Theorems · Theorem · nonassociative algebras

LieAlgebra.IsKilling.apply_coroot_eq_cast

∀ {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 : LieAlgebra.IsKilling K L] [inst_5 : FiniteDimensional K L] {H : LieSubalgebra K L}
  [inst_6 : H.IsCartanSubalgebra] [LieModule.IsTriangularizable K (↥H) L] (α β : LieModule.Weight K (↥H) L),
  β (LieAlgebra.IsKilling.coroot α) = ↑(↑(LieModule.chainBotCoeff (⇑α) β) - ↑(LieModule.chainTopCoeff (⇑α) β))

If β - qα ... β ... β + rα is the α-chain through β, then β (coroot α) = q - r. In particular, it is an integer.

Defined in

Mathlib.Algebra.Lie.Weights.RootSystem

Cited by

0 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldCharZeroLieRingLieAlgebraLieAlgebra.IsKillingFiniteDimensionalLieSubalgebra.IsCartanSubalgebraLieModule.IsTriangularizable