SL2-Triples
An \(\mathfrak{sl}_2\)-triple in a Lie algebra is a triple \((e, h, f)\) satisfying the standard \(\mathfrak{sl}_2\)-commutation relations. Such triples are the algebraic counterpart of representations of \(\mathfrak{sl}_2\) inside \(\mathfrak{g}\), and they package the structure of a nilpotent orbit into a finite list of weights.
Definition
Let \(\mathfrak{g}\) be a Lie algebra over a field \(k\) (typically characteristic 0 or good for the type of \(\mathfrak{g}\)). An \(\mathfrak{sl}_2\)-triple is a triple \((e, h, f)\) of elements of \(\mathfrak{g}\) satisfying: \[ [h, e] = 2e, \qquad [h, f] = -2f, \qquad [e, f] = h. \]
Equivalently: \((e, h, f)\) defines a Lie algebra homomorphism \(\mathfrak{sl}_2(k) \to \mathfrak{g}\), \(\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \mapsto e\), \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \mapsto h\), etc.
Conventions:
- \(e\) (or \(X\)) is the nilpositive element;
- \(f\) (or \(Y\)) is the nilnegative element;
- \(h\) (or \(H\)) is the neutral (semisimple) element.
Jacobson-Morozov theorem
Theorem (Jacobson-Morozov). Let \(\mathfrak{g}\) be semisimple over \(k\) of characteristic 0 (or good positive characteristic). For every nonzero nilpotent \(e \in \mathfrak{g}\), there exist \(h, f \in \mathfrak{g}\) such that \((e, h, f)\) is an \(\mathfrak{sl}_2\)-triple.
Moreover: any two such extensions are conjugate by an element of \(Z_G(e)^\circ\) (the identity component of the centraliser).
Corollary. There is a bijection \[ \{\text{nonzero nilpotent orbits in } \mathfrak{g}\} \;\longleftrightarrow\; \{\text{}\mathfrak{sl}_2\text{-triples in } \mathfrak{g}\}/G. \]
Weights and representation structure
For an \(\mathfrak{sl}_2\)-triple \((e, h, f)\) in \(\mathfrak{g}\), the adjoint representation of the triple on \(\mathfrak{g}\) decomposes \[ \mathfrak{g} \;=\; \bigoplus_{n \in \mathbb{Z}} \mathfrak{g}_n, \qquad \mathfrak{g}_n = \{X : [h, X] = nX\}. \]
\(\mathfrak{g}\) becomes a graded \(\mathfrak{sl}_2\)-module: each \(\mathfrak{g}_n\) is finite-dimensional, and the multiplicities of irreducible \(\mathfrak{sl}_2\)-modules \(V_m\) in \(\mathfrak{g}\) give a finite list of integers — the dimension vector of the triple.
By the structure of \(\mathfrak{sl}_2\)-representations: \(\dim \mathfrak{g}_n = \dim \mathfrak{g}_{-n}\), and \(e\) maps \(\mathfrak{g}_n \xrightarrow{\cong} \mathfrak{g}_{n+2}\) for \(n \ge 0\).
Weighted Dynkin diagram
Given an \(\mathfrak{sl}_2\)-triple, conjugate \(h\) into the dominant Weyl chamber of a fixed Cartan. The weighted Dynkin diagram records \(d_i = \alpha_i(h)\) for each simple root \(\alpha_i\): \[ d_i \in \{0, 1, 2\}. \]
This is the canonical combinatorial invariant of the \(G\)-conjugacy class of the triple (= the orbit \(G \cdot e\)). See Weighted Dynkin Diagrams.
Examples
\(\mathfrak{sl}_2\) standard
\(e = E_{12}\), \(h = E_{11} - E_{22}\), \(f = E_{21}\). Adjoint representation decomposes as the irreducible \(V_3\) (\(3\)-dimensional).
\(\mathfrak{sl}_3\), regular nilpotent
Sum of all simple-root vectors: \(e = E_{12} + E_{23}\). The triple has \(h = 2(\omega_1 + \omega_2)\) in dominant Weyl coordinates; weighted Dynkin = \(2 - 2\).
\(\mathfrak{g}\) decomposes into one \(V_5\) (the regular module) and nothing else.
\(\mathfrak{sl}_n\), Jordan-block triple
For Jordan partition \(\lambda = (\lambda_1, \ldots, \lambda_k)\): each block contributes an irreducible \(\mathfrak{sl}_2\)-module \(V_{\lambda_i - 1}\). The total \(\mathfrak{sl}_2\)-representation of \(\mathfrak{gl}_n\) (under the nilpotent triple) is determined by \(\lambda\).
Kostant's classification
Theorem (Kostant 1959). The \(G\)-conjugacy classes of \(\mathfrak{sl}_2\)-triples in \(\mathfrak{g}\) are in bijection with the \(G\)-conjugacy classes of nonzero nilpotents (via \((e, h, f) \mapsto e\)).
This is the existence-and-uniqueness theorem complementing Jacobson-Morozov.
Slice and transversal
For an \(\mathfrak{sl}_2\)-triple \((e, h, f)\), the Kostant-Slodowy slice \[ S_e \;:=\; e + Z_{\mathfrak{g}}(f) \;\subseteq\; \mathfrak{g} \] is a transverse slice to the nilpotent orbit \(G \cdot e\) inside \(\mathfrak{g}\). The intersection \(S_e \cap \mathcal{N}\) gives a "local model" for the closure ordering of orbits near \(G \cdot e\).
For the subregular nilpotent orbit: \(S_e \cap \mathcal{N}\) is a Kleinian (Du Val) singularity of the same type as the Dynkin diagram of \(\mathfrak{g}\) — a beautiful link between Lie theory and surface singularities.
Mathlib
Not formalised.
References
- Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group (Amer.\ J.\ Math.\ 1959).
- Carter, Finite Groups of Lie Type (1985), §5.5.
- Collingwood–McGovern, Nilpotent Orbits in Semisimple Lie Algebras (1993), Ch. 3.