The Diffiety Institute Preprint Series

Preprint DIPS-8/96 October 1, 1996

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The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Remarks on algebraic Lagrangian formalism

by

Michael M. Vinogradov

REMARKS ON ALGEBRAIC LAGRANGIAN FORMALISM

Michael M. Vinogradov

Diffiety Institute (Pereslavl-Zalessky, Russia), Institute of Economic Forecasting (Moscow, Russia)

Abstract. An axiomatic approach to the notion of adjoint operator is provided within the framework of the algebraic Lagrangian formalism.

Introduction

In 1977 A. Vinogradov [1, 2] suggested an algebraic approach to the Lagrangian formalism over smooth functions algebras. This approach consists of several steps:

(1) The definition of adjoint operators and a proof of the Green formula;

(2) A construction of the Lagrangian formalism;

(3) A proof of the Noether theorem.

A second and a third steps in this approach are purely algebraic. But the notion of adjoint operator is suitable for algebras of smooth functions over finite dimensional manifolds only. In a process of constructing this notion the module of volume forms Λ^{n} is used as a dual object to the basic algebra. It was clear that in order to generalize this approach to graded algebras or to function algebras over infinite dimensional manifolds one needs to generalize the notion of adjoint operator. Recently two papers ([3], [4]) on this topic have appeared. Both of them are based on ideas of I. Penkov [5]. These papers showed that potentially there exist various objects which can play the role of Λ^{n}.

In this paper an axiomatic point of view on this topic is presented. From this point of view the main results of [2], [3] can be considered as existence theorems for dual structure. The meaning of the words dual structure will be explained below.

For simplicity I will consider only the nongraded case, because the jump from our constructions to a graded one is trivial. The following meta-mathematical rule justifies me: if an algebraic nongraded construction is valid, then the jump from it to a graded one is trivial. If such jump is not trivial, then that construction is not valid.

The begining of a proof is marked by ◀, end or absence of the latter is marked by ▶. As a rule, we omit pure technical proofs.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16S32, 16S80, 17B37, 16W30, 18D10, 35A99, 58A99, 58G05, 58G99, 81R50.

Key words and phrases. Differential operator, jets, differential forms, adjoint operator, Berezinian, integral forms, the Green formula.

Necessary notions and notations. Let K be a commutative ring with a unit, A be a commutative associative unitary K-algebra and P be an A-module. We deal with the functors of differential calculus arising in the category M(A) of modules over A. The definition of differential operators over a commutative algebra and other necessary notions and notations may be found in the paper [6] of this issue.

Dual structures and right connections

Definition 1. An A-homomorphism ∇ : Diff_{l}^{+} P → P, l > 0, is called a right connection of order l on P if ∇|_{P} = id.

Proposition 1. Let ∇ : Diff_{l}^{+} P → P, l > 0, be a right connection of order l on P and ∗ : Diff_{l}^{+} P → Hom_{K}(A, P) be a map defined by the rule Δ^{∗}(a) =^{def} ∇(aΔ). Then ∗ is an A-homomorphism from Diff_{l}^{+} P to Diff_{l} P and the following conditions hold: (C1) ∗|_{P} = id; (C2) ∗|_{Diff_{l−1}^{+} P} is a right connection of order l − 1 on P;

(C3) If Δ ∈ Diff_{l}^{+} P, Δ ∉ Diff_{l−1}^{+} P, then Δ − (−1)^{l}Δ^{∗} ∈ Diff_{l−1}^{+} P. The DO Δ − (−1)^{l}Δ^{∗} ∈ Diff_{l−1}^{+} P is called the body of the DO Δ and is denoted by Δ_{(∗)}.

◀ We start with prooving (C3). First note that (Δ ∘ b)^{∗} = b ∘ Δ^{∗}, b ∈ A. Really,

(Δ ∘ b)^{∗}(a) = ∇(a ∘ (Δ ∘ b)) = b∇(aΔ) = (b ∘ Δ^{∗})(a)

Similarly, (b ∘ Δ)^{∗} = Δ^{∗} ∘ b. Consequently

(δ_{a}Δ)^{∗} = (Δ ∘ a)^{∗} − (a ∘ Δ)^{∗} = a ∘ Δ^{∗} − Δ^{∗} ∘ a = −δ_{a}Δ^{∗}

This implies that Δ^{∗} is the DO of order l. From the relation (Δ ∘ b)^{∗} = b ∘ Δ^{∗} it follows that ∗ is an A-homomorphosm.

Now let us prove that Δ_{(∗)} ∈ Diff_{l−1}^{+} P. The proof is by induction on l. For l = 0, there is nothing to prove. In the general case, we have for any a ∈ A

δ_{a}Δ_{(∗)} = δ_{a}Δ − (−1)^{l}δ_{a}Δ^{∗} = δ_{a}Δ − (−1)^{l−1}(δ_{a}Δ)^{∗} = (δ_{a}Δ)_{(∗)}

The DO (δ_{a}Δ)_{(∗)} is the body of the DO δ_{a}Δ. By the inductive assumption, (δ_{a}Δ)_{(∗)} ∈ Diff_{l−2}^{+} P.

It remains to prove statements (C1) and (C2). But (C1) immediately follows from the definition of a right connection. The restriction of ∇ to Diff_{l−1} P is the right connection of order l − 1. This proves statement (C2). ▶

Definition 2. An A-isomorphism ∗ : Diff_{l}^{+} P → Diff_{l} P is called a dual structure of order l on an A-module P if the following conditions hold: (D1) ∗|_{P} = id. (D2) ∗|_{Diff_{l−1}^{+} P} is a dual structure of order l − 1.

(D3) If Δ ∈ Diff_{l}^{+} P, Δ ∉ Diff_{l−1}^{+} P, then Δ_{(∗)} =^{def} Δ − (−1)^{l}Δ^{∗} ∈ Diff_{l−1}^{+} P. (D4) Δ_{(∗)}^{∗} = (−1)^{l−1}Δ_{(∗)}.

Proposition 2. (Δ^{∗})^{∗} = Δ

◀ Let Δ ∈ Diff_{l}^{+} P. By condition (D3) of the definition, Δ^{∗} = (−1)^{l}Δ − (−1)^{l}Δ_{(∗)}, where Δ_{(∗)} ∈ Diff_{l−1}^{+} P. Applying the operator ∗ to both sides and taking into account the relation (D4), we obtain

Δ^{∗∗} = (−1)^{l}Δ^{∗} − (−1)^{l}Δ_{(∗)}^{∗} = Δ − Δ_{(∗)} − (−1)^{l}Δ_{(∗)}^{∗} = Δ. ▶

Proposition 3. Let ∗ be a dual structure of order l. By definition, put ∇(Δ) = Δ^{∗}(1). Then ∇ is a right connection of order l. ◀ It follows immediately from Definitions 1 and 2 ▶

Theorem 1. The following conditions are equivalent:

(1) the A-module P possesses a dual structure of order 1;

(2) there exist a right connection of order 1 on P.

◀ The implication (1) ⟹ (2) follows from Proposition 3.

On the other hand, let ∇ : Diff_{1}^{+} P → P be a right connection of order 1. By definition, put Δ^{∗}(a) = ∇(aΔ), Δ ∈ Diff_{1}^{+} P. From Proposition 1 it follows that conditions (D1), (D2), (D3) of Definition 2 are hold. Since the order of the DO Δ_{(∗)} is equal to zero, condition (D4) of this definition is also true. ▶

This theorem gives a constructive way to obtain a dual structure. If we have a right connection of order 1 on P, then there exists a dual structure of order 1 on P, and, under natural conditions, we can obtain a dual structure of higher order.

Examples

Denoted by the Diff_{1} A ⊗_{in} P tensor product Diff_{1} A ⊗ P equipped with the A-module structure induced by the right multiplication in Diff_{1}^{(+)} A. In other words, the following relations hold for elements of the A-module Diff_{1} A ⊗_{in} P:

(aΔ) ⊗_{in} p = Δ ⊗_{in} ap, a(Δ ⊗_{in} p) = (Δ ∘ a) ⊗_{in} p

There exists a natural A-homomorphism

i_{⊗} : Diff_{1} A ⊗_{in} P → Diff_{1}^{+} P, i_{⊗}(X ⊗ p)(a) = X(a)p, X ∈ Diff_{1} A, a ∈ A, p ∈ P.

If we are given a right connection ∇ : Diff_{1}^{+} P → P, then to any DO X ∈ Diff_{1} A a DO L_{X} = L_{X}^{∇} ∈ Diff_{1}(P, P) can be assosiated by setting

L_{X}(p) = −∇(i_{⊗}(X ⊗ p)), p ∈ P.

Definition 3. A right connection is said to be flat if it preserves the Lie structure, i.e., for any X, Y ∈ D(A), p ∈ P, we have

L_{X}(L_{Y} (p)) − L_{Y} (L_{X}(p)) = L_{[X,Y]}(p)  (1)

In this section we shall consider flat connections only.

Now we consider how the concept of a right connection is realized in the case A = C^{∞}(M), the algebra of smooth functions on a smooth n-dimensional manifold M, and P is the module of sections of a locally trivial one-dimensional vector bundle over M.

In this case there exists a A-isomorphism of A-module Diff_{1}^{+} P to A-module Diff_{1} A⊗_{in}P. For simplicity we identify these modules.

Remark. Some authors, see for example [7], define a right connection as A-homomorphism Diff_{1} A ⊗_{in} P → P.

Let U ⊂ M be an open domain, x = {x_{1},…, x_{n}} be a local coordinate system, P |_{U} be a free C^{∞}(U)-module, and e be a local generating element of the module P |_{U}.

Definition 3. We shall say that e and x are adjoint with respect to the right connection ∇ if L_{∂/∂x_{i}}(e) = 0 for all i.

Proposition 4.

(1) For given e and x there exists a unique local right connection ∇ with respect to which they are adjoint.

(2) If a coordinate system x and a right connection ∇ are given, then there exists a local generator e such that e and x are adjoint relative to ∇.

(3) If e and x are adjoint relative to ∇ and y is another coordinate system on U, then the generator of P |_{U} adjoint to y has the form c·det(I)e, where c ∈ K is a constant and I is the Jacobian of the corresponding coordinate transformation.

◀ (1) Assume that the desired right connection exists, then, by definition of a right connection,

∇((∑ α_{i} \frac{∂}{∂x_{i}} + β) ⊗ ae) = (β − ∑ \frac{∂aα_{i}}{∂x_{i}})e  (2)

It follows that the right connection ∇ is uniquely determined. On the other hand, the desired right connection is well defined by this relation.

(2) Let e be an arbitrary generator of P |_{U}, ∇(\frac{∂}{∂x_{i}} ⊗ e) = β_{i}e, and p = ae. It follows from the relations ∇(\frac{∂}{∂x_{i}} ⊗ p) = 0 that the function a satisfies the following system of equations

\frac{∂ ln a}{∂x_{i}} = β_{i}, i = 1,…, n.

It follows from relation (1) that \frac{∂β_{i}}{∂x_{j}} = \frac{∂β_{j}}{∂x_{i}}. Therefore the function a exists, is uniquely determined up to a constant multiplier and does not vanish at all points of U.

(3) It follows from the previous statement that there exists a generator g of P |_{U} adjoint to y. Let g = ae. Let us prove that a = c · det(I). Any change of coordinates can be represented as a composition of linear changes and changes of one of coordinates. Since the case of linear change is trivial, it is sufficient to consider the later case. Let

y_{i} = \{ \begin{matrix}F (x_{1},…, x_{n}), i = 1, \\ x_{i}, i > 1. \end{matrix}

Using relations (2), we get

\{ \begin{matrix}\frac{∂}{∂x_{1}} & (a(\frac{∂F}{∂x_{1}} )^{−1}) = 0 \\ \frac{∂a}{∂x_{j}} & + \frac{∂}{∂x_{1}} (a − \frac{∂F}{∂x_{j}} (\frac{∂F}{∂x_{1}} )^{−1}) = 0, j = 2,…, n \end{matrix}

It follows from the first equation that a = f(x_{2},…, x_{n})\frac{∂F}{∂x_{1}} . Using the later equations, we get f = const ∈ K. ▶

The proof is adapted from §5 of Chapter 4 [7].

Corollary 1. A global flat right connection ∇ on a one-dimensional module P defines a canonical isomorphism i : P → Λ^{n}(M).

◀ Let x be a local coordinate system on U, e be a local generating element. Suppose e and x are adjoint with respect to the right connection ∇. Then the correspondence e ↦ dx_{1} ∧ ⋯ ∧ dx_{n} establishes an isomorphism i_{U} : P |_{U} → Λ^{n}(U). By statement (3) of Proposition 4

i_{U}|_{U ⋂ V} = i_{U ⋂ V} = i_{V} |_{U ⋂ V} ,

This implies that the set of the local isomorphisms {i_{U}} defines the global isomorphism i : P → Λ_{n}(M). ▶

Also, the C^{∞}(M)-module Λ^{n}(M) possesses a dual structure of order 1. Using methods of [1], one can show that Λ^{n}(M) possesses a dual structure of order l for any positive integer l.

Corollary 2. Under the isomorphism of Corollary 1 the operator L_{X}, X ∈ D(A), coinsides with the usual Lie derivative. ▶

Remark. Of course the super-analogs of all statements of this section hold. In order to give right formulation one needs to replace the determinant det(I) by the berezinian ber(I) and module of n-forms Λ^{n}(M) by berezinian Ber(M) of a super-manifold M only.

The latest known to me example of existence of a dual structure belongs to A. Verbovetsky ([8]): for an arbitrary K-algebra A an A-module Hom_{K}(A, K) possesses such a structure.

Integral forms and the Green formula

Now we fix an A-module P possessing a dual structure of order l and denote it by Σ_{0}. Let Σ_{i} =^{def} D_{i}(Σ_{0}) = Hom(Λ^{i}, Σ_{0}). Elements of A-module Σ_{i} are called integral i-forms. Let us consider the Spenser Diff-complex for Σ_{0}:

0 ⟶ D_{l}(Σ_{0}) ⟶^{S_{l}^{l}} D_{l−1}(Diff_{1}^{+} Σ_{0}) ⟶^{S_{l}^{l−1}} … ⟶^{S_{l}^{1}} Diff_{l}^{+} Σ_{0} ⟶ Σ_{0} ⟶ 0

Its cohomology at the term Diff_{l}^{+} Σ_{0} is denoted by H_{l}^{0}(Σ_{0}). By Proposition 3 there exist a right connection ∇ : Diff_{1}^{+} Σ_{0} → Σ_{0} of order 1. The operator δ_{i} : Σ_{i} = D_{i}(Σ_{0}) → Σ_{i−1} = D_{i−1}(Σ_{0}) is given by the composition

δ_{i} =^{def} D_{i−1}(∇) ∘ S_{l}^{l} : Σ_{i} = D_{i}(Σ_{0}) → D_{i−1}(Diff_{1}^{+} Σ_{0}) → D_{i−1}(Σ_{0}) = Σ_{i−1}

The sequence

… ⟶ Σ_{i} ⟶^{δ_{i}} Σ_{i−1} ⟶ … ⟶ Σ_{1} ⟶^{δ_{1}} Σ_{0} ⟶ 0

is called the sequence of integral forms. If δ_{1} ∘ δ_{2} = 0, then this sequence is a complex.

Definition. For any A-module P the A-module P̂ =^{def} Hom_{A}(P, Σ_{0}) is called the dual module to P.

Theorem 2. For any A-modules P, Q there exist a natural isomorphism

∗ : Diff_{l}^{+}(P, Q̂) → Diff_{l}(Q, P̂)

I should stress that here and below the A-modules Σ_{0}, P, Q may be not projective. ◀ The proof is in 2 steps. Step1: Suppose Q = A and, therefore, Q̂ = Σ_{0}. Then

Diff_{l}(A, P̂) = Diff_{l} P̂ = Hom_{A}(J^{l}, Hom_{A}(P, Σ_{0})) = Hom_{A}(J^{l} ⊗ P, Σ_{0})
= Hom_{A}(P, Hom_{A}(J^{l}, Σ_{0})) = Hom_{A}(P, Diff_{l} Σ_{0})
=^{i_{∗}} Hom_{A}(P, Diff_{l}^{+} Σ_{0}) = Diff_{l}^{+}(P, Σ_{0}) = Diff_{l}^{+}(P, Â)

where the isomorphism i_{∗} is induced by the duality ∗ : Diff_{l}^{+} Σ_{0} → Diff_{l} Σ_{0}.

Note also that under the isomorphism Diff_{l}(A, P̂) = Diff_{l}^{+}(P, Σ_{0}) the right multiplication on the module Diff_{l}(A, P̂) becomes the left multiplication on the module Diff_{l}^{+}(P, Σ_{0}). Consequently there exists an A-isomorphism Diff_{l}^{+} P̂ = Diff_{l}(P, Σ_{0}).

Step 2: Let us consider the general case.

Diff_{l}^{+}(Q, P̂) = Hom_{A}(Q, Diff_{l}^{+} P̂) =^{i_{∗}} Hom_{A}(Q, Diff_{l}(P, Σ_{0}))
= Hom_{A}(Q, Hom_{A}(J^{l}(P), Σ_{0}) = Hom_{A}(Q ⊗ J^{l}(P), Σ_{0})
= Hom_{A}(J^{l}(P), Hom_{A}(Q, Σ_{0})) = Hom_{A}(J^{l}(P), Q̂) = Diff_{l}(P, Q̂). ▶

Lemma 1. For any A-module P there exists a natural isomorphism ψ : D_{i}(Diff_{l}^{+} P̂) → Diff_{l}(P, Σ^{i}).

◀ Taking into account the characteristic properties of the modules Λ^{i} and J^{l}(P), we obtain

D_{i}(Diff_{l}^{+} P̂) = Hom_{A}(Λ^{i}, Diff_{l}^{+} P̂) =^{i_{∗}} Hom_{A}(Λ^{i}, Diff_{l}(P, Σ_{0}))
= Hom_{A}(Λ^{i}, Hom_{A}(J^{l}(P), Σ_{0})) = Hom_{A}(Λ^{i} ⊗ J^{l}(P), Σ_{0})
= Hom_{A}(J^{l}(P), Hom_{A}(Λ^{i}, Σ_{0})) = Hom_{A}(J^{l}(P), Σ_{i})
= Diff_{l}(P, Σ_{i}) ▶

Using this Lemma we immediately obtain the following statement.

Theorem 3. For any A-module P we have the commutative diagram

0  P̂  Diff_{l}^{+} P̂  D_{1}(Diff_{l−1}^{+} P̂)  …  D_{i}(Diff_{l−i}^{+} P̂) ← …

0  P̂  Diff_{l}^{+}(P, Σ_{0})  Diff_{l−1}^{+}(P, Σ_{1})  …  Diff_{l−i}^{+}(P, Σ_{i}) ← …

[figure] S_{l}^{0} [figure]

∗

[figure]

S_{l}^{1} [figure]

ψ

[figure]

S_{l}^{2} [figure] S_{l}^{i} [figure]

ψ

[figure]

[figure] μ [figure] δ_{1}^{∘} [figure] δ_{2}^{∘} [figure] δ_{i}^{∘} [figure]

where the upper row is the Spencer complex of A-module P̂, δ_{i}^{∘}(Δ_{i}) = δ_{i} ∘ Δ_{i}, i > 0, and μ(Δ_{0}) = Δ_{0}^{∗}(1), Δ_{i} ∈ Diff_{l−i}(P, Σ_{i}). ▶

Denote by ⟨ , ⟩ the natural pairing P̂ ⊗ P → Σ_{0}.

Theorem 4 (the Green formula). If H_{l}^{0}(Σ_{0}) = 0, then for any Δ ∈ Diff_{l}(P, Q̂) we have

⟨Δ(p), q⟩ − ⟨p, Δ^{∗}(q)⟩ = δ_{1}^{∘}(σ),

where σ ∈ Σ_{1}.

◀ The short proof can be found in [3]. ▶ Thus one problem is open: how to find modules possessing a dual structure.

Acknowledgments. The author is grateful to A. M. Vinogradov, A. M. Verbovetsky and A. B. Sosinskii for useful discussions and to A. B. Sosinskii and I. S. Krasil’shchik for their help in translating this paper in English.

References

1. Vinogradov A. M., On the algebro-geometric foundations of Lagrangian field theory, Soviet Math. Dokl. 18 (1977), 1200–1204.

2. Vinogradov A. M., The C-spectral sequence, Lagrangian formalism, and conservation laws., Jour. of Math. Analysis and Appl. 100 (1984), no. 1, 1 – 129.

3. Vinogradov M. M., Coconnection and integral forms, Russian Acad. Sci. Dokl. Math. 50 (1995), 229–297.

4. Verbovetsky A. M., Lagrangian formalism over graded algebras, J. Geom. Phys. 18 (1996), 195–214; hep-th/9407037.

5. Penkov I. B., D-modules on supermanifolds, Invent. math. 71 (1983).

6. Krasil’shchik I. S., Calculus over commutative algebras: a concise user guide, Preprint DIPS 1/96.

7. Manin Yu. I., Gauge field theory and complex geometry, Springer-verlag, Berlin and Heidelberg, 1988.

8. Verbovetsky A. M., Private communication.

Institute of Economic Forecasting, 32, Krasikova str., Moscow 117418, Russia E-mail address: vin@ecfor.msk.su