evenToNeg — Mathlib

English

For Q' and h with Q' = −Q, the EvenToNeg map sends the bilinear basis element (even.ι Q).bilin m1 m2 to its negative in the target, i.e., evenToNeg Q Q' h ((even.ι Q).bilin m1 m2) = −(even.ι Q').bilin m1 m2.

Русский

Для Q' и h с условием Q' = −Q отображение EvenToNeg отправляет базисный элемент, задаваемый bilinearity генераторов, в его отрицательный аналог: дажеToNeg( Q) снижает знак bilinear.

LaTeX

$$$ evenToNeg\\ Q\\ Q'\\ h\\bigl((even.ι Q).bilin(m_1,m_2)\\bigr) = -\\bigl((even.ι Q').bilin(m_1,m_2)\\bigr) $$$

Lean4

/-- One direction of `CliffordAlgebra.evenEquivEvenNeg` -/
def evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) : CliffordAlgebra.even Q →ₐ[R] CliffordAlgebra.even Q' :=
  even.lift Q <|
    { bilin := -(even.ι Q' :).bilin
      contract := fun m => by
        simp_rw [LinearMap.neg_apply, EvenHom.contract, h, QuadraticMap.neg_apply, map_neg, neg_neg]
      contract_mid := fun m₁ m₂ m₃ => by
        simp_rw [LinearMap.neg_apply, neg_mul_neg, EvenHom.contract_mid, h, QuadraticMap.neg_apply, smul_neg,
          neg_smul] }

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