Theorems · Definition · nonassociative algebras

LieAlgebra.IsKilling.sl2SubmoduleOfRoot

{K : Type u_2} →
  {L : Type u_3} →
    [inst : LieRing L] →
      [inst_1 : Field K] →
        [inst_2 : LieAlgebra K L] →
          [FiniteDimensional K L] →
            {H : LieSubalgebra K L} →
              [inst_4 : H.IsCartanSubalgebra] →
                [LieAlgebra.IsKilling K L] →
                  [LieModule.IsTriangularizable K (↥H) L] →
                    [CharZero K] → {α : LieModule.Weight K (↥H) L} → α.IsNonZero → LieSubmodule K (↥H) L

The sl₂ subalgebra associated to a root, regarded as a Lie submodule over the Cartan subalgebra.

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

9 results in Mathlib

Foundations

Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingFieldLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieAlgebra.IsKillingLieModule.IsTriangularizableCharZero