Topic Jacobson-Morozov Theorem
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Jacobson-Morozov Theorem

The Jacobson-Morozov theorem asserts that every nonzero nilpotent element in a semisimple Lie algebra extends to an \(\mathfrak{sl}_2\)-triple. This is the foundational existence statement underlying the entire classification of nilpotent orbits.

Statement

Theorem (Jacobson-Morozov). Let \(\mathfrak{g}\) be a finite-dimensional semisimple Lie algebra over a field \(k\) of characteristic 0 (or of good characteristic for the type of \(\mathfrak{g}\)). For every nonzero nilpotent element \(e \in \mathfrak{g}\), there exist \(h, f \in \mathfrak{g}\) such that \((e, h, f)\) is an \(\mathfrak{sl}_2\)-triple: \[ [h, e] = 2e, \qquad [h, f] = -2f, \qquad [e, f] = h. \]

Furthermore, the extension is unique up to \(Z_G(e)^\circ\)-conjugacy: any two \(\mathfrak{sl}_2\)-triples extending the same \(e\) differ by conjugation by an element of \(Z_G(e)^\circ\).

Proof

sketch

The key idea: extend the action of \(e\) (a nilpotent element) to a representation of \(\mathfrak{sl}_2\) on \(\mathfrak{g}\) by adjointly acting.

  1. Construct \(h\). Use \(e\) and a Lie-theoretic argument: \(\mathrm{im}(\mathrm{ad}\, e) \cap \mathrm{ker}(\mathrm{ad}\, e) \supseteq \{0\}\) when \(\mathfrak{g}\) is "non-degenerate enough" (the Killing form condition). Specifically, find \(h\) with \([h, e] = 2e\) using semisimplicity of \(\mathrm{ad}(h)\).
  2. Construct \(f\). Solve the equation \([e, f] = h\) using the surjectivity of \(\mathrm{ad}\, e\) restricted to suitable weight spaces.
  3. Verify \([h, f] = -2f\). Automatic from the construction.

The "non-degenerate enough" condition is automatic in characteristic 0 (Killing form non-degenerate); in positive characteristic, "good" means: not a torsion prime for \(\mathfrak{g}\) (e.g.\ \(p \ne 2\) for \(B_n, C_n, D_n\); \(p \ne 2, 3\) for exceptional types; with specific exceptions for \(G_2, F_4, E_6, E_7, E_8\)).

Failure in bad characteristic

In bad characteristic (e.g.\ \(p = 2\) for \(\mathrm{SO}_n\), \(p = 3\) for \(G_2\)): the theorem can fail. Specific nilpotents may not embed in any \(\mathfrak{sl}_2\)-triple, or extensions may not be unique.

The remedy is the Premet correspondence: replacing \(\mathfrak{sl}_2\)-triples with "associated cocharacters" — the extension still works.

Consequences

Bijection of orbits with triples

\(G\)-conjugacy classes of nonzero nilpotents = \(G\)-conjugacy classes of \(\mathfrak{sl}_2\)-triples (Kostant).

Weighted Dynkin diagrams

Each triple gives a labelling of the simple roots by \(\alpha_i(h) \in \{0, 1, 2\}\) — the weighted Dynkin diagram. Injective on orbits; characterising image is the Bala-Carter analysis.

Finiteness of nilpotent orbits

Because there are finitely many weighted Dynkin diagrams, there are finitely many nilpotent orbits.

Examples

\(\mathfrak{sl}_2\)

For \(e = E_{12}\): \(h = E_{11} - E_{22}\), \(f = E_{21}\). The unique triple (no centraliser).

\(\mathfrak{sl}_3\)

For \(e = E_{12} + E_{23}\) (regular nilpotent): the unique extension gives \(h = 2 E_{11} - 2 E_{33}\), \(f\) determined up to scale by \([e, f] = h\). Centraliser \(Z_G(e)^\circ\) is trivial (regular implies abelian centraliser, here just centre).

Subregular nilpotent in \(E_8\)

\(e\) is a non-trivial sum of simple-root vectors; the triple's \(h\)-weights give a specific weighted Dynkin label that distinguishes the subregular orbit from neighbouring orbits in the closure order.

Constructive aspects

The proof gives an algorithm: given \(e\), one can compute \(h\) and \(f\) by solving linear equations in \(\mathfrak{g}\). This is the basis of computer-algebra implementations (LiE, GAP, Magma) of nilpotent-orbit classification.

Mathlib

Not formalised. Depends on full Lie algebra and reductive group infrastructure.

References

  • Jacobson, Completely reducible Lie algebras of linear transformations (Proc.\ AMS 1951).
  • Morozov, On a nilpotent element in a semisimple Lie algebra (Doklady 1942).
  • Bourbaki, Groupes et algèbres de Lie, Ch. VIII.
  • Collingwood–McGovern, Nilpotent Orbits in Semisimple Lie Algebras (1993), §3.3.
  • Premet, Nilpotent commuting varieties of reductive Lie algebras (Invent.\ Math.\ 2003).

Prerequisite Topics