DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The special linear Lie algebra sl_2

Definition

Write ${g l}_{2} (C) = M_{2} (C)$ for the Lie algebra of complex $2 \times 2$ matrices under the commutator bracket $[A, B] = A B - B A$ (Representations of Lie algebras, Lie algebras over a field). The special linear Lie algebra ${s l}_{2} (C)$ is the Lie subalgebra (Lie subalgebras, ideals, and center) of traceless matrices

${s l}_{2} (C) = {A \in M_{2} (C) : tr A = 0},$

which is closed under the bracket because $tr (A B - B A) = 0$ . Put

$e = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}), f = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}), h = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}) .$

Direct matrix multiplication gives $h e - e h = 2 e$ , $h f - f h = - 2 f$ and $e f - f e = h$ , that is,

$[h, e] = 2 e, [h, f] = - 2 f, [e, f] = h .$

Since ${e, f, h}$ is a basis of the space of traceless matrices, these relations determine the bracket completely, ${s l}_{2} (C)$ is three-dimensional, and $h$ spans a one-dimensional abelian subalgebra. A Lie algebra over $C$ is called a copy of ${s l}_{2}$ if it has a basis $(e, f, h)$ satisfying exactly these three bracket relations.

Depends on

Used by

A nondominant integral highest-weight module can be infinite-dimensional Counterexample
SL₂ and PGL₂ have the same Lie algebra but differ globally Counterexample
The full weight lattice need not integrate through a central quotient Counterexample
Two nonconjugate real cartan subalgebras Counterexample
All irreducible finite-dimensional sl2 modules Example
Cartan subalgebra and roots of sl₂ Example
Clebsch–Gordan decomposition for sl2 Example
Compact and split cartan subalgebras of sl two r Example
Compact and split real forms of sl two c Example
Diagonal Cartan subalgebra and roots of slₙ Example
Root-space brackets for matrix units Example
The root sl₂ triple inside slₙ Example
The Weyl reflection in sl₂ Example
Verma modules for sl2 Example
Weyl character and dimension formulas for sl2 Example
A plain dynkin diagram classifies real forms False statement
A tensor-product top weight does not determine all constituents False statement
All cartan subalgebras of a real semisimple lie algebra are conjugate False statement
All integer multiples of a root are roots False statement
All real forms of a complex semisimple lie algebra are isomorphic False statement
Dominance depends on a positive system False statement
Every element of a complex semisimple Lie algebra is semisimple False statement
Finite-dimensionality requires dominance integrality False statement
Global cartan and iwasawa decompositions hold for every nonlinear cover without modified k False statement
Not every weight vector is highest False statement
Root spaces can have arbitrary dimension in a complex semisimple Lie algebra False statement
The root-space decomposition classifies real semisimple Lie algebras with no extra data False statement
The Weyl quotient requires cancellation or extension False statement
Verma modules need not be finite-dimensional False statement
Chevalley basis and real structure constants Lemma
Real Cartan subalgebras need not be conjugate Proposition
Finite-dimensional representations of sl₂ Theorem
The root sl₂ triple Theorem
The root-string property Theorem

Dependency tree · two levels

8 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)