Theorems · Definition · nonassociative algebras

LieIdeal.toInvtRootSubmodule

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

The invariant root submodule corresponding to a Lie ideal. Given a Lie ideal I, this produces an invariant root submodule by taking the span of all roots whose root spaces are contained in I.

Defined in

Mathlib.Algebra.Lie.Weights.IsSimple

Cited by

2 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLieRingLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraCharZeroLieAlgebra.IsKillingLieModule.IsTriangularizable