English
There is a noncomputable abbreviation that upgrades a bialgebra to a Hopf algebra by specifying an antipode as an algebra homomorphism, together with compatibility equations with comultiplication and counit.
Русский
Существуют требования к аппарату: можно превратить билагебру в гензелеву алгебру, указав antipode как алгебра-гомоморфизм и требования совместимости с коумулей и counit.
LaTeX
$$$\text{ofAlgHom}(antipode, mul\_antipode\_rTensor\_comul, mul\_antipode\_lTensor\_comul) : HopfAlgebra R A$$$
Lean4
/-- Upgrade a bialgebra to a Hopf algebra by specifying the antipode as an algebra map with
appropriate conditions. -/
noncomputable abbrev ofAlgHom {R A : Type*} [CommSemiring R] [CommSemiring A] [Bialgebra R A] (antipode : A →ₐ[R] A)
(mul_antipode_rTensor_comul :
((Algebra.TensorProduct.lift antipode (.id R A) fun _ ↦ .all _).comp (Bialgebra.comulAlgHom R A)) =
(Algebra.ofId R A).comp (Bialgebra.counitAlgHom R A))
(mul_antipode_lTensor_comul :
(Algebra.TensorProduct.lift (.id R A) antipode fun _ _ ↦ .all _ _).comp (Bialgebra.comulAlgHom R A) =
(Algebra.ofId R A).comp (Bialgebra.counitAlgHom R A)) :
HopfAlgebra R A where
antipode := antipode
mul_antipode_rTensor_comul :=
by
rw [← Algebra.TensorProduct.lmul'_comp_map] at mul_antipode_rTensor_comul
exact congr(($mul_antipode_rTensor_comul).toLinearMap)
mul_antipode_lTensor_comul :=
by
rw [← Algebra.TensorProduct.lmul'_comp_map] at mul_antipode_lTensor_comul
exact congr(($mul_antipode_lTensor_comul).toLinearMap)
