Topic Lie Algebras of Reductive Groups
topic admitted

Lie Algebras of Reductive Groups

The Lie algebra \(\mathfrak{g} = \operatorname{Lie}(G)\) of a reductive algebraic group \(G\) is the tangent space at the identity, equipped with the Lie bracket from the group structure. It encodes the infinitesimal symmetries of \(G\) and is the natural setting for the representation theory, the Killing form, root data, and nilpotent orbit theory.

Definition

For an algebraic group \(G\) over a field \(k\): \[ \operatorname{Lie}(G) \;:=\; T_e G \;=\; \ker(G(k[\varepsilon]/\varepsilon^2) \to G(k)), \] the tangent space at the identity \(e\). The Lie bracket \([X, Y]\) is induced from the group commutator \((g, h) \mapsto ghg^{-1}h^{-1}\).

For smooth \(G\): \(\dim \operatorname{Lie}(G) = \dim G\).

Structure for reductive \(G\)

If \(G\) is a connected reductive algebraic group over an algebraically closed field \(k\) of characteristic 0 (or good characteristic), then \(\mathfrak{g}\) is a reductive Lie algebra:

  • \(\mathfrak{g} = \mathfrak{z} \oplus [\mathfrak{g}, \mathfrak{g}]\), with \(\mathfrak{z} = \operatorname{Lie}(Z(G)^\circ)\) abelian and \([\mathfrak{g}, \mathfrak{g}] = \operatorname{Lie}(G_{\mathrm{der}})\) semisimple.
  • Killing form \(\kappa(X, Y) = \mathrm{tr}(\operatorname{ad}\,X \circ \operatorname{ad}\,Y)\) is non-degenerate on \([\mathfrak{g}, \mathfrak{g}]\).

Root space decomposition

For \(T \subseteq G\) a maximal torus and \(\mathfrak{t} = \operatorname{Lie}(T)\): \[ \mathfrak{g} \;=\; \mathfrak{t} \;\oplus\; \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, \] where \(\mathfrak{g}_\alpha\) is the 1-dimensional root space: \(\mathfrak{g}_\alpha = \{X \in \mathfrak{g} : [H, X] = \alpha(H) X \text{ for all } H \in \mathfrak{t}\}\).

The root spaces \(\mathfrak{g}_\alpha\) are 1-dimensional in characteristic 0 (a key part of the root datum classification); in positive characteristic they remain 1-dimensional in good characteristic.

Examples

\(\mathfrak{gl}_n\) and \(\mathfrak{sl}_n\)

  • \(\mathfrak{gl}_n = M_n(k)\), all \(n \times n\) matrices, with bracket \([X, Y] = XY - YX\).
  • \(\mathfrak{sl}_n = \{X \in M_n : \mathrm{tr}\,X = 0\}\), the trace-zero matrices.
  • \(\mathfrak{gl}_n = \mathfrak{sl}_n \oplus k \cdot I\); \(\mathfrak{sl}_n\) semisimple, \(kI\) the abelian centre.

\(\mathfrak{so}_n\)

\(\mathfrak{so}_n = \{X \in M_n : X + X^T = 0\}\), skew-symmetric matrices. Dimension \(\binom{n}{2}\).

\(\mathfrak{sp}_{2n}\)

\(\mathfrak{sp}_{2n} = \{X : X^T J + J X = 0\}\), with \(J\) the standard symplectic form. Dimension \(n(2n+1)\).

Exceptional Lie algebras

  • \(G_2\): 14-dim (derivations of octonions);
  • \(F_4\): 52-dim (derivations of \(H_3(\mathbb{O})\));
  • \(E_6\): 78-dim, \(E_7\): 133-dim, \(E_8\): 248-dim.

The exponential map

In characteristic 0: \(\exp : \mathfrak{g} \to G\) is defined and gives a local diffeomorphism near 0/identity. Restricted to nilpotent elements: \[ \exp : \mathcal{N} \;\xrightarrow{\sim}\; \mathcal{U}, \] a bijection (in fact, an isomorphism of varieties) between the nilpotent cone and the unipotent variety. This is the Springer isomorphism.

In positive characteristic: \(\exp\) is not well-defined globally, but in good characteristic the Springer isomorphism still holds (as a variety isomorphism that may not literally be the exponential).

Adjoint representation

\(G\) acts on \(\mathfrak{g}\) by conjugation: \(\operatorname{Ad}(g)(X) = g X g^{-1}\) (in matrix terms). This is the adjoint representation, a homomorphism \(\operatorname{Ad} : G \to \mathrm{GL}(\mathfrak{g})\), with image the adjoint group \(G_{\mathrm{ad}} = G/Z(G)\) and kernel \(Z(G)\).

The induced map on Lie algebras is \(\mathrm{ad} : \mathfrak{g} \to \operatorname{End}(\mathfrak{g})\), \(X \mapsto [X, -]\).

Killing form and invariants

The Killing form \(\kappa : \mathfrak{g} \otimes \mathfrak{g} \to k\): \[ \kappa(X, Y) \;:=\; \mathrm{tr}_{\mathfrak{g}}(\mathrm{ad}\, X \circ \mathrm{ad}\, Y). \]

Properties:

  • Symmetric, \(G\)-invariant, bilinear;
  • Non-degenerate on a semisimple Lie algebra (Cartan's criterion);
  • Restricts to a positive-definite form on a compact real form;
  • Gives the basic invariant theory of \(G\) acting on \(\mathfrak{g}\).

Mathlib status

Mathlib.Algebra.Lie.* has:

  • Lie algebras (general, semisimple, nilpotent, solvable);
  • Killing form;
  • Root systems via Mathlib.LinearAlgebra.RootSystem.

The Lie algebra of a specific algebraic group (e.g.\ \(\operatorname{Lie}(\mathrm{GL}_n)\)) is not yet directly formalised as such; the path goes via formal group schemes.

References

  • Humphreys, Introduction to Lie Algebras and Representation Theory (1972).
  • Bourbaki, Groupes et Algèbres de Lie I–III, VII–VIII.
  • Borel, Linear Algebraic Groups (1991), §3.
  • Milne, Reductive Groups (notes), §3.

Prerequisite Topics

Used by