PROBLEMS TO THE COURSE “ HOMOLOGICAL ASPECTS OF NONLINEAR DIFFERENTIAL EQUATIONS”

INDEPENDENT UNIVERSITY OF MOSCOW october – december 1998

Joseph Krasil’shchik and Alexander Verbovetsky

Complexes and spectral sequences

Problem 1. Let V be a vector space and W ⊂ V be its subspace. Show that there exists a unique automorphism d of the bigraded algebra S^{∗}(V) ⊗ Λ^{∗}(W)^{1} satisfying the conditions

1. d(S^{p}(V) ⊗ Λ^{q}(W)) ⊂ S^{p+1}(V) ⊗ Λ^{q−1}(W);

2. d^{2} = 0;

3. d(u_{1}u_{2}) = d(u_{1})u_{2} + (−1)^{q}u_{1}d(u_{2}), where u_{1} ∈ S^{∗}(V) ⊗ Λ^{q}(W) and u_{2} ∈ S^{∗}(V) ⊗ Λ^{∗}(W);

4. d(1 ⊗ w) = w ⊗ 1 for w ∈ W. Thus the space S^{∗}(V) ⊗ Λ^{∗}(W) endowed with the grading inherited from the space Λ^{∗}(W) and with the differential d is a complex.

Compute the homology of this complex.

Compute the homology of the complex

… ⟵ Der_{k−1}(V) ⟵ Der_{k}(V) ⟵ …,

where Der_{k}(V) is the set of graded derivations of order k of the algebra S^{∗}(V) ⊗ Λ^{∗}(W), i.e., of the mappings satisfying the Leibniz rule

Δ(u_{1}u_{2}) = Δ(u_{1})u_{2} + (−1)^{kq}u_{1}Δ(u_{2}),

u_{1} ∈ S^{∗}(V) ⊗ Λ^{q}(W), u_{2} ∈ S^{∗}(V) ⊗ Λ^{∗}(W) and Δ(S^{∗}(V) ⊗ Λ^{i}(W)) ⊂ S^{∗}(V) ⊗ Λ^{i+k}(W), while the differential is of the form [d, ·]: Δ ↦ [d, Δ] = dΔ − (−1)^{k}Δd for Δ ∈ Der_{k}(V).

Problem 2. Let M be a smooth manifold. Compute the homology of the complex

… ⟶ GDiff^{k−1}(M) ⟶^{[d,·]} GDiff^{k}(M) ⟶ …,

where GDiff^{k}(M) is the set of graded differential operators Δ: Λ^{∗}(M) → Λ^{∗}(M) of grading k (i.e., Δ(Λ^{i}(M)) ⊂ Λ^{i+k}(M)) acting on the exterior algebra of differential forms Λ^{∗}(M), while the differential acts by the rule

GDiff^{k}(M) ∋ Δ ↦ [d, Δ] ∈ GDiff^{k+1}(M),

1S^{∗}(V) is the symmetric and Λ^{∗}(V) is the external algebra over V.

[d, Δ] = dΔ − (−1)^{k}Δd. Recall that a mapping Δ: Λ^{∗}(M) → Λ^{∗}(M) is called a graded differential operator, if there exists a number r such that for any forms ω_{i} ∈ Λ^{∗}(M), i = 1, …, r + 1, the identity

[ω_{1}, [ω_{2}, …, [ω_{r+1}, Δ] …]] = 0,

holds, while for the form ω ∈ Λ^{l}(M) and the operator ∇ ∈ GDiff^{k}(M) their commutator [ω, ∇] ∈ GDiff^{k+l}(M) is defined by the equality

[ω, ∇](η) = ω ∧ ∇(η) − (−1)^{kl}∇(ω ∧ η), η ∈ Λ^{∗}(M).

Problem 3 (Homological perturbation theory). Consider a bigraded algebra

A = ⨁_{p,q∈ℤ}A_{p,q}, A_{p,q} · A_{p′,q′} ⊂ A_{p+p′,q+q′},

for which A_{p,q} = 0, if p < 0. Let δ and d be two graded derivations of the algebra A such that

1. δ(A_{p,q}) ⊂ A_{p−1,q};

2. d(A_{p,q}) ⊂ A_{p,q−1};

3. for any q, the complex

0 ⟵ A_{0,q} ⟵^{δ} A_{1,q} ⟵^{δ} A_{2,q} ⟵^{δ} A_{3,q} ⟵^{δ} …

is a resolution. i.e., is exact at positive terms;

4. δ^{2} = 0;

5. δd + dδ = 0;

6. there exists a graded derivation d′ of the algebra A such that

d^{2} = [δ, d′] (= δd′ + d′δ).

This means, in particular, that d induces a differential in the homologies with respect to δ:

d: H_{0}(A_{•,q}; δ) ⟶ H_{0}(A_{•,q−1}; δ), (d|_{H_{0}(A_{•,q};δ)})^{2} = 0.

Prove that in the space A endowed with the total grading A_{i} = ⨁_{p+q=i} A_{p,q} there exists a differential

s: A_{i} → A_{i−1}, s^{2} = 0, s = δ + d + s^{(1)} + s^{(2)} + s^{(3)} + …,

where s^{(r)}(A_{p,q}) ⊂ A_{p+r,q−r−1}, such that

H_{i}(A; s) = H_{i}(H_{0}(A; δ); d).

Problem 4. Consider an arbitrary complex

… ⟶ K^{i−1} ⟶^{d^{i−1}} K^{i} ⟶^{d^{i}} K^{i+1} ⟶ …

Compute the spectral sequences associated to the following filtrations … ⊂ K_{p+1}^{•} ⊂ K_{p}^{•} ⊂ … ⊂ K^{•} of the complex K^{•}:

1. K_{p}^{i} = \begin{cases}K^{i} & for i < −p, \\ ker d^{i} & for i = −p, \\ 0 & for i > −p, \end{cases} (canonical filtration); 2. K_{p}^{i} = \begin{cases}0 & for i < p, \\ K^{i} & for i ⩾ p, \end{cases} (“silly” filtration);

3. K_{p}^{i} = \begin{cases}0 & for i < p, \\ im d^{i−1} & for i = p, \\ K^{i} & for i > p. \end{cases}

Schouten bracket

Problem 5. Let A be a commutative 𝕜-algebra and P br an A-module. The inductive definition of the wedge product ∧: D_{∗}(A) ⊗_{A} D_{∗}(P) → D_{∗}(P) is as follows:

a ∧ p = ap,
(X ∧ Y)(a) = X ∧ Y (a) + (−1)^{|Y |}X(a) ∧ Y,

where a ∈ A, p ∈ P, X ∈ D_{∗}(A), Y ∈ D_{∗}(P).

1. Prove that the wedge product is well defined (i.e., show that X ∧ Y lies in D_{∗}(P)).

2. Describe and prove the basic properties of the wedge product.

3. What is a correct definition of the wedge product for polyderivations in the category of ℤ_{n}-graded modules over a ℤ_{n}-graded algebra (cf. with Problem 7)?

Problem 6. Let A be a commutative 𝕜-algebra. The inductive of the Schouten bracket [[· , ·]]: D_{∗}(A) ⊗_{𝕜} D_{∗}(A) → D_{∗}(A) is as follows:

[[a, b]] = 0,
[[X, a]] = X(a),
[[a, X]] = −X(a),
[[Y, Z]](a) = [[Y, Z(a)]] − (−1)^{|Z|}[[Y (a), Z]],

��� a, b ∈ A, X ∈ D_{1}(A), Y, Z ∈ D_{∗}(A).

1. Prove that the Schouten bracket is well defined (i.e., show that [[Y, Z]] lies in D_{∗}(A)).

2. Describe and prove the basic properties of the Schouten bracket.

Problem 7. Let A be a commutative ℤ-graded algebra. This means that A = ∑_{i∈ℤ} A_{i}, A_{i} · A_{j} ⊂ A_{i+j} and ab = (−1)^{ij}ba, if a ∈ A_{i} and b ∈ A_{j}. Over such algebras, differential calculus is constructed almost literally in the same way as in the commutative case. In particular, one can define the notion of the graded Schouten bracket and of the graded Poisson structure. Do it and describe properties of the constructions obtained.

Problem 8. It would be nice to generalize the Schouten bracket in such a way to obtain an operation [[· , ·]]: D_{∗}(A) ⊗_{𝕜} D_{∗}(P) → D_{∗}(P) (cf. with Problem 5). Nevertheless, for an arbitrary A-module P the definition similar to the one given in Problem 6 does not work. What additional structures should a module be endowed with to obtain the desired definition? Describe the category of these modules.

Problem 9. For any commutative algebra A, the corresponding algebra D_{∗}(A) = ⨁_{i⩾0} D_{i}(A) of polyderivations is a commutative ℤ-graded algebra (see Problem 7) with respect to the wedge product of polyderivations.

1. Show that the Schouten bracket over A is a graded Poisson structure in D_{∗}(A).

2. Hence, a (super)Poisson complex corresponds to this structure. What can be said about its cohomology?

Problem 10 (The Magri scheme). Let A be a commutative algebra and P and Q be commuting Poisson structures over A, i.e.,

[[P, P]] = [[P, Q]] = [[Q, Q]] = 0.

Let a_{1}, a_{2} ∈ A be elements such that P (a_{1}) = Q(a_{2}).

1. Show that if the Poisson complex of the algebra A with respect to the structure Q is 1-acyclic, i.e., H^{1}(A; ∂_{Q}) = 0, then there exist elements a_{3}, a_{4}, … ∈ A such that P (a_{i}) = Q(a_{i+1}).

2. Prove that these elements are in involution with respect to both structures, i.e., {a_{i}, a_{j}}_{P} = {a_{i}, a_{j}}_{Q} = 0 for all i, j ⩾ 1.

Problem 11. Let (A, P) be a Poisson algebra.

1. Show that the Schouten bracket [[· , ·]] over A induces a bracket [[· , ·]]_{P} in cohomology groups H^{∗}(A; ∂_{P}).

2. Prove that if P is a nondegenerate structure, then the bracket [[· , ·]]_{P} is trivial.

Problem 12. Let 𝔤 be a semisimple Lie algebra over a field of zero characteristic and S_{𝔤} be the Poisson algebra of left-invariant symbols (i.e., the graded algebra associated with the universal enveloping algebra U_{𝔤}). Prove that the first Poisson cohomology group of the algebra S_{𝔤} vanishes.

Problem 13. The Poisson center of a Poisson algebra (A, P) is called the set

Z(A, P) = { a ∈ A | {a, b}_{P} = 0 ∀ b ∈ A }.

Evidently, Z(A, P) is a subalgebra in A and Z(A, P) = H^{0}(A; ∂_{P}). Describe the Poisson center of the algebra S_{𝔤} (see Problem 12).

Problem 14. Let (A, P) be a Poisson algebra. An ideal 𝔭 of the algebra A is called Poisson, if it is stable with respect to the bracket {· , ·}_{P}: {A, 𝔭}_{P} ⊂ 𝔭. Let I ⊂ A be a primitive ideal of A. Denote by 𝔪_{P} (I) the maximal Poisson ideal lying in I (i.e., maximal in the class of Poisson ideals lying in I).

1. Prove that 𝔪_{P} (I) is a primitive ideal.

2. The quotient algebra A/𝔪_{P} (I) possesses a natural structure of a Poisson algebra. Prove that it Poisson center is a field.

Problem 15. Study relations between maximal Poisson ideals of the algebra S_{𝔤} (see Problem 12) and the orbits of the coadjoint action of 𝔤 in 𝔤^{∗}.

Problem 16. The exterior algebra of polyvector fields on a smooth manifold M can be understood as the algebra of functions on the cotangent manifold T^{∗}(M) considered as a supermanifold of dimension n|n, where n = dim M. This manifold possesses an odd symplectic structure whose Poisson bracket is the Schouten bracket. Prove that in this situation, similar to the standard one, any canonical transformation possesses a generating function.

Problem 17. Let M be a smooth manifold, π: E → M be a smooth vector bundle, E ⊂ J^{k}(π) be a differential equation of order k, and E^{∞} ⊂ J^{∞}(π) be its infinite prolongation. Describe Poisson structures P ∈ D_{2}(E^{∞}) whose Hamiltonian vector fields P (f) ∈ D_{1}(E^{∞}) are symmetries of the Cartan distribution on E^{∞} for any f ∈ F(E).

Nijenhuis bracket

Problem 18. For any unitary commutative algebra A in which one can cancel by 2 (i.e., such that 2a = 0 ⟹ a = 0), prove existence and uniqueness of the Nijenhuis bracket [[D_{1}(Λ^{i}(A)), D_{1}(Λ^{j}(A))]] ⊂ D_{1}(Λ^{i+j}(A)).

Problem 19. Describe and prove the basic properties of the Nijenhuis bracket.

Problem 20. Similar to Problem 7, define the Nijenhuis superbracket and describe its properties.

Problem 21. Let A and B be two commutative unitary algebras, φ: A → B be their homomorphism, and ∇: D_{1}(A, B) → D_{1}(B) be a connection. Prove the identity [[U_{∇}, U_{∇}]](X, Y) = 2R_{∇}(X_{A}, Y_{A}), where U_{∇} is the connection form, R_{∇} is the curvature, X, Y ∈ D_{1}(B) and Z_{A} = Z ∘ φ for any Z ∈ D_{1}(B).

Problem 22. Let A, B, φ be the same as in Problem 21 and the connection ∇ be flat (i.e., R_{∇} = 0). Let L_{∇} = L_{U_{∇}} and d_{h} = d − L_{∇} be the vertical and horizontal differentials in Λ^{∗}(B). Prove that Λ^{1}(B) = Λ_{h}^{1}(B) ⊕ ∇Λ^{1}(B), where Λ_{h}^{1}(B) and ∇Λ^{1}(B) are the submodules in Λ^{1}(B) generated by the images of d_{h} and L_{∇} respectively.

Problem 23. Let the algebras A, B, the homomorphism φ, and the connection ∇ satisfy the conditions of Problem 22. Let d_{h}: Λ^{∗}(B) → Λ^{∗}(B) be the corresponding horizontal differential. Consider the ℤ-graded Lie algebra D(Λ^{∗}(B)) of graded derivations of the graded algebra Λ^{∗}(B) (they consist of 𝕜-linear graded mappings X: Λ^{∗}(B) → Λ^{∗}(B) such that X(ω ∧ θ) = X(ω) ∧ θ + (−1)^{|X|·|ω|}ω ∧ X(θ), ω, θ ∈ Λ^{∗}(B)). Then d_{h} generates the mapping ∂_{h}: D(Λ^{∗}(B)) → D(Λ^{∗}(B)), acting by ∂_{h}(X) = [d_{h}, X].

1. Prove that ∂_{h} is a differential.

2. Describe the cohomology H(∂_{h}).

3. What will be the answer if one takes the de Rham differential

d: Λ^{∗}(B) → Λ^{∗}(B)

instead of ∂_{h}?

Problem 24 (The Vinogradov unifying construction). Let us recall that the Nijenhuis bracket was defined in the modules D_{1}(Λ^{∗}(A)). On the other hand, the Schouten bracket “lives” in D_{∗}(A). A natural intention is to construct a unified bracket in D_{∗}(Λ^{∗}(A)) in such a way that its restriction to D_{∗}(Λ^{0}(A)) = D_{∗}(A) would coincide with the Schouten bracket, while its restriction to D_{1}(Λ^{∗}(A)) would be the Nijenhuis bracket. Try.

A bracket producing machine

Let P be a graded differential A-module, A being a graded algebra, i.e., P = ⨁_{i∈ℤ} P_{i} and d_{P}: P_{i} → P_{i+1} be a differential. Let a graded module M be represented by graded endomorphisms End_{A}^{gr}(P), i.e., a graded homomorphism i: M → End_{A}^{gr}(P) is given. Then one can define Lie derivatives of elements from P along elements of M: L_{m} = d_{P} ∘ i_{m} −(−1)^{| i_{m} |} i_{m} ∘d_{P}. If one is lucky, the graded commutator [L_{m}, L_{n}] will again be of the form L_{o} for some element o ∈ M and, if o is uniquely defined, the equality o = [[m, n]] will define a bracket in the module M. The Schouten and the Nijenhuis brackets were introduced in this manner exactly. In both cases, the module P was Λ^{∗}(A) with the de Rham differential, while the role of i was played by substitution of polyderivations (for Schouten) and of Λ^{∗}-valued derivations (for Nijenhuis) to differential forms.

Problem 25. Let (A, P) be a Poisson algebra. Let us take the differential module (D_{∗}(A), ∂_{P}) for P and substitution of differential forms to polyderivations for i. Describe the bracket arising in Λ^{∗}(A).

Problem 26. Let A be a commutative algebra and U ∈ Λ^{∗}(A) ⊗ D_{1}(A) be an integrable element (i.e., [[U, U]] = 0, where [[· , ·]] is the Nijenhuis bracket. Let P be the differential module (Λ^{∗}(A) ⊗ D_{1}(A), ∂_{U}), and i be the substitution of elements from Λ^{∗}(A) ⊗ D_{1}(A) to elements of the same type. Describe the bracket arising in Λ^{∗}(A) ⊗ D_{1}(A).

Problem 27. Let the module P be the same as in Problem 26, and i is the substitution of polyderivations to elements of Λ^{∗}(A)⊗D_{1}(A). Describe the bracket arising in D_{∗}(A).

Brackets generated by second-order operators

Problem 28. Let (M, θ) be a Poisson manifold and Δ_{θ}: Λ^{∗}(M) → Λ^{∗}(M) be the Poisson homology operator Δ_{θ} = [i_{θ}, d] (= i_{θ} d − d i_{θ}). It is a graded second-order operator and its symbol determines a bracket in Λ^{∗}(M):

{ω_{1}, ω_{2}} = [ω_{1}, [ω_{2}, Δ_{θ}]].

Describe properties of this bracket. Compute {f, g}, {df, g}, {df, dg}, where f, g ∈ C^{∞}(M).

Problem 29. Let M be an orientable manifold. Any volume form on this manifold determines an isomorphism between differential forms Λ^{∗}(M) and polyvector fields D_{∗}(M). Shoe that the de Rham operator is taken under this isomorphism to a second order operator acting on the algebra D_{∗}(M) and its symbol is independent of the volume form choice. Compute this symbol.

Problem 30. Let (M, ω) be a symplectic manifold and ∂_{ω}: D_{∗}(M) → D_{∗}(M) is the corresponding Poisson differential. Compute the symbol of the operator

Δ = [i_{ω}, ∂_{ω}]: D_{∗}(M) → D_{∗}(M).