Of U1Subgroup smul eq smul
StandardModel.HiggsVec.ofU1Subgroup_smul_eq_smul
Project documentation
The subgroup of gaugeGroup := SU(3) × SU(2) × U(1) which preserves every HiggsVec by the action of StandardModel.HiggsVec.rep is given by SU(3) × ℤ₆ where ℤ₆ is the subgroup of SU(2) × U(1) with elements (α^(-3) * I₂, α) where α is a sixth root of unity. -/ informal_lemma stability_group where deps := [``HiggsVec] tag := "6V2MO" /-! ## A.8...
Exact Lean statement
lemma ofU1Subgroup_smul_eq_smul (g : unitary ℂ) (φ : HiggsVec) :
(StandardModel.GaugeGroupI.ofU1Subgroup g) • φ =
(WithLp.toLp 2 <| !![1, 0; 0, g.1 ^ 6] *ᵥ φ.ofLp)Formal artifact
Lean source
lemma ofU1Subgroup_smul_eq_smul (g : unitary ℂ) (φ : HiggsVec) : (StandardModel.GaugeGroupI.ofU1Subgroup g) • φ = (WithLp.toLp 2 <| !![1, 0; 0, g.1 ^ 6] *ᵥ φ.ofLp) := by rw [gaugeGroupI_smul_eq_U1_smul_SU2] simp only [GaugeGroupI.ofU1Subgroup_toU1, GaugeGroupI.ofU1Subgroup_toSU2, SubmonoidClass.coe_pow, star_pow, RCLike.star_def, smul_of, smul_cons, smul_zero, smul_empty, cons_mulVec, cons_dotProduct, zero_mul, dotProduct_of_isEmpty, add_zero, zero_add, empty_mulVec, one_mul, WithLp.toLp.injEq, vecCons_inj, mul_eq_mul_right_iff, and_true] refine ⟨?_, Or.inl ?_⟩ · have h0 : g ^ 3 • (starRingEnd ℂ) ↑g ^ 3 = 1 := by simp only [starRingEnd_apply, Submonoid.smul_def, smul_eq_mul, SubmonoidClass.coe_pow, ← mul_pow, Unitary.mul_star_self_of_mem g.2, one_pow] simp [h0] · show (g : ℂ) ^ 3 * (g : ℂ) ^ 3 = (g : ℂ) ^ 6 ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Particles/StandardModel/HiggsBoson/Basic.lean:349-363
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