Nilpotent Elements
A nilpotent element of a Lie algebra \(\mathfrak{g}\) is one whose adjoint action is a nilpotent linear operator. Together with their conjugacy classification (= nilpotent orbits), they form the combinatorial backbone of representation theory and the geometric input to Langlands.
Definition
Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra over a field \(k\). An element \(X \in \mathfrak{g}\) is nilpotent if \(\operatorname{ad}(X) : \mathfrak{g} \to \mathfrak{g}\), \(Y \mapsto [X, Y]\), is a nilpotent endomorphism, i.e.\ \(\operatorname{ad}(X)^N = 0\) for some \(N\).
For \(\mathfrak{g}\) reductive (or more generally, for \(X\) in any faithful linear representation), this is equivalent to: \(X\) acts nilpotently on some (equivalently, every) faithful linear representation.
Jordan decomposition
Theorem (Jordan decomposition). For \(X \in \mathfrak{g}\) (\(\mathfrak{g}\) semisimple, or more generally with appropriate hypotheses), there exist unique commuting elements \(X_s, X_n \in \mathfrak{g}\) with:
- \(X = X_s + X_n\);
- \(\operatorname{ad}(X_s)\) semisimple, \(\operatorname{ad}(X_n)\) nilpotent;
- \([X_s, X_n] = 0\).
\(X_s\) is the semisimple part and \(X_n\) the nilpotent part of \(X\). They are polynomials in \(X\) with no constant term.
Examples
\(\mathfrak{gl}_n\) and \(\mathfrak{sl}_n\)
For \(X \in \mathfrak{gl}_n\) (a matrix), nilpotent in the Lie-algebra sense iff nilpotent as a matrix (\(X^N = 0\)). The Jordan decomposition agrees with the usual matrix Jordan decomposition.
\(\mathfrak{so}_n\)
\(X \in \mathfrak{so}_n\) (skew-symmetric matrices) nilpotent iff \(X^N = 0\). The Jordan block structure of \(X\) (viewed as an \(n \times n\) matrix) must be compatible with the bilinear form preserved by \(\mathrm{SO}_n\); parity constraints apply (see Bala-Carter).
Nilpotent cone
The set of all nilpotent elements is the nilpotent cone: \[ \mathcal{N} \;:=\; \{X \in \mathfrak{g} : X \text{ is nilpotent}\}. \]
\(\mathcal{N}\) is:
- a closed cone (closed under scalar multiplication);
- an algebraic variety, irreducible for semisimple \(\mathfrak{g}\);
- of dimension \(\dim \mathfrak{g} - \mathrm{rank}\, \mathfrak{g}\);
- the null cone of the adjoint action (image of \(0\) under the Chevalley-Steinberg map \(\mathfrak{g} \to \mathfrak{g} /\!\!/ G\)).
Properties of nilpotent elements
For \(X \in \mathfrak{g}\) nilpotent in a semisimple \(\mathfrak{g}\):
- \(\exp(X) \in G\) is a well-defined unipotent element of the algebraic group \(G\) (in good characteristic; for char \(0\) the exponential converges as a polynomial).
- The map \(X \mapsto \exp(X)\) is a bijection between nilpotent elements and unipotent elements (Springer).
- The orbit \(G \cdot X\) under the adjoint action is finite-dimensional and has a "good" geometry (Springer fibre).
Regular and subregular nilpotents
- Regular nilpotent: \(X \in \mathcal{N}\) is regular if \(\dim Z_\mathfrak{g}(X) = \mathrm{rank}\, \mathfrak{g}\) (minimal possible). The regular nilpotents form a single open dense \(G\)-orbit in \(\mathcal{N}\).
- Subregular: \(\dim Z_\mathfrak{g}(X) = \mathrm{rank}\, \mathfrak{g} + 2\); a unique \(G\)-orbit of codimension 2 in \(\mathcal{N}\).
- Principal nilpotent: synonym for regular nilpotent (for semisimple Lie algebras).
For \(\mathfrak{sl}_n\): the regular nilpotent corresponds to the partition \((n)\) (single Jordan block); subregular = \((n-1, 1)\).
Jacobson-Morozov
Theorem (Jacobson-Morozov). Every nonzero nilpotent \(X \in \mathfrak{g}\) (\(\mathfrak{g}\) semisimple, good characteristic) extends to an \(\mathfrak{sl}_2\)-triple \((X, H, Y)\):
- \([H, X] = 2X\), \([H, Y] = -2Y\), \([X, Y] = H\);
- \(X\), \(Y\) are nilpotent, \(H\) is semisimple.
Two extensions are \(Z_G(X)\)-conjugate. This is the foundation of the [[node:conjugacy_classes.weighted_dynkin_diagrams|weighted Dynkin diagram]] classification.
Mathlib
Mathlib.Algebra.Lie.Nilpotent defines nilpotent Lie algebras and
provides basic properties; nilpotent elements of a fixed Lie algebra
are not (yet) a separate object.
References
- Collingwood–McGovern, Nilpotent Orbits in Semisimple Lie Algebras (1993), Ch. 1.
- Humphreys, Conjugacy Classes in Semisimple Algebraic Groups (Math.\ Surveys 1995).
- Jantzen, Nilpotent Orbits in Representation Theory (2004).