Theorems · Definition · nonassociative algebras

RootPairing.GeckConstruction.equivRootSystem

{ι : Type u_1} →
  {K : Type u_2} →
    {M : Type u_3} →
      {N : Type u_4} →
        [inst : Fintype ι] →
          [inst_1 : DecidableEq ι] →
            [inst_2 : Field K] →
              [inst_3 : CharZero K] →
                [inst_4 : AddCommGroup M] →
                  [inst_5 : Module K M] →
                    [inst_6 : AddCommGroup N] →
                      [inst_7 : Module K N] →
                        {P : RootPairing ι K M N} →
                          [inst_8 : P.IsReduced] →
                            [inst_9 : P.IsCrystallographic] →
                              [inst_10 : P.IsIrreducible] →
                                [inst_11 : P.IsRootSystem] →
                                  (b : P.Base) →
                                    [inst_12 : IsAlgClosed K] →
                                      P.Equiv
                                        (LieAlgebra.IsKilling.rootSystem
                                          (RootPairing.GeckConstruction.cartanSubalgebra' b))

Up to equivalence, LieAlgebra.IsKilling.rootSystem is left inverse to RootPairing.GeckConstruction.lieAlgebra.

Defined in

Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basis

Cited by

0 results in Mathlib

Foundations

Depth 214 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeDecidableEqFieldCharZeroAddCommGroupModuleAddCommGroupModuleRootPairing.IsReducedRootPairing.IsCrystallographicRootPairing.IsIrreducibleRootPairing.IsRootSystemIsAlgClosed