Formal Pseudodifferential Operators in One and Several Variables, Central Extensions, and Integrable Systems
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.
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 . 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  and Adler , 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 , 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 . Also, a 2-cocycle of the algebra of differential operators on the circle was constructed in , 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 .
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,  or ) 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  on hierarchies of Lie algebras of differential operators on the circle. Finally, motivated by [11, 19, 20] and the later paper , 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  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  and references therein) and the hierarchies considered by Parshin, . 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  by Dzhumadil’daev, in which he presents a classification of central extensions. The paper  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  using an inductive argument. We present a full inductive proof of Dzhumadil’daev’s theorem using some technical homological tools elsewhere; see .
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 , 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
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 , 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 .
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  (in the case of one-dimensional central extensions induced by fields of characteristic ) and in  (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 , 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, ).
Lemma 12. Let and be integers and . Then
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 , and therefore cocycle (37) is nontrivial . It is also known (see, for instance, ) that the restriction of 2-cocycle (37) to the subalgebra of differential operator is a multiple of the Kac-Peterson cocycle :Interestingly, the Lie algebra has exactly one central extension  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 )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 .
2.4. A Hierarchy of Centrally Extended Lie Algebras
is not unique in admitting nontrivial central extensions. In fact, a whole hierarchy of Lie algebras does. This fact was first observed by Khesin, , in the case .
For any positive integer , we let be the subalgebra of consisting of differential operators of the form for some nonnegative integer .
Theorem 17. Suppose that the bilinear form defined in Corollary 5 is nondegenerate on . The restriction of the 2-cocycle to defines a nontrivial central extension of this subalgebra.
Proof. Using the bilinear form defined in Corollary 5, we identify with the dual space and we have . If we assume that is a coboundary, then (22) implies that for , , and a fixed we have Then, the coefficient of satisfiesfor , . On the other hand, we have that Then, the coefficient of satisfiescomparing (45) and (47) for any , , and we find , and this contradicts the condition for and appearing after (45).
We note that in this case we cannot use Corollary 11 to prove that the 2-cocycle determines a nontrivial central extension because is not a derivation on .
2.5. The Algebra of Twisted Pseudodifferential Symbols in One Variable
We consider the algebra of twisted pseudodifferential symbols and its corresponding logarithmic cocycle following . Particular examples have appeared much earlier; see, for instance, [11, 19, 20].
Definition 18. Let be an automorphism of fixed algebra , and let .(1)A -derivation on is a linear map such that .(2)A -trace on is a linear map such that .
Given a triplet as above, the algebra of twisted formal pseudodifferential symbols is the set of all formal Laurent series in with coefficients in :equipped with a multiplication determined by the rulesFor example, for each we havewhere is a noncommutative polynomial in and with terms of total degree such that the degree of is . If , for instance, we get . We extend (50) for . We obtainand it follows that if and , thenwhere is an -tuple of integers and . The next proposition and theorem are proved in .
Proposition 19. Let be algebra, an automorphism of , a -trace on , and a -derivation on . If , then for any and any -tuple of nonnegative integers, we have
Theorem 20. Let be algebra, an automorphism of , a -trace on , and a -derivation on . If , then the linear functional defined byis a trace on .
As pointed out in , if , formulae (50) and (51) simplify toFor example, the twisted pseudodifferential operators considered in  satisfy (55). We introduce a twisted logarithmic cocycle assuming that and commute. Let be a 1-parameter group of automorphisms of with . We formally replace the integer by in (55) and obtainTaking derivatives with respect to at , as in Section 2.3 we obtain the commutation relationWe note that . The following two results are also proven in .
Proposition 21. The map defined byis a derivation.
Theorem 22. The 2-cochain is a Lie algebra 2-cocycle.
Now we go beyond . The algebra has a direct sum decomposition as a vector space,whereMoreover, we can prove a result analogous to Proposition 6, and we can also produce hierarchies of centrally extended algebras of twisted pseudodifferential symbols as in Theorem 17.
Proposition 23. for all . This implies that the bilinear form is invariant; that is, it satisfies . Also, the subalgebras and are isotropic subspaces of ; that is, the restrictions of the form to both and vanish.
Theorem 24. Suppose that the bilinear form defined in Proposition 23 is nondegenerate on . The restriction of the 2-cocycle to defines a nontrivial central extension of this subalgebra.
Proof. In addition to and , we note that it is also possible to define subalgebras of byThe proof now follows along the lines of the demonstration of Theorem 17.
3. The Algebra of Formal Pseudodifferential Symbols in Several Variables
Let be an algebra on and let , with , be (commuting) derivations on . The algebra of formal differential symbols in several variables is, by definition, the algebra generated by and symbols with the relationsfor all and . Elements of are of the form . Using (64), we can prove that
We extend the algebra to the algebra of formal pseudodifferential operators by introducing differentiations with negative exponents viaand we define a structure of Lie algebra on by the usual commutatorwhere and are determined by linearity and the rules
3.2. The Logarithm of a Symbol and the Logarithmic 2-Cocycle
Proposition 25. The linear operator defines a derivation of the (both associative and Lie) algebra for all . The Lie algebra derivation is outer.
Proof. One verifies, as in Section 2, that for any two symbols and in so that is a derivation of the associative algebra. Equation (67) implies that is also a derivation of the Lie algebra structure. That is an outer derivation is proven as in Proposition 13.
Let be a -invariant trace, , so that for all . Then, as in Section 2, we have
Proposition 26. Let be an algebra, derivations on , and a -invariant trace, . Then, the linear functional defined byis a trace on .
Proof. It suffices to prove that, for any and ,Let such that for some , , we have or ; then (73) holds by Proposition 3. Thus, without loss of generality, we assume that, for all , and . If , then using Proposition 3 we have (73) again. Now suppose that and let . We haveOn the other hand, applying (12) times and using (71), we obtain
As in Section 2.1, we define the bilinear form for . We can prove that it is symmetric and invariant, and we will assume hereafter that it is nondegenerate. Examples of nondegenerate bilinear forms as above appear in [22, 23] for special choices of algebras .
Theorem 27. The 2-cochains , , are Lie algebra 2-cocycles of .
Proof. Skew-symmetry is proven as in Theorem 14. Also, as before, a straightforward computation using that is a Lie algebra derivation yields .
3.3. Hierarchies of Centrally Extended Algebras of Pseudodifferential Symbols in Several Variables
We present two examples of hierarchies of subalgebras of and we prove that they admit nontrivial central extensions.
For our first example, we set for fixed , and
Theorem 28. The cocycle defined by is a nontrivial cocycle for the subalgebra . Hence, it defines a nontrivial central extension of .
Proof. We identify with making use of the nondegenerate bilinear form . We also make the identification . Let and assume that is a coboundary. Then there exists such that for we have The coefficient of verifies thaton the th position. On the other hand, we have that The coefficient of satisfies thaton the th position. Comparing (78) with (80) for arbitrary we have that , and this is a contradiction, because and .
For our second example, let us say that if for all . Fix , and define as the subalgebra of with elements of the form such that .
Theorem 29. The cocycle generated by for each is a nontrivial cocycle in . Hence, it defines a nontrivial central extension of .
Proof. The proof is similar to the proof of Theorem 28. We see that if the cocycle generated by were trivial, then (78) and (80) would be true for all components, and we would have for all , which is impossible.
3.4. The Dzhumadil’daev Classification Theorem
We sketch a new proof of the principal theorem of . It is a new proof, in the sense that we argue by induction and we use tools from homological algebra to perform the inductive step. It is a sketch, in the sense that full homological details are left for the paper . We continue using the notation introduced in Section 3.1.
Let be the algebra of polynomials in variables , and let be the algebra of Laurent power series of the form such that the number of ’s with nonzero is finite. The action of the derivation on this algebra is determined by , with .
Let be another algebra with derivations . The tensor product becomes an associative algebra if we endow it with the multiplication ruleClearly this algebra contains as a subalgebra and we have the following.
Proposition 30. There is an isomorphism between the associative algebra of formal pseudodifferential symbols and determined by the correspondence This isomorphism determines a Lie algebra isomorphism between and .
Now, following , we let be the Lie algebra associated to with . Then, identifying , we see that an element of is of the formand that the Lie bracket on is given bywhere are derivations acting on asand .
Definition 31. We define a linear function on via and if .
It is not hard to check that is an outer derivation of the Lie algebra . The following result is the main theorem of this section.
Theorem 32. The first group of cohomology of the Lie algebra with coefficients in , , is generated (as space of outer derivations, see Proposition 7) by and , .
Proof. We sketch an inductive proof of Theorem 32. We consider first the one variable case; that is, the basic elements of the Lie algebra are and . We write instead of . Proposition 25 implies that and are outer derivations, and it is easy to check that the derivations , , and are linearly independent. Now we have the following.
Lemma 33. Let be a derivation on with . Then, there is such thatfor some .
Proof. We know that and then for all . This implies that , and thus we can write . Now, let and . We haveEquation (87) implies that except when . Thus , and so . Analogously, we have that . Also, (87) implies thatexcept when , , and . We rewrite (88) asThenNow we writefor some to be determined. Comparing (90) and (92), we have that andwith . On the other hand,Comparing (91) and (94) and using (93), we have that and