[Merged by Bors] - feat(Counterexamples/CliffordAlgebra_not_injective): Some quadratic forms cannot be constructed from bilinear forms #9670
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…orms cannot be constructed from bilinear forms
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It shouldn't be too hard to use LinearMap.toQuadraticForm instead, right? |
| @@ -260,6 +264,16 @@ is not injective, as it sends the non-zero `α * β * γ` to zero. -/ | |||
| theorem algebraMap_not_injective : ¬Function.Injective (algebraMap K <| CliffordAlgebra Q) := | |||
| fun h => αβγ_ne_zero <| h <| by rw [algebraMap_αβγ_eq_zero, RingHom.map_zero] | |||
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| /-- Bonus counterexample: `Q` is a quadratic form that has no bilinear form. -/ | |||
| theorem Q_not_in_range_toQuadraticForm : Q ∉ Set.range BilinForm.toQuadraticForm := by | |||
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It seems there is a positive result: if a quadratic form is induced by a bilinear form, then the algebraMap to the clifford algebra is injective.
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Correct; this is in fact a consequence of a (trivial) generalization of equivExterior; for now, that definition assumes Invertible (2 : R) instead of IsInducedByBilinear Q, as the latter does not exist as a typeclass.
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…orms cannot be constructed from bilinear forms (#9670) Turns out that this follows trivially from the previous counterexample. Perhaps there's a slicker proof that doesn't go via Clifford algebras at all, but it seemed worth recording this one anyway.
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…orms cannot be constructed from bilinear forms (#9670) Turns out that this follows trivially from the previous counterexample. Perhaps there's a slicker proof that doesn't go via Clifford algebras at all, but it seemed worth recording this one anyway.
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…orms cannot be constructed from bilinear forms (#9670) Turns out that this follows trivially from the previous counterexample. Perhaps there's a slicker proof that doesn't go via Clifford algebras at all, but it seemed worth recording this one anyway.
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…orms cannot be constructed from bilinear forms (#9670) Turns out that this follows trivially from the previous counterexample. Perhaps there's a slicker proof that doesn't go via Clifford algebras at all, but it seemed worth recording this one anyway.
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…orms cannot be constructed from bilinear forms (#9670) Turns out that this follows trivially from the previous counterexample. Perhaps there's a slicker proof that doesn't go via Clifford algebras at all, but it seemed worth recording this one anyway.
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Turns out that this follows trivially from the previous counterexample.
Perhaps there's a slicker proof that doesn't go via Clifford algebras at all, but it seemed worth recording this one anyway.