Theorems · Definition · nonassociative algebras

LieAlgebra.IsKilling.cartanEquivDual

{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) → [H.IsCartanSubalgebra] → [LieAlgebra.IsKilling K L] → ↥H ≃ₗ[K] Module.Dual K ↥H

The restriction of the Killing form to a Cartan subalgebra, as a linear equivalence to the dual.

Defined in

Mathlib.Algebra.Lie.Weights.Killing

Cited by

17 results in Mathlib

Foundations

Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LieRingFieldLieAlgebraFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieAlgebra.IsKilling