}$ .

Proof

technique · direct

1.1

In the ranges $n \geq 1$ for $A_{n}$ , $n \geq 2$ for $B_{n}$ , $n \geq 3$ for $C_{n}$ , and $n \geq 4$ for $D_{n}$ , the algebras in the Statement have the asserted root systems by [L1], are semisimple by [L2], and have connected diagrams by [L3]; hence they are simple and have the asserted classical types.

L1L2L3algebra

1.2

The algebras ${s o}_{3}, {s p}_{2}, {s o}_{5}, {s p}_{4}, {s o}_{6}$ and ${s l}_{4}$ are in the nondegenerate Killing-form ranges of [L2]. Their based root systems agree in the pairs prescribed by [L4], so [L3] gives ${s o}_{3} ≅ {s l}_{2} ≅ {s p}_{2}$ , ${s o}_{5} ≅ {s p}_{4}$ , and ${s o}_{6} ≅ {s l}_{4}$ .

L1L2L3L4A1algebra

1.3

For the remaining $D_{2}$ case, let $V, W$ be two-dimensional complex vector spaces with nondegenerate alternating forms. Their product defines a nondegenerate symmetric form on $V \otimes W$ . Since $s l (V) = s p (V)$ and likewise for $W$ , the map $(A, B) \mapsto A \otimes I + I \otimes B$ is a homomorphism $s l (V) \oplus s l (W) \to s o (V \otimes W)$ . It is injective: taking the partial trace over $W$ in $A \otimes I + I \otimes B = 0$ gives $2 A = 0$ , and similarly $2 B = 0$ . Both sides have dimension $6$ , so it is an isomorphism ${s l}_{2} \oplus {s l}_{2} ≅ {s o}_{4}$ .

algebra

2.1

For each type in the stable ranges, the complex simple Lie algebra with that based root system is unique up to isomorphism by [L3]. Thus ${s l}_{n + 1} (C)$ , ${s o}_{2 n + 1} (C)$ , ${s p}_{2 n} (C)$ and ${s o}_{2 n} (C)$ realize $A_{n}, B_{n}, C_{n}, D_{n}$ , respectively.

L3step 1.1algebra

3.1

Apart from the coincidences in [L4], the connected classical diagrams in [L3] are distinct. Therefore the classification gives no further isomorphisms among these four classical families.

L3L4step 2.1step 1.2step 1.3algebra∎

Depends on

Used by

Vogan diagrams for real forms of sl three c Example
A plain dynkin diagram classifies real forms False statement
Classical real forms of the classical complex lie algebras Proposition

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I, Lectures 19-24 (standard reference, not scraped)
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)