Theorems · Theorem · nonassociative algebras

LieAlgebra.mem_corootSpace

∀ {R : Type u_1} {L : Type u_2} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
  {H : LieSubalgebra R L} [inst_3 : LieRing.IsNilpotent ↥H] [inst_4 : H.IsCartanSubalgebra] [inst_5 : IsNoetherian R L]
  (α : ↥H → R) {x : ↥H},
  x ∈ LieAlgebra.corootSpace α ↔
    ↑x ∈ Submodule.span R {x | ∃ y ∈ LieAlgebra.rootSpace H α, ∃ z ∈ LieAlgebra.rootSpace H (-α), ⁅y, z⁆ = x}

Defined in

Mathlib.Algebra.Lie.Weights.Cartan

Cited by

4 results in Mathlib

Foundations

Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLieRingLieAlgebraLieRing.IsNilpotentLieSubalgebra.IsCartanSubalgebraIsNoetherian