The Diffiety Institute Preprint Series

Preprint DIPS–5/2006 October 30, 2006

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Iterated Differential Forms IV: C–Spectral Sequence

by

A. M. Vinogradov and L. Vitagliano

DIPS-5/2006

                                                 arXiv:math.DG/0610917

Iterated Differential Forms IV: C–Spectral Sequence

A. M. VINOGRADOV AND L. VITAGLIANO

Abstract. For the multiple differential algebra of iterated differential forms [1, 2, 3, 4] on a diffiety (O, C) [5, 6] an analogue of C–spectral sequence [7, 8, 6] is constructed. The first term of it is naturally interpreted as the algebra of secondary [5, 6, 9] iterated differential forms on (O, C). This allows to develop secondary tensor analysis on generic diffieties, some simplest elements of which are sketched here. The presented here general theory will be specified to infinite jet spaces and infinitely prolonged PDEs in subsequent notes.

1. Introduction

Let (O, C) be a diffiety. Recall that an elementary diffiety is a pair (E^{∞}, C(E)), where E^{∞} is the infinite prolongation of a system E of partial differential equations and C(E) the Cartan distribution on it (see [5, 7, 8]). More generally, a diffiety is a smooth, usually pro–finite manifold O endowed with a finite dimensional integrable distribution C, locally identical to an elementary diffiety (see [6, 9]). Recall that Secondary Calculus on the diffiety (E^{∞}, C(E)) is the proper formalization of the idea of differential calculus on the “space of all solutions of a system of partial differential equations E”. By this reason the search for analogues in secondary calculus of objects, concepts and constructions of “traditional” mathematics is an actual and important problem, called the secondarization problem (see [6, 9]). In this note this problem is solved for iterated differential forms (in the sequel, IDFs) on the basis of the algebraic approach developed in preceding notes [1, 2] of this series. Below we use the notation, definitions and results of these notes. It is worth noticing that the alternative approach to iterated forms proposed in [3, 4] is not very secondarization friendly. Recall that the solution of the secondarization problem for usual, i.e., one time iterated forms, is given in [1, 2] (see also [6, 9]). Namely, the algebra of secondary differential forms is realized as the first term of the C–spectral sequence. In this note an analogue of this sequence is constructed and its first term is interpreted as the algebra of secondary IDFs.

2000 Mathematics Subject Classification. 58A10, 58A50, 13N99.

Key words and phrases. Differential Calculus, Graded Algebras, Differential Forms, Integral Forms.

2. Preliminaries on IDFs

Let O be a smooth, generally, pro–finite manifold, i.e., the inverse limit of a sequence of finite–dimensional submersions

⋯ ⟵ O_{s−1} ⟵ O_{s} ⟵ O_{s+1} ⟵ · · · .  (1)

Denote by (Λ_{k}(O_{s}), d_{1}, …, d_{k}) the ℤ^{k}–graded multiple differential algebra of geometric k–times IDFs on the (usual) manifold O_{s} (see [1, 2]). Sequence (1) induces the sequence of embeddings of graded multiple differential algebras

⋯ ⊂ Λ_{k}(O_{s−1}) ⊂ Λ_{k}(O_{s}) ⊂ Λ_{k}(O_{s+1}) ⊂ · · · .

Then Λ_{k}(O) =^{def} _{⋃s}Λ_{k}(O_{s}) is a ℤ^{k}–graded multiple differential algebra.

Definition 1. Elements of the algebra Λ_{k}(O) are called k–times IDFs on the pro–finite manifold O.

The theory of IDFs on pro–finite manifolds is a straightforward generalization of the theory presented in [1, 2] and by this reason we shall not go into its details here. Denote by D(Λ_{k}(O), Λ_{m}(O)) the Λ_{m}(O)–module of graded Λ_{m}(O)–valued derivations of Λ_{k}(O), k ≤ m.

Proposition 2. Let X ∈ D(C^{∞}(O), Λ_{k}(O)) and K ⊂ {1, …, k}. Then there exists a unique i_{X}^{K} ∈ D(Λ_{k}(O), Λ_{k}(O)) such that

i_{X}^{K}(d_{K′}f) = \{ \begin{matrix}d_{K′∖K}(X(f)), & if K ⊂ K′ \\ 0, & otherwise \end{matrix} , K′ ⊂ \{1, …, k\}.

For instance, if i_{m}^{K} =^{def} i_{d_{m}|_{C^{∞}(O)}}^{K} ∈ D(Λ_{k}(O), Λ_{k}(O)), m ≤ k, K ⊂ {1, …, k}, then i_{m}^{∅} = d_{m}. Recall that, according to proposition 3 of [1] covariant k–tensors on O are naturally interpreted as k–times IDFs. Namely, for any k there exists an injective morphism of C^{∞}(O)–modules ι_{k} : T_{k}^{0}(O) ↪ Λ_{k}(O). Then proposition 4 of [1] characterizes those IDFs that represent covariant tensors. Below we give another useful characterization of such IDFs.

Proposition 3. A homogeneous element ω ∈ Λ_{k}(O) of multi–degree (1, …, 1, 1) ∈ ℤ^{k} is a covariant tensor (i.e., ω ∈ im ι_{k}) iff i_{m}^{K}(ω) = 0 for any m < k and K = {k_{1}, …, k_{s}} ⊂ {1, …, k}, s ≥ 2.

3. Secondary IDFs

ITERATED DIFFERENTIAL FORMS IV: C–SPECTRAL SEQUENCE 3

Let (Λ_{k}(O), d_{1}, …, d_{k}) be the multiple complex of k–times IDFs on O and κ_{1k} : Λ_{k}(O) ⟶ Λ_{k}(O) the involution that interchanges differentials d_{1} and d_{k}, k ≥ 1. The ideal CΛ(O) ⊂ Λ(O) of (ordinary) differential forms on O vanishing on C(O) is embedded naturally into Λ_{k}(O) due to the embedding Λ(O) ≡ Λ_{1}(O) ⊂ Λ_{k}(O). Denote by CΛ_{k}(O) ⊂ Λ_{k}(O) the ideal generated by elements in the form

(d_{K} ∘ κ_{1k})(ω), ω ∈ CΛ(O), K ⊂ {1, …, k − 1}.

It is multi–differential, i.e.,

d_{m}(CΛ_{k}(O)) ⊂ CΛ_{k}(O), m ≤ k.  (2)

Definition 4. Elements of CΛ_{k}(O) are called k–Cartan k–times IDFs.

Let C^{p}Λ_{k}(O) be the p–th power of CΛ_{k}(O). Then the sequence

Λ_{k}(O) ⊃ CΛ_{k}(O) ⊃ C^{2}Λ_{k}(O) ⊃ ⋯ ⊃ C^{p}Λ_{k}(O) ⊃ ⋯,  (3)

is a filtration of the multi–graded differential algebra (Λ_{k}(O), d_{k}) by means of multi– graded differential ideals.

Definition 5. The spectral sequence Λ_{k−1}CE(O) = {(Λ_{k−1}CE_{r}(O), d_{k,r})}_{r} associated with filtration (3) is called the Λ_{k−1}C–spectral sequence of the diffiety (O, C).

In particular, Λ_{0}CE(O) = CE(O). Filtration (3) is regular and, therefore, the Λ_{k−1}C–spectral sequence converges to the cohomology H(Λ_{k}(O), d_{k}). Hence,

Λ_{k−1}CE_{∞}(O) ≃ H(Λ_{k}(O), d_{k}) ≃ H(O).

It follows from (2) and commutation relations [d_{m}, d_{k}] = 0, m < k, that d_{m} induces a well defined differential in each term of the Λ_{k−1}C–spectral sequence. Denote by d_{m,r} the so-obtained differential in the term Λ_{k−1}CE_{r}(O). Obviously, [d_{m,r}, d_{n,r}] = 0, m, n ≤ k, so that (Λ_{k−1}CE_{r}(O), d_{1,r}, …, d_{k,r}) is a multiple differential algebra. The zeroth term of the Λ_{k−1}C–spectral sequence is described as follows. Let C^{p}Λ_{k}^{p}(O) denote the Λ_{k−1}(O)–module of k–times iterated p–forms in C^{p}Λ_{k}(O). Note that C^{p}Λ_{k}(O) = C^{p}Λ_{k}^{p}(O) ∧ Λ_{k}(O) and put

HΛ_{k}(O) =^{def} Λ_{k}(O) / CΛ_{k}(O) = Λ_{k−1}CE_{0}(O).

Definition 6. (HΛ_{k}(O), d_{k,0}) is called the differential algebra of horizontal k–times IDFs on the diffiety (O, C).

Proposition 7. There exists a natural isomorphism

Λ_{k−1}CE_{0}^{p,•}(O) ≃ C^{p}Λ_{k}^{p}(O) ⊗_{Λ_{k−1}} HΛ_{k}(O).

The isomorphism of proposition 7 supplies C^{p}Λ_{k}^{p}(O)⊗_{Λ_{k−1}}HΛ_{k}(O) with a structure of a differential algebra. The corresponding differential \overline{d}_{k} looks as follows. Let ω ∈ C^{p}Λ_{k}^{p}(O) and \overline{ρ} = [ρ]_{CΛ_{k}(O)} ∈ HΛ_{k}(O), ρ ∈ Λ_{k}(O). In its turn d_{k}ω ∈ C^{p}Λ_{k}(O). Therefore, d_{k}ω = ∑_{α} ω_{α} ∧ σ_{α} for some ω_{α} ∈ C^{p}Λ_{k}^{p}(O) and σ_{α} ∈ Λ_{k}(O). Then

\overline{d}_{k}(ω ⊗ \overline{ρ}) = _{∑α}ω_{α} ⊗ \overline{σ_{α} ∧ ρ} + (−1)^{p}ω ⊗ d_{k,0}\overline{ρ} ∈ C^{p}Λ_{k}^{p}(O) ⊗_{Λ_{k−1}} HΛ_{k}(O),

where \overline{σ_{α} ∧ ρ} = [σ_{α} ∧ ρ]_{CΛ_{k}(O)} ∈ HΛ_{k}(O).

Definition 8. (Λ_{k−1}CE_{1}(O), d_{1,1}, …, d_{k,1}) is called the multiple differential algebra of secondary k–times IDFs. Accordingly, elements of Λ_{k−1}CE_{1}^{p,•}(O) are called secondary k–times iterated differential p–forms.

It takes place the secondary analogue of the fact that (k + 1)–times iterated 0–forms coincide with k–times iterated ones. Indeed, it holds the

Proposition 9. There exists a natural isomorphism of graded multiple differential algebras

φ_{k−1} : (Λ_{k−1}CE_{1}(O), d_{1,1}, …, d_{k,1}) ⟶ (Λ_{k}CE_{1}^{0,•}(O), d_{1,1}, …, d_{k,1}).

Since Λ_{k−1}CE_{1}^{0,•}(O) = H(HΛ_{k}(O), d_{k,0}), according to the above proposition secondary k–times IDFs may be understood either as elements in the first term of the Λ_{k−1}C– spectral sequence or as cohomologies of the differential algebra of horizontal (k + 1)– times IDFs. The standard action of the permutation group S_{k} on IDFs is secondarized as follows. Let σ ∈ S_{k−1} be a permutation of {1, …, k − 1}. The associated automorphism κ_{σ} : Λ_{k}(O) ⟶ Λ_{k}(O) respects both the differential d_{k} and the filtration (3), i.e., κ_{σ} ∘ d_{k} = d_{k} ∘ κ_{σ} and κ_{σ}(CΛ_{k}(O)) ⊂ CΛ_{k}(O). Therefore, κ_{σ} induces an automorphism of the Λ_{k−1}C–spectral sequence. Abusing the notation we continue denoting by κ_{σ} the induced automorphism in the first term of Λ_{k−1}CE(O). In particular, κ_{σ} : Λ_{k−1}CE_{1}(O) ⟶ Λ_{k−1}CE_{1}(O) is an isomorphism of the multiple differential algebras (Λ_{k−1}CE_{1}(O), d_{1,1}, …, d_{k,1}) and (Λ_{k−1}CE_{1}(O), d_{σ(1),1}, …, d_{σ(k−1),1}, d_{k,1}). Proposition 9 allows to define the action of the transposition τ exchanging k and m < k on Λ_{k−1}CE_{1}(O). Namely, observe that the action of κ_{τ} on Λ_{k}CE_{1}^{0,•}(O) is well defined as it was explained above and the isomorphism φ_{k−1} transfers it then to Λ_{k−1}CE_{1}(O). Since CΛ_{k}(O) is a multi–graded differential ideal, Λ_{k−1}CE_{1}^{p,•}(O) inherits a multi– graded algebra structure. Namely, let θ = [[ω]_{C^{p+1}Λ_{k}(O)}]_{imd_{k,0}} ∈ Λ_{k−1}CE_{1}^{p,•}(O), ω ∈ C^{p}Λ_{k}(O), d_{k}ω ∈ C^{p+1}Λ_{k}(O). Then

θ ∈ Λ_{k−1}CE_{1}^{(p_{1},...,p_{k−1},p),q}(O) ⟺^{def} ω has multi–degree (p_{1}, …, p_{k−1}, p + q) ∈ ℤ^{k}, (4)

p_{1}, …, p_{k−1}, q ≥ 0. Definition (4) is correct and

Λ_{k−1}CE_{1}^{p,•}(O) = ⨁_{p_{1},...,p_{k−1},q} Λ_{k−1}CE_{1}^{(p_{1},...,p_{k−1},p),q}(O).

Proposition 10. Let σ ∈ S_{k} be a permutation of {1, …, k}. Then

κ_{σ^{−1}}(Λ_{k−1}CE_{1}^{(p_{1},...,p_{k−1},p_{k}),q}) ⊂ Λ_{k−1}CE_{1}^{(p_{σ(1)},...,p_{σ(k−1)},p_{σ(k)}),q}.

4. IDF–symmetries of a diffiety

Iterated analogues of contact and, respectively, trivial contact vector fields on a diffiety are defined as follows (see [5, 6]):

D_{C}(Λ_{k−1}(O)) =^{def} {X ∈ D(Λ_{k−1}(O), Λ_{k−1}(O)) | L_{X}^{{k}}(CΛ_{k}(O)) ⊂ CΛ_{k}(O)}

and

CD(Λ_{k−1}(O)) =^{def} {X ∈ D(Λ_{k−1}(O), Λ_{k−1}(O)) | i_{X}^{{k}}(CΛ_{k}(O)) = 0}.

D_{C}(Λ_{k−1}(O)) is a graded sub–algebra of the Lie algebra D(Λ_{k−1}(O)) and CD(Λ_{k−1}(O)) is an its graded ideal. Denote by

Λ_{k−1}Sym(O) =^{def} D_{C}(Λ_{k−1}(O)) / CD(Λ_{k−1}(O)).

the quotient Lie algebra. In particular, Λ_{0}Sym(O) = Sym(O).

Definition 11. Elements in Λ_{k−1}Sym(O) are called (k − 1)–IDF–symmetries of the diffiety (O, C).

Just as symmetries of (O, C) act naturally on secondary differential forms, IDF– symmetries of (O, C) act naturally on secondary IDFs. Namely, let χ = [X] ∈ Λ_{k−1}Sym(O), X ∈ D_{C}(Λ_{k−1}(O)). The Lie derivative L_{X}^{{k}} : Λ_{k}(O) ⟶ Λ_{k}(O) respects both the differential d_{k} and the filtration (3), i.e., L_{X}^{{k}} ∘ d_{k} = d_{k} ∘ L_{X}^{{k}} and L_{X}^{{k}}(CΛ_{k}(O)) ⊂ CΛ_{k}(O). Therefore, L_{X}^{{k}} induces a morphism of the Λ_{k−1}C–spectral sequence. Denote by

L_{χ}^{{k}} : Λ_{k−1}CE_{1}^{p,•}(O) ⟶ Λ_{k−1}CE_{1}^{p,•}(O)

its action on the first term since it doesn’t depend on the choice of the representative X of χ. L_{χ}^{{k}} is a graded derivation of Λ_{k−1}CE_{1}(O). Moreover, if χ_{1}, χ_{2} ∈ Λ_{k−1}Sym(O), then

[L_{χ_{1}}^{{k}}, d_{k,1}] = 0, [L_{χ_{1}}^{{k}}, L_{χ_{2}}^{{k}}] = L_{[χ_{1},χ_{2}]}^{{k}}.

The definition of L_{χ}^{{k}} is consistent with the natural isomorphism Λ_{k−1}CE_{1}(O) ≃ Λ_{k}CE_{1}^{0,•}(O). Namely, let χ and X be as above, so that L_{X}^{{k}} ∈ D_{C}(Λ_{k}(O)). Put χ′ =^{def} [L_{X}^{{k}}] ∈ Λ_{k}Sym(O). Then

L_{χ}^{{k}} = φ_{k−1}^{−1} ∘ L_{χ′}^{{k+1}}|_{Λ_{k}CE_{1}^{0,•}(O)} ∘ φ_{k−1}.

Now observe that i_{m}^{K} ∈ D_{C}(Λ_{k−1}(O)) for any m < k and K ⊂ {1, …, k − 1} and put

I_{m}^{K} =^{def} [i_{m}^{K}] ∈ Λ_{k−1}Sym(O).

Then d_{m,1} = L_{I_{m}^{∅}}^{{k}}. IDF–symmetries of (O, C) can be also inserted into secondary IDFs as follows. Indeed, let χ and X be as above and θ = [[ω]_{C^{p+1}Λ_{k}(O)}]_{im d_{k,0}} ∈ Λ_{k−1}CE_{1}^{p,•}(O), ω ∈ C^{p}Λ_{k}(O), d_{k}ω ∈ C^{p+1}Λ_{k}(O), be a secondary IDF. Then i_{X}^{{k}}ω ∈ C^{p−1}Λ_{k}(O) and d_{k}(i_{X}^{{k}}ω) = L_{X}^{{k}}ω − i_{X}^{{k}}(d_{k}ω) ∈ C^{p}Λ_{k}(O) so that

i_{χ}^{{k}}θ =^{def} [[i_{X}^{{k}}ω]_{C^{p}Λ_{k}(O)}]_{im d_{k,0}} ∈ Λ_{k−1}CE_{1}^{p−1,•}(O)

is a well defined secondary IDF. Moreover, the map

i_{χ}^{{k}} : Λ_{k−1}CE_{1}^{p,•}(O) ⟶ Λ_{k−1}CE_{1}^{p−1,•}(O)

is a graded derivation of Λ_{k−1}CE_{1}(O) not depending on the choice of X in χ. For χ_{1}, χ_{2} ∈ Λ_{k−1}Sym(O) the following secondary analogues of well-known (graded) commutation relations hold

[i_{χ_{1}}^{{k}}, d_{k,1}] = L_{χ_{1}}^{{k}}, [i_{χ_{1}}^{{k}}, i_{χ_{2}}^{{k}}] = 0, [i_{χ_{1}}^{{k}}, L_{χ_{2}}^{{k}}] = i_{[χ_{1},χ_{2}]}^{{k}}.

The definition of i_{χ}^{{k}} is consistent with the natural isomorphism Λ_{k−1}CE_{1}(O) ≃ Λ_{k}CE_{1}^{0,•}(O). Namely, if χ and X be as above, then i_{X}^{{k}} ∈ D_{C}(Λ_{k}(O)) and

i_{χ}^{{k}} = φ_{k−1}^{−1} ∘ L_{χ′′}^{{k+1}}|_{Λ_{k}CE_{1}^{0,•}(O)} ∘ φ_{k−1}.

with χ′′ =^{def} [i_{X}^{{k}}] ∈ Λ_{k}Sym(O).

Definition 12. Let τ ∈ Λ_{k−1}CE_{1}^{(1,...,1,1),•}(O) be a secondary IDF of multi–degree (1, …, 1, 1). τ is called a secondary covariant k–tensor iff L_{I_{m}^{K}}^{{k}}τ = i_{I_{m}^{K′}}^{{k}}τ = 0 for any m < k and K = {k_{1}, …, k_{s}}, K′ = {k′_{1}, …, k′_{s′}} ⊂ {1, …, k − 1}, s ≥ 2, s′ ≥ 1.

Definition 12 mimics characterization of covariant tensors given in proposition 3.

5. Other related C–Spectral Sequences

Now on put C_{k}Λ_{k}(O) = CΛ_{k}(O) and observe that C_{k}Λ_{k}(O) is not unique canonical multi–differential ideal in Λ_{k}(O). Indeed, such are ideals

C_{m}Λ_{k}(O) =^{def} κ_{mk}(C_{k}Λ_{k}(O)),

for any m ≤ k. Here κ_{mk} : Λ_{k}(O) ⟶ Λ_{k}(O) is the involution that interchanges differentials d_{m} and d_{k}. More generally, let K = {m_{1}, …, m_{r}} ⊂ {1, …, k}. Put

C_{K}Λ_{k}(O) =^{def} C_{m_{1}}Λ_{k}(O) + ⋯ + C_{m_{r}}Λ_{k}(O).

Then C_{K}Λ_{k}(O) ⊂ Λ_{k}(O) is a multi–differential for any K ⊂ {1, …, k}.

Definition 13. Elements in C_{K}Λ_{k}(O) are called K–Cartan k–times IDFs.

All these ideals transform (Λ_{k}(O), d_{k}) into a multi–filtered complex. Homological algebra of such kind complexes and appearing this way numerous C–spectral–like sequences will be considered separately. They enrich noteworthy secondary calculus and, as a consequence, geometrical theory of nonlinear PDEs with new powerful instruments.

References

[1] A. M. Vinogradov, L. Vitagliano, Dokl. Math. 73, n^{∘} 2 (2006) 169, see also The Diffiety

   Inst. Preprint Series, DIPS 1/06 ( http://diffiety.ac.ru/preprint/2006/01-06abs.htm ) or
   arXiv:math.DG/0605113 .

[2] A. M. Vinogradov, L. Vitagliano, Dokl. Math. 73, n^{∘} 2 (2006) 182, see also The Diffiety

   Inst. Preprint Series, DIPS 3/06 ( http://diffiety.ac.ru/preprint/2006/03-06abs.htm ) or
   arXiv:math.DG/06009287 .

[3] D. Kochan, P. Severa, arXiv:math.DG/0307303 .

[4] P. Severa, arXiv:math.DG/0606645 .

[5] A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor’kova, I. S. Krasil’shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, and A. M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, AMS “Translation of Mathematical Monographs ” vol. 182 (1999).

[6] A. M. Vinogradov, Cohomological Analysis of Partial Differential Equations and Secondary Calculus, AMS “Translation of Mathematical Monographs ” vol. 204 (2001).

[7] A. M. Vinogradov, Soviet Math. Dokl. 19 (1978) 144.

[8] A. M. Vinogradov, J. Math. Anal. Appl. 100 (1984) 1.

[9] A. M. Vinogradov, in M. Henneaux, I. Krasil’shchik, and A. Vinogradov (Eds.), Secondary Calculus and Cohomological Physics, Contemporary Mathematics 219, AMS (1998). See also The Diffiety Inst. Preprint Series, DIPS 5/98.

Alexandre Vinogradov, DMI, Facoltà di Scienze MM. FF. NN. Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA) and INFN, sez. Napoli–Salerno, Italy

  E-mail address: vinograd@unisa.it

Luca Vitagliano, DMI, Facoltà di Scienze MM. FF. NN. Università degli Studi di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA) and INFN, sez. Napoli–Salerno, Italy

  E-mail address: luca vitagliano@fastwebnet.it