Theorems · Inductive type · nonassociative algebras

LieRinehartAlgebra.Hom

{R : Type u_1} →
  {A₁ : Type u_2} →
    {A₂ : Type u_4} →
      [inst : CommRing R] →
        [inst_1 : CommRing A₁] →
          [inst_2 : CommRing A₂] →
            [inst_3 : Algebra R A₁] →
              [inst_4 : Algebra R A₂] →
                (A₁ →ₐ[R] A₂) →
                  (L₁ : Type u_8) →
                    (L₂ : Type u_9) →
                      [inst_5 : LieRing L₁] →
                        [Module A₁ L₁] →
                          [LieRingModule L₁ A₁] →
                            [LieAlgebra R L₁] →
                              [inst_9 : LieRing L₂] →
                                [Module A₂ L₂] → [LieRingModule L₂ A₂] → [LieAlgebra R L₂] → Type (max u_8 u_9)

A morphism of Lie-Rinehart algebras, from (A₁, L₁) to (A₂, L₂), consists of a pair of maps (σ, F) where σ : A₁ → A₂ is a morphism of algebras and F is a morphism of Lie algebras, which respect the module structures. Here we define the type of such morphisms with fixed σ (which can be regarded as functions L₁ → L₂). In the future it may be useful to define the type of such morphisms with fixed F (which can be regarded as functions A₁ → A₂) and the type of all such morphisms (which can be regarded as functions A₁ × L₁ → A₂ × L₂).

Defined in

Mathlib.Algebra.LieRinehartAlgebra.Defs

Cited by

9 results in Mathlib

Foundations

Depth 11 from the axioms · uses no axioms

Assumes

CommRingCommRingCommRingAlgebraAlgebraLieRingModuleLieRingModuleLieAlgebraLieRingModuleLieRingModuleLieAlgebra