Theorems · Definition · nonassociative algebras

RootPairing.RootFormIn

{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) →
                    (S : Type u_5) →
                      [inst_5 : CommRing S] →
                        [inst_6 : Algebra S R] →
                          [FaithfulSMul S R] →
                            [inst_8 : Module S M] →
                              [IsScalarTower S R M] →
                                [P.IsValuedIn S] → [Fintype ι] → LinearMap.BilinForm S ↥(P.rootSpan S)

A canonical bilinear form on the span of roots in a finite root pairing, taking values in a commutative ring, where the root-coroot pairing takes values in that ring.

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear

Cited by

13 results in Mathlib

Foundations

Depth 45 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleCommRingAlgebraFaithfulSMulModuleIsScalarTowerRootPairing.IsValuedInFintype