Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.isSimple_iff_isIrreducible

∀ {K : Type u_1} {L : Type u_2} [inst : Field K] [inst_1 : CharZero K] [inst_2 : LieRing L] [inst_3 : LieAlgebra K L]
  [inst_4 : FiniteDimensional K L] [inst_5 : LieAlgebra.IsKilling K L] {H : LieSubalgebra K L}
  [inst_6 : H.IsCartanSubalgebra] [inst_7 : LieModule.IsTriangularizable K (↥H) L],
  (LieAlgebra.IsKilling.rootSystem H).IsIrreducible ↔ LieAlgebra.IsSimple K L

A Killing Lie algebra is simple if and only if its root system is irreducible.

Defined in

Mathlib.Algebra.Lie.Weights.IsSimple

Cited by

0 results in Mathlib

Foundations

Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldCharZeroLieRingLieAlgebraFiniteDimensionalLieAlgebra.IsKillingLieSubalgebra.IsCartanSubalgebraLieModule.IsTriangularizable