Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.span_weight_eq_top

∀ {K : Type u_2} {L : Type u_3} [inst : LieRing L] [inst_1 : Field K] [inst_2 : LieAlgebra K L]
  [inst_3 : FiniteDimensional K L] {H : LieSubalgebra K L} [inst_4 : H.IsCartanSubalgebra]
  [inst_5 : LieAlgebra.IsKilling K L] [LieModule.IsTriangularizable K (↥H) L],
  Submodule.span K (Set.range (LieModule.Weight.toLinear K (↥H) L)) = ⊤

Given a splitting Cartan subalgebra H of a finite-dimensional Lie algebra with non-singular Killing form, the corresponding roots span the dual space of H.

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

2 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingFieldLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieAlgebra.IsKillingLieModule.IsTriangularizable