Change of connection for curved differential graded algebras

Let (A, d, w) be a curved differential graded algebra and let a ∈ A¹. Adding the graded commutator with a to the differential,

d^a b = d b + a * b - (-1) ^ |b| • (b * a),

gives a new degree-one derivation, and (A, d^a, w^a) is again a curved differential graded algebra with the changed curvature

w^a = w - d a - a * a.

This is the algebraic form of changing a connection by a one-form: the curvature changes by the covariant derivative and the square of the connection form. Applied to an ordinary differential graded algebra, it produces a curved one of curvature -(d a + a * a), which need not vanish. Twisting by a connection form is not a strict morphism of curved differential graded algebras; strict morphisms preserve both d and w.

Main definitions

• TauCeti.oddInnerDerivation 𝒜 a: the graded commutator b ↦ a * b - (-1) ^ |b| • (b * a) with an element a of odd degree.
• TauCeti.connectionChange 𝒜 d a: the differential d + [a, -] twisted by the connection form a.

Main results

• TauCeti.IsCurvedDGAlgebra.connectionChange: for a ∈ A¹, the twisted differential d^a and the curvature w - d a - a * a form a curved differential graded algebra.
• TauCeti.IsDGAlgebra.connectionChange: twisting a differential graded algebra gives a curved one of curvature -(d a + a * a).

References

• L. Positselski, Differential graded Koszul duality: an introductory survey, Section 6.2, for the change-of-connection formula; the sign of the curvature follows the right-module convention d (d a) = a * w - w * a of TauCeti.IsCurvedDGAlgebra, so it is the negative of his.

The twisted differential

The twisted differential only uses the homogeneous decomposition of A, through the Koszul sign of TauCeti.InternalGrading.koszulTwist; the graded multiplication enters only in the change-of-connection theorems below.

noncomputable def TauCeti.oddInnerDerivation {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (𝒜 : ℤ → Submodule R A) [DirectSum.Decomposition 𝒜] (a : A) : A →ₗ[R] A

The graded commutator with an element a of odd degree: on a homogeneous element b, a * b - (-1) ^ |b| • (b * a). For a of degree one this is the inner derivation by a.

Equations

• TauCeti.oddInnerDerivation 𝒜 a = LinearMap.mulLeft R a - LinearMap.mulRight R a ∘ₗ (TauCeti.InternalGrading.ofDecomposition 𝒜).koszulTwist 1

theorem TauCeti.oddInnerDerivation_apply_of_mem {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (𝒜 : ℤ → Submodule R A) [DirectSum.Decomposition 𝒜] {p : ℤ} {b : A} (hb : b ∈ 𝒜 p) (a : A) : (oddInnerDerivation 𝒜 a) b = a * b - p.negOnePow • (b * a)

The value of the odd inner derivation on a homogeneous element.

noncomputable def TauCeti.connectionChange {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (𝒜 : ℤ → Submodule R A) [DirectSum.Decomposition 𝒜] (d : A →ₗ[R] A) (a : A) : A →ₗ[R] A

The differential twisted by a connection form a: d + [a, -], where [a, -] is the graded commutator with a.

Equations

• TauCeti.connectionChange 𝒜 d a = d + TauCeti.oddInnerDerivation 𝒜 a

@[simp]

theorem TauCeti.connectionChange_apply {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (𝒜 : ℤ → Submodule R A) [DirectSum.Decomposition 𝒜] (d : A →ₗ[R] A) (a b : A) : (connectionChange 𝒜 d a) b = d b + (oddInnerDerivation 𝒜 a) b

The twisted differential is the differential plus the odd inner derivation.

theorem TauCeti.connectionChange_apply_of_mem {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] (𝒜 : ℤ → Submodule R A) [DirectSum.Decomposition 𝒜] {p : ℤ} {b : A} (hb : b ∈ 𝒜 p) (d : A →ₗ[R] A) (a : A) : (connectionChange 𝒜 d a) b = d b + a * b - p.negOnePow • (b * a)

The value of the twisted differential on a homogeneous element.

theorem TauCeti.IsCurvedDGAlgebra.connectionChange {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] {𝒜 : ℤ → Submodule R A} [GradedAlgebra 𝒜] {d : A →ₗ[R] A} {w : A} (h : IsCurvedDGAlgebra 𝒜 d w) {a : A} (ha : a ∈ 𝒜 1) : IsCurvedDGAlgebra 𝒜 (TauCeti.connectionChange 𝒜 d a) (w - d a - a * a)

Change of connection. Twisting the differential of a curved differential graded algebra by a connection form a of degree one changes the curvature to w - d a - a * a.

theorem TauCeti.IsDGAlgebra.connectionChange {R : Type u_1} {A : Type u_2} [CommRing R] [Ring A] [Algebra R A] {𝒜 : ℤ → Submodule R A} [GradedAlgebra 𝒜] {d : A →ₗ[R] A} (h : IsDGAlgebra 𝒜 d) {a : A} (ha : a ∈ 𝒜 1) : IsCurvedDGAlgebra 𝒜 (TauCeti.connectionChange 𝒜 d a) (-(d a + a * a))

Twisting the differential of a differential graded algebra by a connection form a of degree one gives a curved differential graded algebra of curvature -(d a + a * a).