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ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Tactic.Abel
import Mathlib.Algebra.MonoidAlgebra.MapDomain
import Mathlib.Basic.Complex.Basic
import Mathlib.Tactic.NormNum
import OAI.Algebra.Finitistic.Selection.GroupSelection
namespace OAI
namespace LittleFinitistic
open MonoidAlgebra
abbrev SelectionRing := MonoidAlgebra ℂ Presentation.AuxiliaryGroup
noncomputable def algebraZ (q : ℤ) : SelectionRing := single (Presentation.z q) 1
@[simp] theorem algebraZ_square (q : ℤ) : algebraZ q * algebraZ q = 1 := by
simp [algebraZ,single_mul_single,Presentation.z_involution]
rfl
theorem algebraZ_central (q : ℤ) (a : SelectionRing) : Commute (algebraZ q) a :=
MonoidAlgebra.single_commute (Presentation.z_central q) Commute.one_left a
noncomputable def selectionIdempotent (q : ℤ) : SelectionRing :=
(1/2 : ℂ) • (1 + algebraZ q)
theorem selectionIdempotent_square (q : ℤ) :
selectionIdempotent q * selectionIdempotent q = selectionIdempotent q := by
have hz : (1 + algebraZ q)*(1 + algebraZ q) = (1+algebraZ q)+(1+algebraZ q) := by
simp only [mul_add,add_mul,one_mul,mul_one,algebraZ_square]
abel
unfold selectionIdempotent
rw [smul_mul_smul,hz,smul_add,← add_smul]
norm_num
theorem selectionIdempotent_central (q : ℤ) (a : SelectionRing) :
Commute (selectionIdempotent q) a := by
show selectionIdempotent q * a = a * selectionIdempotent q
unfold selectionIdempotent
rw [smul_mul_assoc,mul_smul_comm]
congr 1
simp only [add_mul,mul_add,one_mul,mul_one,(algebraZ_central q a).eq]
noncomputable def selectionAutomorphism (k : ℤ) : SelectionRing ≃ₐ[ℂ] SelectionRing :=
MonoidAlgebra.domCongr ℂ ℂ (Presentation.shiftAut k)
@[simp] theorem selectionAutomorphism_z (k q : ℤ) :
selectionAutomorphism k (algebraZ q) = algebraZ (q+k) := by
simp [selectionAutomorphism,algebraZ,Presentation.shiftAut_z]
@[simp] theorem selectionAutomorphism_idempotent (k q : ℤ) :
selectionAutomorphism k (selectionIdempotent q) = selectionIdempotent (q+k) := by
simp [selectionIdempotent]
end LittleFinitistic
end OAI
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