The basic perturbation lemma #
Let c be a special contraction of (M, dM) onto (N, dN), with inclusion i, projection p
and homotopy h, and let δ be a perturbation of dM: an endomorphism with
(dM + δ)² = dM², so that dM + δ is again a differential whenever dM is. The basic
perturbation lemma transports the contraction to the perturbed differential. Its engine is the
operator
X = (1 + δ h)⁻¹ δ = δ (1 + h δ)⁻¹,
which is the geometric series ∑ₙ (-1)ⁿ (δ h)ⁿ δ whenever that series is pointwise finite; the
lemma is stated for any δ making 1 + δ h invertible, and
Module.End.isUnit_one_add_of_forall_exists_pow_apply_eq_zero supplies the pointwise-finite case.
The operator satisfies the identity
dM X + X dM + X i p X = 0 (LinearSpecialContraction.perturbationSeries_maurerCartan),
and the perturbed contraction is
i' = i - h X i, p' = p - p X h, h' = h - h X h, dN' = dN + p X i.
The square of the perturbed differential of the retract is the square of the original one, so it
is a differential whenever dN is.
With the convention 1 - i p = dM h + h dM fixed in TauCeti.Contraction, the homotopy is the
negative of the one in Crainic's account, whence the signs 1 + δ h and - h X i in place of
Crainic's 1 - δ h and + h X i.
Main definitions #
TauCeti.LinearSpecialContraction.perturbationSeries: the operatorX = (1 + δ h)⁻¹ δ.TauCeti.LinearSpecialContraction.perturbedDifferential: the endomorphismdN + p X i.TauCeti.LinearSpecialContraction.perturb: the basic perturbation lemma, as a special contraction of(M, dM + δ)onto(N, dN + p X i).
Main results #
TauCeti.LinearSpecialContraction.perturbationSeries_maurerCartan: the identitydM X + X dM + X i p X = 0.TauCeti.LinearSpecialContraction.perturbedDifferential_comp_perturbedDifferential: the perturbed differential of the retract squares to the square ofdN.TauCeti.LinearSpecialContraction.perturb_proj_comp_one_add_mul: the perturbed projection satisfiesp' (1 + δ h) = p; withTauCeti.LinearSpecialContraction.perturb_proj_comp_homotopyandTauCeti.LinearSpecialContraction.perturb_proj_comp_incl, these are the identities makingp'a coalgebra morphism in the tensor trick.
References #
- E. H. Brown, Twisted tensor products, I, Annals of Mathematics 69 (1959), 223--246.
- V. K. A. M. Gugenheim, L. A. Lambe, and J. D. Stasheff, Perturbation theory in differential homological algebra II, Illinois Journal of Mathematics 35 (1991), 357--373.
- M. Crainic, On the perturbation lemma, and deformations, Section 2.
The perturbation operator X = (1 + δ h)⁻¹ δ of the basic perturbation lemma. When
δ h is locally nilpotent it is the pointwise finite geometric series ∑ₙ (-1)ⁿ (δ h)ⁿ δ. It is
defined through Ring.inverse, so it is meaningful for every δ, and satisfies its defining
equations once 1 + δ h is a unit.
Equations
- c.perturbationSeries δ = Ring.inverse (1 + δ * c.homotopy) * δ
Instances For
The perturbation operator is Ring.inverse (1 + δ h) composed with δ.
(1 + δ h) X = δ.
X (1 + h δ) = δ.
The fixed-point equation δ h X = δ - X of the perturbation operator.
The fixed-point equation X h δ = δ - X of the perturbation operator.
The fixed-point equation δ h X = δ - X, with a further factor on the right.
The fixed-point equation X h δ = δ - X, with a further factor on the right.
On an element z killed by (δ h)^k δ, the perturbation operator is the finite geometric
series ∑_{j < k} (-1)^j (δ h)^j δ.
The perturbation operator satisfies the Maurer--Cartan-type identity
dM X + X dM + X i p X = 0; this single identity drives every equation of the basic
perturbation lemma.
The perturbed endomorphism dN + p X i of the retract.
Equations
- c.perturbedDifferential δ = dN + c.proj ∘ₗ c.perturbationSeries δ ∘ₗ c.incl
Instances For
The perturbed differential of the retract is dN + p X i.
The perturbed differential of the retract squares to the square of dN; in particular it is
a differential whenever dN is.
If dM squares to zero, so does the perturbed differential of the retract.
The basic perturbation lemma. A special contraction of (M, dM) onto (N, dN) and a
perturbation δ of dM with 1 + δ h invertible yield a special contraction of (M, dM + δ)
onto (N, dN + p X i), with
i' = i - h X i, p' = p - p X h, h' = h - h X h,
where X = (1 + δ h)⁻¹ δ is the perturbation operator.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The inclusion of the perturbed contraction is i - h X i.
The projection of the perturbed contraction is p - p X h.
The homotopy of the perturbed contraction is h - h X h.
The perturbed projection annihilates the unperturbed homotopy, p' h = 0.
The perturbed projection is a left inverse of the unperturbed inclusion, p' i = 1.
The perturbed projection solves the fixed-point equation p' (1 + δ h) = p.
The perturbed homotopy solves the fixed-point equation h' (1 + δ h) = h.
The perturbed inclusion satisfies i' + h' δ i = i.