English
Let S be a commutative semiring and S → R an algebra structure. Then the quaternion algebra ℍ[R,c1,c2,c3] carries a natural structure of an S-algebra compatible with the given algebra maps, with smul and algebraMap defined via the base maps.
Русский
Пусть S — коммутативное полупрямое полувынумное полусреднее кольцо и дано отображение S → R как алгебра. Тогда кватернионовая алгебра ℍ[R,c1,c2,c3] естественным образом становится S-алгеброй совместимой с данными отображениями базового кольца.
LaTeX
$$$\mathbb{H}(R;c_1,c_2,c_3) \text{ is an } S\text{-algebra with structure maps induced by } S \to R.$$$
Lean4
instance [CommSemiring S] [Algebra S R] : Algebra S ℍ[R,c₁,c₂,c₃]
where
smul := (· • ·)
algebraMap :=
{ toFun s := coe (algebraMap S R s)
map_one' := by simp only [map_one, coe_one]
map_zero' := by simp only [map_zero, coe_zero]
map_mul' x y := by simp only [map_mul, coe_mul]
map_add' x y := by simp only [map_add, coe_add] }
smul_def' s x := by ext <;> simp [Algebra.smul_def]
commutes' s x := by ext <;> simp [Algebra.commutes]
