Theorems · Definition · nonassociative algebras

RootPairing.RootPositiveForm.posForm

{ι : Type u_1} →
  {R : Type u_2} →
    {S : Type u_3} →
      {M : Type u_4} →
        {N : Type u_5} →
          [inst : CommRing S] →
            [inst_1 : LinearOrder S] →
              [inst_2 : CommRing R] →
                [inst_3 : Algebra S R] →
                  [inst_4 : AddCommGroup M] →
                    [inst_5 : Module R M] →
                      [inst_6 : AddCommGroup N] →
                        [inst_7 : Module R N] →
                          {P : RootPairing ι R M N} →
                            [inst_8 : P.IsValuedIn S] →
                              RootPairing.RootPositiveForm S P →
                                [FaithfulSMul S R] →
                                  [inst_10 : Module S M] →
                                    [IsScalarTower S R M] →
                                      LinearMap.BilinForm S ↥(Submodule.span S (Set.range ⇑P.root))

Given a root-positive form associated to a root pairing with coefficients in R but taking values in S, this is the associated S-bilinear form on the S-span of the roots.

Defined in

Mathlib.LinearAlgebra.RootSystem.RootPositive

Cited by

22 results in Mathlib

Foundations

Depth 40 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLinearOrderCommRingAlgebraAddCommGroupModuleAddCommGroupModuleRootPairing.IsValuedInFaithfulSMulModuleIsScalarTower