Theorems · Definition · nonassociative algebras

RootPairing.GeckConstruction.lieAlgebra

{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} →
                    [P.IsCrystallographic] →
                      (b : P.Base) →
                        [Finite ι] →
                          [IsDomain R] →
                            [CharZero R] →
                              [inst_9 : Fintype ι] →
                                [inst_10 : DecidableEq ι] → LieSubalgebra R (Matrix (↥b.support ⊕ ι) (↥b.support ⊕ ι) R)

Geck's construction of the Lie algebra associated to a root system with distinguished base. Note that it is convenient to include range h in the Lie span, to make it elementary that it contains RootPairing.GeckConstruction.cartanSubalgebra, and not depend on RootPairing.GeckConstruction.lie_e_f_same.

Defined in

Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic

Cited by

14 results in Mathlib

Foundations

Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleRootPairing.IsCrystallographicFiniteIsDomainCharZeroFintypeDecidableEq