Theorems · Theorem · nonassociative algebras

RootPairing.Base.cartanMatrix_mul_diagonal_eq

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M]
  [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] {P : RootPairing ι R M N} (b : P.Base)
  [inst_5 : P.IsCrystallographic] [inst_6 : CharZero R] [inst_7 : Fintype ι] [inst_8 : DecidableEq ι] [P.IsRootSystem],
  (b.cartanMatrix * Matrix.diagonal fun i => ((P.RootFormIn ℤ) (P.rootSpanMem ℤ ↑i)) (P.rootSpanMem ℤ ↑i)) =
    2 • (LinearMap.BilinForm.toMatrix b.toWeightBasisInt) (P.posRootForm ℤ).posForm

Defined in

Mathlib.LinearAlgebra.RootSystem.CartanMatrix

Cited by

1 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleRootPairing.IsCrystallographicCharZeroFintypeDecidableEqRootPairing.IsRootSystem