Abstract

We review some aspects of the theory of Lie algebras of (twisted and untwisted) formal pseudodifferential operators in one and several variables in a general algebraic context. We focus mainly on the construction and classification of nontrivial central extensions. As applications, we construct hierarchies of centrally extended Lie algebras of formal differential operators in one and several variables, Manin triples and hierarchies of nonlinear equations in Lax and zero curvature form.

1. Introduction

This paper is on some cohomological aspects of Lie algebras of formal pseudodifferential operators in one and several independent variables, motivated by previous works on algebras of importance for integrable systems and symplectic geometry such as the Lie algebra of vector fields on the circle and their deformations [1, 2] or the Lie algebra of differential operators; see, for example, [3–6].

A very well-known infinite-dimensional Lie algebra of interest for physics is the Virasoro algebra [7]. This algebra is a one-dimensional central extension of the Lie algebra of vector fields on the circle (also called centerless Virasoro algebra) which also appears naturally in applications. We mention, as a recent example, that the centerless Virasoro algebra can be realized as an algebra of nonlocal symmetries for the Camassa-Holm and Hunter-Saxton equations [8, 9].

Now, the Lie algebra of vector fields on the circle is included naturally in the Lie algebra of differential operators on the circle. This algebra, in turn, is included in the Lie algebra of formal pseudodifferential operators on the circle which has been studied very carefully, for example, by Dickey [5] and Adler [10], in connection with the algebraic and geometric theory of the famous Korteweg-de Vries (KdV) equation and other integrable systems. Moreover, there exist nontrivial twisted versions of these works related to “twisted” and “quantum” analogs of classical integrable systems; see, for instance, [11–14]. It is certainly reasonable to consider formal pseudodifferential operators as a general arena for integrable systems.

Our aim in this work is to study central extensions of Lie algebras of formal pseudodifferential operators in a general algebraic setting and to apply this study to the construction of hierarchies of centrally extended Lie algebras, Manin triples [15], and nonlinear integrable equations in one and several independent variables.

We mention four examples of relevant central extensions. Centerless Virasoro has a unique central extension; see [3]. Also, a 2-cocycle of the algebra of differential operators on the circle was constructed in [6], and a 2-cocycle of the algebra of pseudodifferential operators on the circle was constructed by Kravchenko and Khesin using logarithms; see [16, 17]. Finally, a 2-cocycle for a quantum analog of the algebra of pseudodifferential operators was considered in [11].

In this paper we consider algebraic versions of these results, and we present classifications of central extensions. In particular, we show that many of the constructions of cocycles appearing in the literature (see, for instance, [17] or [4]) are valid well beyond their original framework.

We have divided our work in three main sections.

In Section 2 we introduce the main objects considered in this work. We define formal pseudodifferential operators (or, “pseudodifferential symbols”) in one variable on an arbitrary associative and commutative algebra, and we construct the Kravchenko-Khesin logarithmic 2-cocycle in this general context. We also construct a hierarchy of centrally extended Lie algebras of differential operators via the logarithmic 2-cocycle. This construction generalizes a theorem by Khesin [18] on hierarchies of Lie algebras of differential operators on the circle. Finally, motivated by [11, 19, 20] and the later paper [12], we consider twisted pseudodifferential symbols on arbitrary associative and commutative algebras and, in analogy with the untwisted case, we construct central extensions and hierarchies of twisted centrally extended algebras.

In Section 3 we consider pseudodifferential symbols in several independent variables on an arbitrary associative and commutative algebra. Our main motivation for considering the several variables case in detail is the relative absence of examples of integrable equations in this context. Indeed, besides the equations of the standard KP hierarchy [5] and their cousins, there are not very many general constructions of integrable equations in several independent variables. Important exceptions are the equations introduced by Tenenblat and her coworkers (see [21] and references therein) and the hierarchies considered by Parshin, [22]. We construct central extensions in the several variables case using logarithmic cocycles, and we also exhibit hierarchies of Lie algebras of differential operators in several independent variables admitting central extensions. We then consider the work [23] by Dzhumadil’daev, in which he presents a classification of central extensions. The paper [23] is quite technical and it requires a very careful and critical reading, and so we have decided to explain how to prove the main result of [23] using an inductive argument. We present a full inductive proof of Dzhumadil’daev’s theorem using some technical homological tools elsewhere; see [24].

Finally, in Section 4 we introduce Manin triples, we define double extensions for the algebras of (twisted) pseudodifferential symbols in one and several independent variables and, using a general algebraic theorem [25], we construct Manin triples for these algebras, thereby putting [17, 26–29] in a very general framework. We also apply our algebraic results to the construction of integrable systems in one and several independent variables, roughly following the techniques of [4, 11, 22].

2. The Algebra of Formal Pseudodifferential Symbols in One Variable

2.1. Basic Definitions and Preliminary Results

Let be an associative and commutative algebra, and let be a derivation on ; that is, is a linear map such that for all . The algebra of formal differential symbols is generated by and a symbol with the relationfor all . The algebra is a subalgebra of and we can prove inductively thatfor all and .

We extend the algebra to obtain the algebra of pseudodifferential symbols by introducing differentiations with negative exponents. A general element of is a formal series of the formWe setand so (2) generalizes tofor all , in which the binomial coefficient is defined byfor . If and , then

The Lie algebra structure on is given by the usual commutator , so that, for instance,

Lemma 1. For any nonnegative integer and , we have

Proof. Proof is by induction.

Let be a -invariant trace on ; that is, is a linear map satisfying and for all . For example, if , the linear functional given by is a -invariant trace.

Lemma 2. Let be an algebra, a derivation on , a -invariant trace on , and a positive integer. Then we can “integrate by parts”; that is, for all , we have

Proof. We have that , and, so, ; then . We now proceed by induction on .

Proposition 3. Let be an algebra, a derivation on , and a -invariant trace. Then the linear map defined byis a trace on ; that is, is linear and it satisfies for all . This is the Adler-Manin noncommutative residue introduced in [10, 30].

Proof. We use the elementary identityfor and . Since is linear, it is sufficient to show that for any and We consider several cases.
Let and this is identically zero since we obtain the coefficient of when , and Similarly,Now we let ; then ( a positive integer), andBut, as , the coefficient of is 0, and then . Analogously we find .
Now assume that ,  . If thenbecause and then . Analogously, .
Finally, we let ; set . Then,and, on the other hand, (10) and (12) imply

Remark 4. Our definition of residue follows the conventions of [23]. For example, if , , and is given by , then, with our convention,which is slightly different from the notation used in [4, 16–18].

Corollary 5. The bilinear form is -invariant; that is, it satisfies .

Now, if is a pseudodifferential symbol such that and for all , we say that is the order or degree of . The following observation has resulted to be fundamental for the theory; see, for instance, [5, 10, 17, 31]. The algebra can be decomposed as a (vector space) direct sum , where

Proposition 6. The subalgebras (of differential operators) and (of pseudodifferential symbols of order ) are isotropic subspaces of with respect to the bilinear form defined in Corollary 5; that is, the restrictions of this form to both and vanish.

2.2. On Cohomology of Lie Algebras

Having reviewed the elementary properties of , we now summarize some basic facts on the cohomology of Lie algebras in order to fix our notation. We will study the cohomology of in Section 2.3.

2.2.1. Basic Definitions

Suppose that is a Lie algebra and that is a module over . A -dimensional cochain of the algebra with coefficients in is a skew-symmetric -linear functional on with values in ; the space of all such cochains is denoted by and we also set . The differential is defined by the formulawhere and . We also set for and for . We can check that for all and therefore is an algebraic complex. This complex is denoted by , while denotes the -cohomology space of the algebra with coefficients in . If is a trivial -module, then the second sum of in the right-hand side of formula (22) vanishes and it may be ignored. If is a field, the notations , are abbreviated to .

2.2.2. Algebraic Interpretations of Cohomology

A derivation of the Lie algebra is a linear map such that . A derivation is inner if , where is a fixed element. Outer derivations are by definition elements of the quotient space of all derivations module the subspace of inner derivations. The proof of the following proposition is in [32], Chapter 1, Section  4.

Proposition 7. can be interpreted as the space of outer derivations of the algebra .

Definition 8. A central extension of a Lie algebra by a vector space is a Lie algebra whose underlying vector space is equipped with the following Lie bracket:for some bilinear map .

Note that depends only on and but not on and . This implies that is the center of the Lie algebra .

The skew-symmetry, bilinearity, and the Jacoby identity on the Lie algebra are equivalent to the antisymmetry, bilinearity, and the following 2-cocycle identity for the map :for any .

Two 2-cocycles on with values in differing by a 2-coboundary give rise to isomorphic central extensions of . We have the following result, see [32, page 33].

Proposition 9. There is a one-to-one correspondence between equivalence classes of central extensions of by and elements of .

The referee has pointed out that Proposition 9 combined with Proposition 10 below allows us to effectively calculate central extensions.

Proposition 10. Let be a Lie algebra. The space of one-dimensional central extensions is isomorphic to a subspace of the first cohomology space . Specifically, if we denote by the subspace of generated by cohomology classes of cocycles such thatfor , then .

This general proposition is due to Dzhumadil’daev, who used it in [33] (in the case of one-dimensional central extensions induced by fields of characteristic ) and in [34] (in the zero characteristic case) for the study of central extensions of Lie algebras of Cartan type; see, for example, [33–35]. Proposition 10 has the following corollary, also pointed out by the referee (see also [23, 33]).

Corollary 11. If the Lie algebra admits an invariant, symmetric, and nondegenerate bilinear form , then the space of one-dimensional central extensions is isomorphic to the space of outer derivations such that for all .

Proof. The existence of allows us to identify with . Now, if we take so that the cohomology class of is in , we obtain a set of derivations of the form for some . Then, invariance of implies that if and only if for all .

2.3. Outer Derivations and Central Extensions of  

We go back to the algebra considered in Section 2.1. Following [16], we write formally the identity . This implies thatHence, setting in (8) and differentiating at using , we obtainIt follows that if , then is also an element of , even though itself is not.

In the next proposition we will make use of the following combinatorial identity (see, for instance, [12]).

Lemma 12. Let and be integers and . Then

Proposition 13 and Theorem 14 below were proved by Kravchenko and Khesin [16] in the case .

Proposition 13. defines a (resp., an outer) derivation of the associative (resp., Lie) algebra .

Proof. We note that the proposition does not follow from the fact that for any associative algebra the map determines a derivation, since in our case is not an element of .
First of all, it is not difficult to see using (27) that belongs to for any . Now, assuming that is a derivation, it is trivial to prove that it is outer derivation of the Lie algebra : if for some , then belongs to the center of , and so , a contradiction.
We show that is a derivation. It is sufficient to prove that, for any ,  ,Indeed, for the left side of (29) we haveOn the other hand, for the right side of (29) we haveNow, for any integer , the coefficient of in (30) iswhere the summation is over all integers , such that . Using (9) we havewhere both summations are over all integers ,   such that . On the other hand, for , the coefficient of (31) iswhere the sum is over all integers ,   such that . Therefore, (33) and (34) are equal if for fixed integers , as above we havewhere the sum is over all integers , , and such that , . This amounts to showing thatand this is consequence of (28).

Theorem 14. The map given bydefines a Lie algebra 2-cocycle on .

Proof. It is easy to see that for , while for we haveand so for all . It follows thatand so is skew-symmetric. It remains to prove the cocycle identity (24). This a direct calculation using Corollary 5 and the fact that is a Lie algebra derivation.

Remark 15. In the case of the Lie algebra of pseudodifferential symbols on , the restriction of the cocycle (37) to the subalgebra of vector fields on is the Gelfand-Fuchs cocyclesee, for instance, [17, Prop. 4.12]. This cocycle is nontrivial; see [3], and therefore cocycle (37) is nontrivial [16]. It is also known (see, for instance, [17]) that the restriction of 2-cocycle (37) to the subalgebra of differential operator is a multiple of the Kac-Peterson cocycle [6]:Interestingly, the Lie algebra has exactly one central extension [36] but has two independent central extensions [17, 37]: in addition to (37), the following expression defines a nontrivial cocycle:We will reprove this result as a corollary of our study of central extensions of Lie algebras of formal pseudodifferential operators in several variables; see Lemma  33 in Section 3.4.

Remark 16. Let be a compact Riemann surface and let be the space of meromorphic functions on . Fix a meromorphic vector field on and denote by the operator of Lie derivative along the field : locally, if , and , then . The associative algebra of meromorphic pseudodifferential symbols is (see [4])with multiplication defined as in (7). We consider the Lie algebra structure of and the residue map , where is understood as a meromorphic differential on . We further define the trace associated to the point by . Then as a consequence of Theorem 14 we have a nontrivial 2-cocycle on given by . This cocycle first appeared in [4].