# Formal Pseudodifferential Operators in One and Several Variables, Central Extensions, and Integrable Systems

Authors: Jarnishs Beltran; Enrique G. Reyes
Journal: Advances in Mathematical Physics
 (2015)

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## 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.


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## Body

## 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 ofnonlocal 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 aunique 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 Section2 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 Section3 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 aclassification 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 Section4 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

LetA be an associative and commutative algebra, and let δ:A→A be a derivation on A; that is, δ is a linear map such that δ(ab)=aδ(b)+δ(a)b for all a,b∈A.The algebra of formal differential symbolsDO is generated by A and a symbol ξ with the relation(1)ξ∘a=aξ+δafor all a∈A. The algebra A is a subalgebra of DO and we can prove inductively that(2)ξn∘a=∑j=0nnjδjaξn-jfor all a∈A and n≥0.We extend the algebraDO to obtain thealgebra of pseudodifferential symbolsΨDO by introducing differentiations with negative exponents. A general element of ΨDO is a formal series P of the form(3)P=∑-∞naiξi,whereai∈A.We set(4)ξ-1∘a=∑i=0∞-1iδiaξ-1-i,and so (2) generalizes to(5)ξn∘a=∑j=0∞njδjaξn-jfor all n∈Z, in which the binomial coefficient is defined by(6)nj=nn-1⋯n-j+1j!for n,j∈Z. If P1=∑i=-∞Maiξi and P2=∑j=-∞Nbjξj, then(7)P1∘P2=∑k=0∞1k!∂k∂ξkP1δkP2.The Lie algebra structure onΨDO is given by the usual commutator [A,B]=A∘B-B∘A, so that, for instance,(8)ξα,aξn=∑j=1∞njδjaξα+n-j.Lemma 1.
For any nonnegative integerm and a,b∈A, we have(9)δmab=∑j=0mmjδm-jaδjb.Proof.
Proof is by induction.Letτ:A→C be a δ-invariant traceonA; that is, τ is a linear map satisfying τ(ab)=τ(ba) and τ(δ(a))=0 for all a,b∈A. For example, if A=Diff(S1),δ=d/dx, the linear functional τ:Diff(S1)→C given by τ(f)=∫S1‍f(x)dx is a δ-invariant trace.Lemma 2.
LetA be an algebra, δ a derivation on A, τ a δ-invariant trace on A, and m a positive integer. Then we can “integrate by parts”; that is, for all a,b∈A, we have(10)τbδma=-1mτaδmb.Proof.
We have thatδab=aδb+δab, and, so, 0=τδab=τaδb+τδab; then -τaδb=τδab. We now proceed by induction on m.Proposition 3.
LetA be an algebra, δ a derivation on A, and τ a δ-invariant trace. Then the linear map res:ΨDO→C defined by(11)res∑i=-∞Maiξi=τa-1is a trace on ΨDO; that is, res is linear and it satisfies res(AB)=res(BA) for all A,B∈ΨDO. This is theAdler-Manin noncommutative residue introduced in [10, 30].Proof.
We use the elementary identity(12)ab=-1bb-a-1bfor a<0 and b≥0. Since res is linear, it is sufficient to show that for any a,b∈A and m,n∈Z(13)resaξn∘bξm=resbξm∘aξn.We consider several cases.
Letm·n⩾0(14)resaξn∘bξm=∑j=0∞resnjaδjbξm+n-j=τnm+n+1aδm+n+1band this is identically zero since we obtain the coefficient of ξ-1 when j=m+n+1, and nm+n+1=0. Similarly,(15)resbξm∘aξn=τmm+n+1aδm+n+1b=0.Now we let m,n<0; then m+n-j≤n+m<-1 (j a positive integer), and(16)resaξn∘bξm=res∑j=0∞njaδjbξm+n-j.But, as m+n-j<-1, the coefficient of ξ-1 is 0, and then resaξnbξm=0. Analogously we find resbξm∘aξn=0.
Now assume thatn⩾0,  m<0. If n+m<-1 then(17)resaξn∘bξm=res∑j=0∞njaδjbξm+n-j=0because n+m-j<-1-j≤-1 and then n+m-j<-1. Analogously, resbξm∘aξn=0.
Finally, we letn+m⩾-1; set k=m+n. Then,(18)resaξn∘bξm=τnk+1aδk+1b=nk+1τaδk+1band, on the other hand, (10) and (12) imply (19)resbξm∘aξn=τmk+1bδk+1a=k-nk+1τbδk+1a=-1k+1nk+1τbδk+1a=-1k+1-1k+1nk+1τaδk+1b=nk+1τaδk+1b.Remark 4.
Our definition of residue follows the conventions of [23]. For example, if A=Diff(S1), δ=d/dx, and τ:Diff(S1)→C is given by τ(f)=∫S1f(x)dx, then, with our convention,(20)resA=res∑i=-∞naix∂i=τa-1x,which is slightly different from the notation used in [4, 16–18].Corollary 5.
The bilinear formA,B=res(A∘B) is ad-invariant; that is, it satisfies [A,B],C=A,[B,C].Now, ifP=anξn+an-1ξn-1+⋯ is a pseudodifferential symbol such that an≠0 and am=0 for all m>n, we say that n is theorder ordegree of P. The following observation has resulted to be fundamental for the theory; see, for instance, [5, 10, 17, 31]. The algebra ΨDO can be decomposed as a (vector space) direct sum ΨDO=DO⊕INT, where(21)DO=∑i=0naiξi∣ai∈A,INT=∑i=-∞-1aiξi∣ai∈A.Proposition 6.
The subalgebrasDO (of differential operators) and INT (of pseudodifferential symbols of order ≤-1) are isotropic subspaces of ΨDO with respect to the bilinear form defined in Corollary 5; that is, the restrictions of this form to both DO and INT vanish.


### 2.2. On Cohomology of Lie Algebras

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


#### 2.2.1. Basic Definitions

Suppose thatg is a Lie algebra and that A is a module over g. A q-dimensional cochain of the algebra g with coefficients in A is a skew-symmetric q-linear functional on g with values in A; the space of all such cochains is denoted by Cq(g,A) and we also set C0(g,A)=A. The differential d=dq:Cq(g,A)→Cq+1(g,A) is defined by the formula(22)dcg1,…,gq+1=∑1≤s<t≤q+1-1s+t-1·cgs,gt,g1,…,g^s,…,g^t,…,gq+1+∑1≤s≤q+1-1sgscg1,…,g^s,…,gq+1,where c∈C(g,A) and g1,…,gq+1∈g. We also set Cq(g,A)=0 for q<0 and dq=0 for q<0. We can check that dq+1∘dq=0 for all q and therefore Cq(g,A),dq∈Z is an algebraic complex. This complex is denoted by C•(g,A), while Hq(g,A) denotes the q-cohomology space of the algebra g with coefficients in A. If A is a trivial g-module, then the second sum of in the right-hand side of formula (22) vanishes and it may be ignored. If A is a field, the notations Cq(g,A), Hq(g,A) are abbreviated to Cq(g),Hq(g).


#### 2.2.2. Algebraic Interpretations of Cohomology

Aderivationδ of the Lie algebra g is a linear map δ:g→g such that δ([x,y])=[δ(x),y]+[x,δ(y)]. A derivation isinner if δ=δx(·)=[x,·], where x∈g 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.
H
1
(
g
,
g
) can be interpreted as the space of outer derivations of the algebra g.Definition 8.
A central extension of a Lie algebrag by a vector space n is a Lie algebra g~ whose underlying vector space g~=g⊕n is equipped with the following Lie bracket:(23)X,u,Y,v=X,Y,cX,Yfor some bilinear map c:g×g→n.Note thatc depends only on X and Y but not on u and v. This implies that n is the center of the Lie algebra g~.The skew-symmetry, bilinearity, and the Jacoby identity on the Lie algebrag~ are equivalent to the antisymmetry, bilinearity, and the following 2-cocycle identity for the map c:(24)cX,Y,Z+cZ,X,Y+cY,Z,X=0for any X,Y,Z∈g.Two 2-cocyclesc on g with values in n differing by a 2-coboundary give rise to isomorphic central extensions of g. We have the following result, see [32, page 33].Proposition 9.
There is a one-to-one correspondence between equivalence classes of central extensions ofg by n and elements of H2(g,n).The referee has pointed out that Proposition9 combined with Proposition 10 below allows us to effectively calculate central extensions.Proposition 10.
Letg be a Lie algebra. The space of one-dimensional central extensions H2(g,C) is isomorphic to a subspace of the first cohomology space H1(g,g∗). Specifically, if we denote by H¯1(g) the subspace of H1(g,g∗) generated by cohomology classes of cocycles ψ:g→g∗ such that(25)ψx·x=0for x∈g, then H2(g,C)≅H¯1(g).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 p>0) 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 algebrag admits an invariant, symmetric, and nondegenerate bilinear form ,, then the space of one-dimensional central extensions H2(g,C) is isomorphic to the space of outer derivations δ:g→g such that δ(x),x=0 for all x∈g.Proof.
The existence of, allows us to identify H1(g,g) with H1(g,g∗). Now, if we take ψ so that the cohomology class of ψ is in H¯1(g), we obtain a set of derivations of the form δψ+[a,·] for some a∈g. Then, invariance of , implies that ψ∈H¯1(g) if and only if δψ(x),x=0 for all x∈g.


### 2.3. Outer Derivations and Central Extensions ofΨDO

We go back to the algebraΨDO considered in Section 2.1. Following [16], we write formally the identity ξt=etlog⁡ξ. This implies that(26)ddtt=0ξt=log⁡ξ.Hence, setting α=t in (8) and differentiating at t=0 using (d/dt)t=0tj=(-1)j+1/j, we obtain(27)log⁡ξ,aξn=∑j=1∞-1j+1jδjaξn-j.It follows that if P∈ΨDO, then log⁡ξ,P is also an element of ΨDO, even though log⁡ξ itself is not.In the next proposition we will make use of the following combinatorial identity (see, for instance, [12]).Lemma 12.
Lets≥1 and k≥0 be integers and β∈Z. Then(28)-1s-1sβ-sk=∑j=ss+k-1j-1jβs+k-jjj-s.Proposition13 and Theorem 14 below were proved by Kravchenko and Khesin [16] in the case A=Diff(S1).Proposition 13.
[
log
⁡
ξ
,
·
] defines a (resp., an outer) derivation of the associative (resp., Lie) algebra ΨDO.Proof.
We note that the proposition does not follow from the fact that for any associative algebra the mapa↦[a,·] determines a derivation, since in our case log⁡ξ is not an element of ΨDO.
First of all, it is not difficult to see using (27) that log⁡ξ,Q belongs to ΨDO for any Q∈ΨDO. Now, assuming that [log⁡ξ,·] is a derivation, it is trivial to prove that it is outer derivation of the Lie algebra ΨDO: if [log⁡ξ,·]=[A,·] for some A∈ΨDO, then log⁡ξ-A belongs to the center of ΨDO, and so log⁡ξ∈ΨDO, a contradiction.
We show that[log⁡ξ,·] is a derivation. It is sufficient to prove that, for any a,b∈A,  m,n∈Z,(29)log⁡ξ,aξn∘bξm=log⁡ξ,aξn∘bξm+aξn∘log⁡ξ,bξm.Indeed, for the left side of (29) we have(30)log⁡ξ,aξn∘bξm=log⁡ξ,∑k=o∞nkaδjbξn+m-k=∑k=o∞∑j=1∞nk-1j+1jδjaδkbξn+m-k-j.On the other hand, for the right side of (29) we have(31)log⁡ξ,aξn∘bξm+aξn∘log⁡ξ,bξm=∑j=o∞-1j+1jδjaξn-jbξm+aξn∑j=o∞-1j+1jδjbξm-j=∑j=1∞∑k=0∞n-jk-1j+1jδjaδkbξn+m-k-j+∑j=1∞∑k=0∞nk-1j+1jaδkδjbξn+m-k-j=∑j=1∞∑k=0∞n-jk-1j+1jδjaδkbξn+m-k-j+∑j=1∞∑k=0∞nk-1j+1jaδk+jbξn+m-k-j.Now, for any integer r⩾1, the coefficient of ξn+k-r in (30) is(32)∑nk-1j+1jδjaδkb,where the summation is over all integers j⩾1, k⩾0 such that j+k=r. Using (9) we have(33)∑∑i=ojnkij-1j+1jδj-iaδi+kb=∑∑i=oj-1nkij-1j+1jδj-iaδi+kb+∑nk-1j+1jaδrb,where both summations are over all integers j⩾1,  k⩾0 such that j+k=r. On the other hand, for r⩾1, the coefficient ξn+k-r of (31) is(34)∑n-sl-1s+1sδsaδlb+∑nl-1s+1saδrb,where the sum is over all integers s⩾1,  l⩾0 such that s+l=r. Therefore, (33) and (34) are equal if for fixed integers s, l as above we have(35)∑nkji-1j+1j=n-sl-1s+1s,where the sum is over all integers j⩾1, k⩾0, and i⩾0 such that i=j-s, i+k=l. This amounts to showing that(36)∑j=sl+snl+s-jjj-s-1j+1j=n-sl-1s+1s,and this is consequence of (28).Theorem 14.
The mapc:ΨDO×ΨDO→C given by(37)cA,B=reslog⁡ξ,A∘Bdefines a Lie algebra 2-cocycle on ΨDO.Proof.
It is easy to see thatres([log⁡ξ,aξn])=0 for n≤-1, while for n≥0 we have(38)reslog⁡ξ,aξn=τ-1nn+1δn+1a=0,and so res([log⁡ξ,P])=0 for all P∈ΨDO. It follows that(39)cB,A=reslog⁡ξ,BA=reslog⁡ξ,BA-Blog⁡ξ,A=-cA,B,and so c is skew-symmetric. It remains to prove the cocycle identity (24). This a direct calculation using Corollary 5 and the fact that [log⁡ξ,·] is a Lie algebra derivation.Remark 15.
In the case of the Lie algebra of pseudodifferential symbols onS1, the restriction of the cocycle (37) to the subalgebra of vector fields on S1 is the Gelfand-Fuchs cocycle(40)cfx∂,gx∂=16∫f′xg′′xdx;see, 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 DO(S1)⊂ΨDO(S1) is a multiple of the Kac-Peterson cocycle [6]:(41)cf∂n,g∂m=n!m!m+n+1!∫S1fm+1gndθ.Interestingly, the Lie algebra DO(S1) has exactly one central extension [36] but ΨDO(S1) has two independent central extensions [17, 37]: in addition to (37), the following expression defines a nontrivial cocycle:(42)c~A,B=∫S1x,A∘B.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.
LetΣ be a compact Riemann surface and let M be the space of meromorphic functions on Σ. Fix a meromorphic vector field v on Σ and denote by Dv the operator of Lie derivative Lv along the field v: locally, if v(x)=f(x)∂/∂x, and g∈M, then Dv(g)=Lv(g(x))=f(x)g′(x). The associative algebra of meromorphic pseudodifferential symbols is (see [4])(43)MΨDO=∑i=-∞naiDvi∣ai∈M,with multiplication defined as in (7). We consider the Lie algebra structure of MΨDO and the residue map resD∑i=-∞n‍aiDvi=a-1Dv-1, where Dv-1 is understood as a meromorphic differential on Σ. We further define the trace associated to the point P∈Σ by TrA=resP(resD(A)). Then as a consequence of Theorem 14 we have a nontrivial 2-cocycle on MΨDO given by cv(A,B)=Tr⁡([log⁡Dv,A]B). This cocycle first appeared in [4].


### 2.4. A Hierarchy of Centrally Extended Lie Algebras

Ψ
D
O is not unique in admitting nontrivial central extensions. In fact, a whole hierarchy of Lie algebras does. This fact was first observed by Khesin, [18], in the case A=Diff(S1).For any positive integerm, we let ΨDOm be the subalgebra of ΨDO consisting of differential operators of the form ∑i=mM‍aiξi for some nonnegative integer M.Theorem 17.
Suppose that the bilinear form defined in Corollary5 is nondegenerate on ΨDO. The restriction of the 2-cocycle c(A,B)=res([log⁡ξ,A]B) to ΨDOm defines a nontrivial central extension of this subalgebra.Proof.
Using the bilinear form, defined in Corollary 5, we identify ΨDO with the dual space ΨDO∗ and we have ΨDOm∗≃∑i=-∞-m-1‍aiξi:ai∈A. If we assume that c(A,B) is a coboundary, then (22) implies that for A=∑i=mM‍aiξi, B=∑i=mM‍biξi∈ΨDOm, and a fixed L=∑i=-∞-m-1‍liξi∈ΨDOm∗ we have (44)
c
A
,
B
=
r
e
s
A
,
B
L
=
r
e
s
∑
i
=
m
M
a
i
ξ
i
,
∑
j
=
m
M
b
j
ξ
j
·
∑
k
=
-
∞
-
m
-
1
l
k
ξ
k
=
r
e
s
∑
i
,
j
,
t
,
k
,
s
,
u
=
m
,
m
,
0
,
-
∞
,
0,0
coefficient
δ
s
a
i
·
δ
t
b
j
δ
u
l
k
ξ
i
+
j
+
k
-
t
+
u
+
s
=
τ
coefficient
-
1
.Then, the coefficient of ξ-1 satisfies(45)i+j+k-t+u+s=-1for s,t,u≥0,  k≤-m-1. On the other hand, we have that (46)
c
A
,
B
=
r
e
s
log
⁡
ξ
,
A
B
=
r
e
s
log
⁡
ξ
,
∑
i
=
m
M
a
i
ξ
i
·
∑
j
=
m
M
b
j
ξ
j
=
r
e
s
∑
i
,
j
,
t
,
s
=
m
,
m
,
0,0
coefficient
δ
s
a
i
·
δ
t
b
j
ξ
i
+
j
-
s
+
t
=
τ
coefficient
-
1
.Then, the coefficient of ξ-1 satisfies(47)i+j-s+t=-1comparing (45) and (47) for any A, B, and we find k=u, and this contradicts the condition for k and u appearing after (45).We note that in this case we cannot use Corollary11 to prove that the 2-cocycle c determines a nontrivial central extension because [log⁡ξ,·] is not a derivation on ΨDOm.


### 2.5. The Algebra of Twisted Pseudodifferential Symbols in One Variable

We consider the algebra oftwisted pseudodifferential symbols and its corresponding logarithmic cocycle following [12]. Particular examples have appeared much earlier; see, for instance, [11, 19, 20].Definition 18.
Letσ:A→A be an automorphism of fixed algebra A, and let a,b∈A.(1)
Aσ-derivation on A is a linear map δ such that δ(ab)=δ(a)b+σ(a)δ(b).(2)
Aσ-trace on A is a linear map τ:A→C such that τ(ab)=τ(σ(b)a).Given a triplet(A,δ,σ) as above, the algebra of twisted formal pseudodifferential symbols ΨDOσ is the set of all formal Laurent series in ξ with coefficients in A:(48)ΨDOσ=∑i=-∞Naiξi:N∈Z,an∈A,equipped with a multiplication determined by the rules(49)ξa=σaξ+δa,ξξ-1=ξ-1ξ=1.For example, for each n≥0 we have(50)ξna=∑i=0nPi,nσ,δaξi,where Pi,n(σ,δ) is a noncommutative polynomial in σ and δ with ni terms of total degree n such that the degree of σ is i. If n=2, for instance, we get ξ2a=δ2(a)+(δσ(a)+σδ(a))ξ+σ2(a)ξ2. We extend (50) for n<0. We obtain(51)ξna=∑i1=0∞⋯∑in=0∞-1i1+⋯+in·σ-1δσ-1in⋯σ-1δσ-1i1aξn-i1-⋯-1n,and it follows that if A=∑n=-∞Nanξn and B=∑m=-∞Mbmξm, then(52)AB=∑m=-∞M∑n<0∑i≥0-1ianσ-1δσ-1i-n⋯σ-1δσ-1i1bmξm+n+i+∑m=-∞M∑n=0N∑j=0nanPj,nσ,δbmξm+j,where i=(i1,…,in) is an n-tuple of integers and |i|=i1+⋯+in. The next proposition and theorem are proved in [12].Proposition 19.
LetA be algebra, σ an automorphism of A, τ a σ-trace on A, and δ a σ-derivation on A. If τ∘δ=0, then for any a,b∈A and any m-tuple i=(i1,…,im) of nonnegative integers, we have(53)τbσ-1δσ-1i1⋯σ-1δσ-1ima=-1i1+⋯+imτaδimσδim-1⋯σδi1b.Theorem 20.
LetA be algebra, σ an automorphism of A, τ a σ-trace on A, and δ a σ-derivation on A. If τ∘δ=0, then the linear functional res:ΨDOσ→C defined by(54)res∑i=-∞naiξi=τa-1is a trace on ΨDOσ.As pointed out in [12], if δ∘σ=σ∘δ, formulae (50) and (51) simplify to(55)ξna=∑j=0∞njδjσn-jaξn-j.For example, the twisted pseudodifferential operators considered in [11] satisfy (55). We introduce a twisted logarithmic cocycleassuming that δ and σ commute. Let σt be a 1-parameter group of automorphisms of A with σ1=σ. We formally replace the integer n by t∈R in (55) and obtain(56)ξta-σtaξt=∑j=1∞tjδjσt-jξt-j.Taking derivatives with respect to t at t=0, as in Section 2.3 we obtain the commutation relation(57)log⁡ξ,a=ddtt=0σta+∑j=1∞-1j-1jσ-jδjaξ-j.We note that [log⁡ξ,ξ]=0. The following two results are also proven in [12].Proposition 21.
The map[log⁡ξ,·]:ΨDOσ→ΨDOσ defined by(58)log⁡ξ,aξn=ddtt=0σtaξn+∑j=1∞-1j-1jσ-jδjaξ-jis a derivation.Theorem 22.
The 2-cochainc(A,B)=res([log⁡ξ,A]B) is a Lie algebra 2-cocycle.Now we go beyond [12]. The algebra Ψσ has a direct sum decomposition as a vector space,(59)ΨDOσ=DOσ⊕INTσ,where(60)DOσ=∑i=0naiξi∣ai∈A,INTσ=∑i=-∞-1aiξi∣ai∈A.Moreover, 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.
r
e
s
(
[
A
,
B
]
)
=
0 for all A,B∈ΨDOσ. This implies that the bilinear form A,Bσ=res(AB) is invariant; that is, it satisfies [A,B],Cσ=A,[B,C]σ. Also, the subalgebras DOσ and INTσ are isotropic subspaces of ΨDO; that is, the restrictions of the form ,σ to both DOσ and INTσ vanish.Theorem 24.
Suppose that the bilinear form defined in Proposition23 is nondegenerate on ΨDOσ. The restriction of the 2-cocycle c(A,B)=res([log⁡ξ,A]B) to ΨDOσ,m defines a nontrivial central extension of this subalgebra.Proof.
In addition toDOσ and INTσ, we note that it is also possible to define subalgebras ΨDOσ,m of ΨDOσ by(61)ΨDOσ,m=∑i=mMaiξi∣ai∈A.The proof now follows along the lines of the demonstration of Theorem 17.


## 2.1. Basic Definitions and Preliminary Results

LetA be an associative and commutative algebra, and let δ:A→A be a derivation on A; that is, δ is a linear map such that δ(ab)=aδ(b)+δ(a)b for all a,b∈A.The algebra of formal differential symbolsDO is generated by A and a symbol ξ with the relation(1)ξ∘a=aξ+δafor all a∈A. The algebra A is a subalgebra of DO and we can prove inductively that(2)ξn∘a=∑j=0nnjδjaξn-jfor all a∈A and n≥0.We extend the algebraDO to obtain thealgebra of pseudodifferential symbolsΨDO by introducing differentiations with negative exponents. A general element of ΨDO is a formal series P of the form(3)P=∑-∞naiξi,whereai∈A.We set(4)ξ-1∘a=∑i=0∞-1iδiaξ-1-i,and so (2) generalizes to(5)ξn∘a=∑j=0∞njδjaξn-jfor all n∈Z, in which the binomial coefficient is defined by(6)nj=nn-1⋯n-j+1j!for n,j∈Z. If P1=∑i=-∞Maiξi and P2=∑j=-∞Nbjξj, then(7)P1∘P2=∑k=0∞1k!∂k∂ξkP1δkP2.The Lie algebra structure onΨDO is given by the usual commutator [A,B]=A∘B-B∘A, so that, for instance,(8)ξα,aξn=∑j=1∞njδjaξα+n-j.Lemma 1.
For any nonnegative integerm and a,b∈A, we have(9)δmab=∑j=0mmjδm-jaδjb.Proof.
Proof is by induction.Letτ:A→C be a δ-invariant traceonA; that is, τ is a linear map satisfying τ(ab)=τ(ba) and τ(δ(a))=0 for all a,b∈A. For example, if A=Diff(S1),δ=d/dx, the linear functional τ:Diff(S1)→C given by τ(f)=∫S1‍f(x)dx is a δ-invariant trace.Lemma 2.
LetA be an algebra, δ a derivation on A, τ a δ-invariant trace on A, and m a positive integer. Then we can “integrate by parts”; that is, for all a,b∈A, we have(10)τbδma=-1mτaδmb.Proof.
We have thatδab=aδb+δab, and, so, 0=τδab=τaδb+τδab; then -τaδb=τδab. We now proceed by induction on m.Proposition 3.
LetA be an algebra, δ a derivation on A, and τ a δ-invariant trace. Then the linear map res:ΨDO→C defined by(11)res∑i=-∞Maiξi=τa-1is a trace on ΨDO; that is, res is linear and it satisfies res(AB)=res(BA) for all A,B∈ΨDO. This is theAdler-Manin noncommutative residue introduced in [10, 30].Proof.
We use the elementary identity(12)ab=-1bb-a-1bfor a<0 and b≥0. Since res is linear, it is sufficient to show that for any a,b∈A and m,n∈Z(13)resaξn∘bξm=resbξm∘aξn.We consider several cases.
Letm·n⩾0(14)resaξn∘bξm=∑j=0∞resnjaδjbξm+n-j=τnm+n+1aδm+n+1band this is identically zero since we obtain the coefficient of ξ-1 when j=m+n+1, and nm+n+1=0. Similarly,(15)resbξm∘aξn=τmm+n+1aδm+n+1b=0.Now we let m,n<0; then m+n-j≤n+m<-1 (j a positive integer), and(16)resaξn∘bξm=res∑j=0∞njaδjbξm+n-j.But, as m+n-j<-1, the coefficient of ξ-1 is 0, and then resaξnbξm=0. Analogously we find resbξm∘aξn=0.
Now assume thatn⩾0,  m<0. If n+m<-1 then(17)resaξn∘bξm=res∑j=0∞njaδjbξm+n-j=0because n+m-j<-1-j≤-1 and then n+m-j<-1. Analogously, resbξm∘aξn=0.
Finally, we letn+m⩾-1; set k=m+n. Then,(18)resaξn∘bξm=τnk+1aδk+1b=nk+1τaδk+1band, on the other hand, (10) and (12) imply (19)resbξm∘aξn=τmk+1bδk+1a=k-nk+1τbδk+1a=-1k+1nk+1τbδk+1a=-1k+1-1k+1nk+1τaδk+1b=nk+1τaδk+1b.Remark 4.
Our definition of residue follows the conventions of [23]. For example, if A=Diff(S1), δ=d/dx, and τ:Diff(S1)→C is given by τ(f)=∫S1f(x)dx, then, with our convention,(20)resA=res∑i=-∞naix∂i=τa-1x,which is slightly different from the notation used in [4, 16–18].Corollary 5.
The bilinear formA,B=res(A∘B) is ad-invariant; that is, it satisfies [A,B],C=A,[B,C].Now, ifP=anξn+an-1ξn-1+⋯ is a pseudodifferential symbol such that an≠0 and am=0 for all m>n, we say that n is theorder ordegree of P. The following observation has resulted to be fundamental for the theory; see, for instance, [5, 10, 17, 31]. The algebra ΨDO can be decomposed as a (vector space) direct sum ΨDO=DO⊕INT, where(21)DO=∑i=0naiξi∣ai∈A,INT=∑i=-∞-1aiξi∣ai∈A.Proposition 6.
The subalgebrasDO (of differential operators) and INT (of pseudodifferential symbols of order ≤-1) are isotropic subspaces of ΨDO with respect to the bilinear form defined in Corollary 5; that is, the restrictions of this form to both DO and INT vanish.


## 2.2. On Cohomology of Lie Algebras

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


### 2.2.1. Basic Definitions

Suppose thatg is a Lie algebra and that A is a module over g. A q-dimensional cochain of the algebra g with coefficients in A is a skew-symmetric q-linear functional on g with values in A; the space of all such cochains is denoted by Cq(g,A) and we also set C0(g,A)=A. The differential d=dq:Cq(g,A)→Cq+1(g,A) is defined by the formula(22)dcg1,…,gq+1=∑1≤s<t≤q+1-1s+t-1·cgs,gt,g1,…,g^s,…,g^t,…,gq+1+∑1≤s≤q+1-1sgscg1,…,g^s,…,gq+1,where c∈C(g,A) and g1,…,gq+1∈g. We also set Cq(g,A)=0 for q<0 and dq=0 for q<0. We can check that dq+1∘dq=0 for all q and therefore Cq(g,A),dq∈Z is an algebraic complex. This complex is denoted by C•(g,A), while Hq(g,A) denotes the q-cohomology space of the algebra g with coefficients in A. If A is a trivial g-module, then the second sum of in the right-hand side of formula (22) vanishes and it may be ignored. If A is a field, the notations Cq(g,A), Hq(g,A) are abbreviated to Cq(g),Hq(g).


### 2.2.2. Algebraic Interpretations of Cohomology

Aderivationδ of the Lie algebra g is a linear map δ:g→g such that δ([x,y])=[δ(x),y]+[x,δ(y)]. A derivation isinner if δ=δx(·)=[x,·], where x∈g 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.
H
1
(
g
,
g
) can be interpreted as the space of outer derivations of the algebra g.Definition 8.
A central extension of a Lie algebrag by a vector space n is a Lie algebra g~ whose underlying vector space g~=g⊕n is equipped with the following Lie bracket:(23)X,u,Y,v=X,Y,cX,Yfor some bilinear map c:g×g→n.Note thatc depends only on X and Y but not on u and v. This implies that n is the center of the Lie algebra g~.The skew-symmetry, bilinearity, and the Jacoby identity on the Lie algebrag~ are equivalent to the antisymmetry, bilinearity, and the following 2-cocycle identity for the map c:(24)cX,Y,Z+cZ,X,Y+cY,Z,X=0for any X,Y,Z∈g.Two 2-cocyclesc on g with values in n differing by a 2-coboundary give rise to isomorphic central extensions of g. We have the following result, see [32, page 33].Proposition 9.
There is a one-to-one correspondence between equivalence classes of central extensions ofg by n and elements of H2(g,n).The referee has pointed out that Proposition9 combined with Proposition 10 below allows us to effectively calculate central extensions.Proposition 10.
Letg be a Lie algebra. The space of one-dimensional central extensions H2(g,C) is isomorphic to a subspace of the first cohomology space H1(g,g∗). Specifically, if we denote by H¯1(g) the subspace of H1(g,g∗) generated by cohomology classes of cocycles ψ:g→g∗ such that(25)ψx·x=0for x∈g, then H2(g,C)≅H¯1(g).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 p>0) 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 algebrag admits an invariant, symmetric, and nondegenerate bilinear form ,, then the space of one-dimensional central extensions H2(g,C) is isomorphic to the space of outer derivations δ:g→g such that δ(x),x=0 for all x∈g.Proof.
The existence of, allows us to identify H1(g,g) with H1(g,g∗). Now, if we take ψ so that the cohomology class of ψ is in H¯1(g), we obtain a set of derivations of the form δψ+[a,·] for some a∈g. Then, invariance of , implies that ψ∈H¯1(g) if and only if δψ(x),x=0 for all x∈g.


## 2.2.1. Basic Definitions

Suppose thatg is a Lie algebra and that A is a module over g. A q-dimensional cochain of the algebra g with coefficients in A is a skew-symmetric q-linear functional on g with values in A; the space of all such cochains is denoted by Cq(g,A) and we also set C0(g,A)=A. The differential d=dq:Cq(g,A)→Cq+1(g,A) is defined by the formula(22)dcg1,…,gq+1=∑1≤s<t≤q+1-1s+t-1·cgs,gt,g1,…,g^s,…,g^t,…,gq+1+∑1≤s≤q+1-1sgscg1,…,g^s,…,gq+1,where c∈C(g,A) and g1,…,gq+1∈g. We also set Cq(g,A)=0 for q<0 and dq=0 for q<0. We can check that dq+1∘dq=0 for all q and therefore Cq(g,A),dq∈Z is an algebraic complex. This complex is denoted by C•(g,A), while Hq(g,A) denotes the q-cohomology space of the algebra g with coefficients in A. If A is a trivial g-module, then the second sum of in the right-hand side of formula (22) vanishes and it may be ignored. If A is a field, the notations Cq(g,A), Hq(g,A) are abbreviated to Cq(g),Hq(g).


## 2.2.2. Algebraic Interpretations of Cohomology

Aderivationδ of the Lie algebra g is a linear map δ:g→g such that δ([x,y])=[δ(x),y]+[x,δ(y)]. A derivation isinner if δ=δx(·)=[x,·], where x∈g 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.
H
1
(
g
,
g
) can be interpreted as the space of outer derivations of the algebra g.Definition 8.
A central extension of a Lie algebrag by a vector space n is a Lie algebra g~ whose underlying vector space g~=g⊕n is equipped with the following Lie bracket:(23)X,u,Y,v=X,Y,cX,Yfor some bilinear map c:g×g→n.Note thatc depends only on X and Y but not on u and v. This implies that n is the center of the Lie algebra g~.The skew-symmetry, bilinearity, and the Jacoby identity on the Lie algebrag~ are equivalent to the antisymmetry, bilinearity, and the following 2-cocycle identity for the map c:(24)cX,Y,Z+cZ,X,Y+cY,Z,X=0for any X,Y,Z∈g.Two 2-cocyclesc on g with values in n differing by a 2-coboundary give rise to isomorphic central extensions of g. We have the following result, see [32, page 33].Proposition 9.
There is a one-to-one correspondence between equivalence classes of central extensions ofg by n and elements of H2(g,n).The referee has pointed out that Proposition9 combined with Proposition 10 below allows us to effectively calculate central extensions.Proposition 10.
Letg be a Lie algebra. The space of one-dimensional central extensions H2(g,C) is isomorphic to a subspace of the first cohomology space H1(g,g∗). Specifically, if we denote by H¯1(g) the subspace of H1(g,g∗) generated by cohomology classes of cocycles ψ:g→g∗ such that(25)ψx·x=0for x∈g, then H2(g,C)≅H¯1(g).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 p>0) 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 algebrag admits an invariant, symmetric, and nondegenerate bilinear form ,, then the space of one-dimensional central extensions H2(g,C) is isomorphic to the space of outer derivations δ:g→g such that δ(x),x=0 for all x∈g.Proof.
The existence of, allows us to identify H1(g,g) with H1(g,g∗). Now, if we take ψ so that the cohomology class of ψ is in H¯1(g), we obtain a set of derivations of the form δψ+[a,·] for some a∈g. Then, invariance of , implies that ψ∈H¯1(g) if and only if δψ(x),x=0 for all x∈g.


## 2.3. Outer Derivations and Central Extensions ofΨDO

We go back to the algebraΨDO considered in Section 2.1. Following [16], we write formally the identity ξt=etlog⁡ξ. This implies that(26)ddtt=0ξt=log⁡ξ.Hence, setting α=t in (8) and differentiating at t=0 using (d/dt)t=0tj=(-1)j+1/j, we obtain(27)log⁡ξ,aξn=∑j=1∞-1j+1jδjaξn-j.It follows that if P∈ΨDO, then log⁡ξ,P is also an element of ΨDO, even though log⁡ξ itself is not.In the next proposition we will make use of the following combinatorial identity (see, for instance, [12]).Lemma 12.
Lets≥1 and k≥0 be integers and β∈Z. Then(28)-1s-1sβ-sk=∑j=ss+k-1j-1jβs+k-jjj-s.Proposition13 and Theorem 14 below were proved by Kravchenko and Khesin [16] in the case A=Diff(S1).Proposition 13.
[
log
⁡
ξ
,
·
] defines a (resp., an outer) derivation of the associative (resp., Lie) algebra ΨDO.Proof.
We note that the proposition does not follow from the fact that for any associative algebra the mapa↦[a,·] determines a derivation, since in our case log⁡ξ is not an element of ΨDO.
First of all, it is not difficult to see using (27) that log⁡ξ,Q belongs to ΨDO for any Q∈ΨDO. Now, assuming that [log⁡ξ,·] is a derivation, it is trivial to prove that it is outer derivation of the Lie algebra ΨDO: if [log⁡ξ,·]=[A,·] for some A∈ΨDO, then log⁡ξ-A belongs to the center of ΨDO, and so log⁡ξ∈ΨDO, a contradiction.
We show that[log⁡ξ,·] is a derivation. It is sufficient to prove that, for any a,b∈A,  m,n∈Z,(29)log⁡ξ,aξn∘bξm=log⁡ξ,aξn∘bξm+aξn∘log⁡ξ,bξm.Indeed, for the left side of (29) we have(30)log⁡ξ,aξn∘bξm=log⁡ξ,∑k=o∞nkaδjbξn+m-k=∑k=o∞∑j=1∞nk-1j+1jδjaδkbξn+m-k-j.On the other hand, for the right side of (29) we have(31)log⁡ξ,aξn∘bξm+aξn∘log⁡ξ,bξm=∑j=o∞-1j+1jδjaξn-jbξm+aξn∑j=o∞-1j+1jδjbξm-j=∑j=1∞∑k=0∞n-jk-1j+1jδjaδkbξn+m-k-j+∑j=1∞∑k=0∞nk-1j+1jaδkδjbξn+m-k-j=∑j=1∞∑k=0∞n-jk-1j+1jδjaδkbξn+m-k-j+∑j=1∞∑k=0∞nk-1j+1jaδk+jbξn+m-k-j.Now, for any integer r⩾1, the coefficient of ξn+k-r in (30) is(32)∑nk-1j+1jδjaδkb,where the summation is over all integers j⩾1, k⩾0 such that j+k=r. Using (9) we have(33)∑∑i=ojnkij-1j+1jδj-iaδi+kb=∑∑i=oj-1nkij-1j+1jδj-iaδi+kb+∑nk-1j+1jaδrb,where both summations are over all integers j⩾1,  k⩾0 such that j+k=r. On the other hand, for r⩾1, the coefficient ξn+k-r of (31) is(34)∑n-sl-1s+1sδsaδlb+∑nl-1s+1saδrb,where the sum is over all integers s⩾1,  l⩾0 such that s+l=r. Therefore, (33) and (34) are equal if for fixed integers s, l as above we have(35)∑nkji-1j+1j=n-sl-1s+1s,where the sum is over all integers j⩾1, k⩾0, and i⩾0 such that i=j-s, i+k=l. This amounts to showing that(36)∑j=sl+snl+s-jjj-s-1j+1j=n-sl-1s+1s,and this is consequence of (28).Theorem 14.
The mapc:ΨDO×ΨDO→C given by(37)cA,B=reslog⁡ξ,A∘Bdefines a Lie algebra 2-cocycle on ΨDO.Proof.
It is easy to see thatres([log⁡ξ,aξn])=0 for n≤-1, while for n≥0 we have(38)reslog⁡ξ,aξn=τ-1nn+1δn+1a=0,and so res([log⁡ξ,P])=0 for all P∈ΨDO. It follows that(39)cB,A=reslog⁡ξ,BA=reslog⁡ξ,BA-Blog⁡ξ,A=-cA,B,and so c is skew-symmetric. It remains to prove the cocycle identity (24). This a direct calculation using Corollary 5 and the fact that [log⁡ξ,·] is a Lie algebra derivation.Remark 15.
In the case of the Lie algebra of pseudodifferential symbols onS1, the restriction of the cocycle (37) to the subalgebra of vector fields on S1 is the Gelfand-Fuchs cocycle(40)cfx∂,gx∂=16∫f′xg′′xdx;see, 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 DO(S1)⊂ΨDO(S1) is a multiple of the Kac-Peterson cocycle [6]:(41)cf∂n,g∂m=n!m!m+n+1!∫S1fm+1gndθ.Interestingly, the Lie algebra DO(S1) has exactly one central extension [36] but ΨDO(S1) has two independent central extensions [17, 37]: in addition to (37), the following expression defines a nontrivial cocycle:(42)c~A,B=∫S1x,A∘B.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.
LetΣ be a compact Riemann surface and let M be the space of meromorphic functions on Σ. Fix a meromorphic vector field v on Σ and denote by Dv the operator of Lie derivative Lv along the field v: locally, if v(x)=f(x)∂/∂x, and g∈M, then Dv(g)=Lv(g(x))=f(x)g′(x). The associative algebra of meromorphic pseudodifferential symbols is (see [4])(43)MΨDO=∑i=-∞naiDvi∣ai∈M,with multiplication defined as in (7). We consider the Lie algebra structure of MΨDO and the residue map resD∑i=-∞n‍aiDvi=a-1Dv-1, where Dv-1 is understood as a meromorphic differential on Σ. We further define the trace associated to the point P∈Σ by TrA=resP(resD(A)). Then as a consequence of Theorem 14 we have a nontrivial 2-cocycle on MΨDO given by cv(A,B)=Tr⁡([log⁡Dv,A]B). This cocycle first appeared in [4].


## 2.4. A Hierarchy of Centrally Extended Lie Algebras

Ψ
D
O is not unique in admitting nontrivial central extensions. In fact, a whole hierarchy of Lie algebras does. This fact was first observed by Khesin, [18], in the case A=Diff(S1).For any positive integerm, we let ΨDOm be the subalgebra of ΨDO consisting of differential operators of the form ∑i=mM‍aiξi for some nonnegative integer M.Theorem 17.
Suppose that the bilinear form defined in Corollary5 is nondegenerate on ΨDO. The restriction of the 2-cocycle c(A,B)=res([log⁡ξ,A]B) to ΨDOm defines a nontrivial central extension of this subalgebra.Proof.
Using the bilinear form, defined in Corollary 5, we identify ΨDO with the dual space ΨDO∗ and we have ΨDOm∗≃∑i=-∞-m-1‍aiξi:ai∈A. If we assume that c(A,B) is a coboundary, then (22) implies that for A=∑i=mM‍aiξi, B=∑i=mM‍biξi∈ΨDOm, and a fixed L=∑i=-∞-m-1‍liξi∈ΨDOm∗ we have (44)
c
A
,
B
=
r
e
s
A
,
B
L
=
r
e
s
∑
i
=
m
M
a
i
ξ
i
,
∑
j
=
m
M
b
j
ξ
j
·
∑
k
=
-
∞
-
m
-
1
l
k
ξ
k
=
r
e
s
∑
i
,
j
,
t
,
k
,
s
,
u
=
m
,
m
,
0
,
-
∞
,
0,0
coefficient
δ
s
a
i
·
δ
t
b
j
δ
u
l
k
ξ
i
+
j
+
k
-
t
+
u
+
s
=
τ
coefficient
-
1
.Then, the coefficient of ξ-1 satisfies(45)i+j+k-t+u+s=-1for s,t,u≥0,  k≤-m-1. On the other hand, we have that (46)
c
A
,
B
=
r
e
s
log
⁡
ξ
,
A
B
=
r
e
s
log
⁡
ξ
,
∑
i
=
m
M
a
i
ξ
i
·
∑
j
=
m
M
b
j
ξ
j
=
r
e
s
∑
i
,
j
,
t
,
s
=
m
,
m
,
0,0
coefficient
δ
s
a
i
·
δ
t
b
j
ξ
i
+
j
-
s
+
t
=
τ
coefficient
-
1
.Then, the coefficient of ξ-1 satisfies(47)i+j-s+t=-1comparing (45) and (47) for any A, B, and we find k=u, and this contradicts the condition for k and u appearing after (45).We note that in this case we cannot use Corollary11 to prove that the 2-cocycle c determines a nontrivial central extension because [log⁡ξ,·] is not a derivation on ΨDOm.


## 2.5. The Algebra of Twisted Pseudodifferential Symbols in One Variable

We consider the algebra oftwisted pseudodifferential symbols and its corresponding logarithmic cocycle following [12]. Particular examples have appeared much earlier; see, for instance, [11, 19, 20].Definition 18.
Letσ:A→A be an automorphism of fixed algebra A, and let a,b∈A.(1)
Aσ-derivation on A is a linear map δ such that δ(ab)=δ(a)b+σ(a)δ(b).(2)
Aσ-trace on A is a linear map τ:A→C such that τ(ab)=τ(σ(b)a).Given a triplet(A,δ,σ) as above, the algebra of twisted formal pseudodifferential symbols ΨDOσ is the set of all formal Laurent series in ξ with coefficients in A:(48)ΨDOσ=∑i=-∞Naiξi:N∈Z,an∈A,equipped with a multiplication determined by the rules(49)ξa=σaξ+δa,ξξ-1=ξ-1ξ=1.For example, for each n≥0 we have(50)ξna=∑i=0nPi,nσ,δaξi,where Pi,n(σ,δ) is a noncommutative polynomial in σ and δ with ni terms of total degree n such that the degree of σ is i. If n=2, for instance, we get ξ2a=δ2(a)+(δσ(a)+σδ(a))ξ+σ2(a)ξ2. We extend (50) for n<0. We obtain(51)ξna=∑i1=0∞⋯∑in=0∞-1i1+⋯+in·σ-1δσ-1in⋯σ-1δσ-1i1aξn-i1-⋯-1n,and it follows that if A=∑n=-∞Nanξn and B=∑m=-∞Mbmξm, then(52)AB=∑m=-∞M∑n<0∑i≥0-1ianσ-1δσ-1i-n⋯σ-1δσ-1i1bmξm+n+i+∑m=-∞M∑n=0N∑j=0nanPj,nσ,δbmξm+j,where i=(i1,…,in) is an n-tuple of integers and |i|=i1+⋯+in. The next proposition and theorem are proved in [12].Proposition 19.
LetA be algebra, σ an automorphism of A, τ a σ-trace on A, and δ a σ-derivation on A. If τ∘δ=0, then for any a,b∈A and any m-tuple i=(i1,…,im) of nonnegative integers, we have(53)τbσ-1δσ-1i1⋯σ-1δσ-1ima=-1i1+⋯+imτaδimσδim-1⋯σδi1b.Theorem 20.
LetA be algebra, σ an automorphism of A, τ a σ-trace on A, and δ a σ-derivation on A. If τ∘δ=0, then the linear functional res:ΨDOσ→C defined by(54)res∑i=-∞naiξi=τa-1is a trace on ΨDOσ.As pointed out in [12], if δ∘σ=σ∘δ, formulae (50) and (51) simplify to(55)ξna=∑j=0∞njδjσn-jaξn-j.For example, the twisted pseudodifferential operators considered in [11] satisfy (55). We introduce a twisted logarithmic cocycleassuming that δ and σ commute. Let σt be a 1-parameter group of automorphisms of A with σ1=σ. We formally replace the integer n by t∈R in (55) and obtain(56)ξta-σtaξt=∑j=1∞tjδjσt-jξt-j.Taking derivatives with respect to t at t=0, as in Section 2.3 we obtain the commutation relation(57)log⁡ξ,a=ddtt=0σta+∑j=1∞-1j-1jσ-jδjaξ-j.We note that [log⁡ξ,ξ]=0. The following two results are also proven in [12].Proposition 21.
The map[log⁡ξ,·]:ΨDOσ→ΨDOσ defined by(58)log⁡ξ,aξn=ddtt=0σtaξn+∑j=1∞-1j-1jσ-jδjaξ-jis a derivation.Theorem 22.
The 2-cochainc(A,B)=res([log⁡ξ,A]B) is a Lie algebra 2-cocycle.Now we go beyond [12]. The algebra Ψσ has a direct sum decomposition as a vector space,(59)ΨDOσ=DOσ⊕INTσ,where(60)DOσ=∑i=0naiξi∣ai∈A,INTσ=∑i=-∞-1aiξi∣ai∈A.Moreover, 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.
r
e
s
(
[
A
,
B
]
)
=
0 for all A,B∈ΨDOσ. This implies that the bilinear form A,Bσ=res(AB) is invariant; that is, it satisfies [A,B],Cσ=A,[B,C]σ. Also, the subalgebras DOσ and INTσ are isotropic subspaces of ΨDO; that is, the restrictions of the form ,σ to both DOσ and INTσ vanish.Theorem 24.
Suppose that the bilinear form defined in Proposition23 is nondegenerate on ΨDOσ. The restriction of the 2-cocycle c(A,B)=res([log⁡ξ,A]B) to ΨDOσ,m defines a nontrivial central extension of this subalgebra.Proof.
In addition toDOσ and INTσ, we note that it is also possible to define subalgebras ΨDOσ,m of ΨDOσ by(61)ΨDOσ,m=∑i=mMaiξi∣ai∈A.The proof now follows along the lines of the demonstration of Theorem 17.


## 3. The Algebra of Formal Pseudodifferential Symbols in Several Variables

### 3.1. Preliminaries

Our notation mainly follows [23]. We fix n∈Z+ and we set(62)Γn=α=α1,…,αi,…,αn:αi∈Z;Γn+=α∈Γn:αi∈Z+,1≤i≤n;εi=0,…,0,1,0,…,0;ϑ=∑i=1nεi∈Γn;α!=∏i=1nαi!,αi∈Γn+.Also, ξα=ξ1α1⋯ξnαn,  δγ(b)=δ1γ1∘⋯∘δnγn(b) for linear maps δ1,…,δn, and(63)αγ=∏iαiγi,in which the binomial coefficient is defined as in Section 2.LetA be an algebra on C and let δi, with i=1,…,n, be (commuting) derivations on A. The algebra of formal differential symbols in several variables DOn is, by definition, the algebra generated by A and symbols ξi with the relations(64)ξia=aξi+δiafor all a∈A and i=1,…,n. Elements of DOn are of the form A=∑α∈Γn+aαξα. Using (64), we can prove that(65)aξα·bξβ=∑γ∈Γn+αγaδγbξα+β-γ.We extend the algebraDOn to the algebra ΨnDO of formal pseudodifferential operators by introducing differentiations with negative exponents via(66)ξi-1a=∑j≥0-1jδijaξi-1-j,and we define a structure of Lie algebra on ΨnDO by the usual commutator(67)A,B=AB-BA=∑γ∈Γn+1γ!∂γ∂ξγAδγB-∂γ∂ξγBδγA,where ∂γ/∂ξγ and δγ are determined by linearity and the rules(68)∂γ∂ξγaξα=γ!αγaξα-γ,δγaξα=δγaξα.


### 3.2. The Logarithm of a Symbol and the Logarithmic 2-Cocycle

We definelog⁡ξi for all i=1,…,n. As in Section 2, the action of log⁡ξi on the algebra ΨnDO is via the commutator [log⁡ξi,·] considered in (67):(69)log⁡ξi,X=∑γ∈Γn+1γ!∂γ∂ξγlog⁡ξiδγX=∑γi≥01γi!∂iγ∂ξiγilog⁡ξiδγiXfor all X∈ΨDOn.Proposition 25.
The linear operator[log⁡ξi,·] defines a derivation of the (both associative and Lie) algebra ΨnDO for all i=1,…,n. The Lie algebra derivation [log⁡ξi,·] is outer.Proof.
One verifies, as in Section2, that for any two symbols A and B in ΨDO(70)log⁡ξi,AB=log⁡ξi,AB+Alog⁡ξi,B,so that [log⁡ξi,·] is a derivation of the associative algebra. Equation (67) implies that [log⁡ξi,·] is also a derivation of the Lie algebra structure. That [log⁡ξi,·] is an outer derivation is proven as in Proposition 13.Letτ be a δi-invariant trace, i=1,…,n, so that τ(δi(a))=0 for all a∈A. Then, as in Section 2, we have(71)τaδγb=-1γτbδγa.Proposition 26.
LetA be an algebra, δi derivations on A, and τ a δi-invariant trace, i=1,…,n. Then, the linear functional res:ΨnDO→CΨnDO→C defined by(72)res∑α∈Γnaαξα=τa-ϑis a trace on ΨnDO.Proof.
It suffices to prove that, for anya,b∈A and α,β∈Γn,(73)resaξαbξβ=resbξβaξα.Let α,β∈Γn such that for some αi, βj, we have αi,βj≥0 or αi,βj<0; then (73) holds by Proposition 3. Thus, without loss of generality, we assume that, for all i, αi≥0 and βi<0. If αi+βi<-1, then using Proposition 3 we have (73) again. Now suppose that αi+βi≥-1 and let ki=αi+βi. We have(74)resaξαbξβ=αk+ϑτaδk+ϑ.On the other hand, applying (12) n times and using (71), we obtain (75)resbξβaξα=τβk+ϑbδk+ϑa=k-αk+ϑτbδk+ϑa=-1k+ϑαk+ϑτbδk+ϑa=-1k+ϑ-1k+ϑαk+ϑτaδk+ϑb=αk+ϑτaδk+ϑb.As in Section2.1, we define the bilinear form A,B=res(AB) for A,B∈ΨnDO. 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 A.Theorem 27.
The 2-cochainsci(A,B)=res([log⁡ξi,A]B), i=1,…,n, are Lie algebra 2-cocycles of ΨnDO.Proof.
Skew-symmetry is proven as in Theorem14. Also, as before, a straightforward computation using that [log⁡ξi,·] is a Lie algebra derivation yields ci([A,B],C)+ci([C,A],B)+ci([B,C],A)=0.


### 3.3. Hierarchies of Centrally Extended Algebras of Pseudodifferential Symbols in Several Variables

We present two examples of hierarchies of subalgebras ofΨnDO and we prove that they admit nontrivial central extensions.For our first example, we setαi=m>0 for fixed 1≤i≤n, and(76)
Ψ
n
DO
α
i
=
∑
β
∈
Γ
n
b
β
ξ
β
=
∑
β
1
,
…
,
β
n
∈
Γ
n
a
β
ξ
1
β
1
⋯
ξ
n
β
n
∈
Ψ
n
DO
:
β
i
≥
α
i
.Theorem 28.
The cocycleci defined by log⁡ξi is a nontrivial cocycle for the subalgebra ΨnDOαi. Hence, it defines a nontrivial central extension of ΨnDOαi.Proof.
We identifyΨnDO∗ with ΨnDO making use of the nondegenerate bilinear form ,. We also make the identification ΨnDOαi∗≃∑θ∈Γn‍lθξθ:θi≤-αi-1. Let A,B∈ΨnDOαi and assume that ci is a coboundary. Then there exists L∈ΨnDOαi∗ such that for A,B∈ΨnDOαi we have (77)
c
A
,
B
=
r
e
s
A
,
B
L
=
r
e
s
∑
α
∈
Γ
n
a
α
ξ
α
,
∑
β
∈
Γ
n
b
β
ξ
β
∑
θ
∈
Γ
n
θ
i
≤
-
α
i
-
1
l
θ
ξ
θ
=
r
e
s
∑
coefficient
δ
φ
a
α
δ
ν
b
β
δ
μ
l
θ
·
ξ
α
+
β
+
θ
-
ν
+
μ
+
φ
=
τ
coefficient
-
ϑ
.The coefficient of ξ-ϑ verifies that(78)αi+βi+θi+νi+μi+φi=-1on the ith position. On the other hand, we have that (79)cA,B=reslog⁡ξi,AB=reslog⁡ξi,∑α∈Γnaαξα∑β∈Γnbβξβ=res∑coefficientδiψiaαδτibβξα+β-ψi+τ=τcoefficient-ϑ.The coefficient of ξ-ϑ satisfies that(80)αi+βi+τi+φi=-1on the ith position. Comparing (78) with (80) for arbitrary A,B we have that μi=θi, and this is a contradiction, because θi≤-αi-1 and μi≥0.For our second example, let us say thatα≥β if αi≥βi for all i. Fix α∈Γn+, and define ΨnDOα as the subalgebra of ΨnDO with elements of the form ∑β∈Γnbβξβ such that β≥α.Theorem 29.
The cocycle generated bylog⁡ξi for each i is a nontrivial cocycle in ΨnDOα. Hence, it defines a nontrivial central extension of ΨnDOα.Proof.
The proof is similar to the proof of Theorem28. We see that if the cocycle generated by log⁡ξi were trivial, then (78) and (80) would be true for all n components, and we would have μi=θi for all i, which is impossible.


### 3.4. The Dzhumadil’daev Classification Theorem

We sketch a new proof of the principal theorem of [23]. 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 [24]. We continue using the notation introduced in Section 3.1.LetPn+=∑α‍λαxα∣α∈Γn+ be the algebra of polynomials in variables x1,…,xn, and let Pn be the algebra of Laurent power series of the form ∑α∈Γn‍λαxα such that the number of α∈Γn+’s with nonzero λα is finite. The action of the derivation δi on this algebra is determined by δi(xα)=αixα-ϵi, with α∈Γn.LetU be another algebra with derivations δ~1,…,δ~n. The tensor product U⊗Pn becomes an associative algebra if we endow it with the multiplication rule(81)u⊗fv⊗g=∑α∈Γn+1α!uδ~αv⊗δαfg.Clearly this algebra contains U⊗Pn+ as a subalgebra and we have the following.Proposition 30.
There is an isomorphism between the associative algebra of formal pseudodifferential symbolsΨnDO and U⊗Pn determined by the correspondence (82)ΨnDO⟶U⊗Pn,uξα⟼u⊗xα.This isomorphism determines a Lie algebra isomorphism between ΨnDO and U⊗Pn.Now, following [23], we let Hn be the Lie algebra associated to U⊗Pn with U=Pn. Then, identifying x+αx-β≃xα⊗xβ, we see that an element of Hn is of the form(83)∑α,β∈Γnλα,βx+αx-βand that the Lie bracket on Hn is given by(84)x+αx-β,x+α¯x-β¯=∑γ∈Γn+1γ!∂-∧∂+x+αx-β,x+α¯x-β¯,where ∂+,∂- are derivations acting on Hn as(85)∂-γx+αx-β=γ!βγx+αx-β-γ,∂+γx+αx-β=γ!αγx+α-γx-β,and ∂-∧∂+=∂-⊗∂+-∂+⊗∂-.Definition 31.
We define a linear functionf0 on Hn via f0(x-ϑ)=1 and f0(xα)=0 if α≠-ϑ.It is not hard to check thatf0 is an outer derivation of the Lie algebra Hn. The following result is the main theorem of this section.Theorem 32.
The first group of cohomology of the Lie algebraHn with coefficients in Hn, H1Hn,Hn, is generated (as space of outer derivations, see Proposition 7) by f0,log⁡x+i,· and log⁡x-i,·, i=1,…,n.Proof.
We sketch an inductive proof of Theorem32. We consider first the one variable case; that is, the basic elements of the Lie algebra are x+=x+1 and x-=x-1. We write H instead of H1. Proposition 25 implies that [log⁡x+,·] and [log⁡x-,·] are outer derivations, and it is easy to check that the derivations log⁡x+,·, [log⁡x-,·], and f0 are linearly independent. Now we have the following.
Lemma  33.LetDbe a derivation onHwithD(x±)≠0. Then, there isω∈Hsuch that(86)D=λlog⁡x+,·+βlog⁡x-,·-ω,·for someλ,β.
Proof. We know that [u,1]=0 and then [D(1),u]=0 for all u∈H. This implies that D(1)∈Z(H)=C, and thus we can write D(1)=λ. Now, let D(x+)=∑‍dn,m+x+nx-m and D(x-)=∑‍dn,m-x+nx-m. We have(87)1=x-,x+⟹λ=Dx-,x++x-,Dx+⟹λ=∑dn,m-x+nx-m,x++∑dn,m+x-,x+nx-m⟹λ=∑dn,m-mx+nx-m-1+∑dn,m+nx+n-1x-m.Equation (87) implies that dn,m+=0 except when n=0. Thus d0,m≠0, and so D(x+)=∑‍d0,m+x-m. Analogously, we have that D(x-)=∑‍dn,0-x+n. Also, (87) implies that(88)dn,m-mx+nx-m-1=-dn,mnx+n-1x-mexcept when n=-1, m=-1, and (n,m)=(0,0). We rewrite (88) as(89)m+1dn,m+1-=-n+1dn+1,m.Then(90)Dx+=∑m≠0,-1-d0,m-x-m+d0,-1x--1+d0,0+,(91)Dx-=∑n≠0,-1dn,0-x+n+d-1,0x+-1+d0,0-.Now we write(92)Dx+=λlog⁡x+,x++βlog⁡x-,x+-ω,x+=βx--1-ω,x+for some ω to be determined. Comparing (90) and (92), we have that β=d0,-1+ and(93)ω,x+=∑m≠0,-1d0,m-x-m+d0,0+⟹∂-ω=∑m≠0,-1d0,m-x-m+d0,0+⟹ω=∑m≠0,-1d0,m-m+1x-m+1+d0,0+x-+ω~with [ω~,x+]=0. On the other hand,(94)Dx-=λlog⁡x+,x-+βlog⁡x-,x--ω,x-=λx+-1-ω,x-.Comparing (91) and (94) and using (93), we have that λ=d0,1- and(95)ω,x-=-∑n≠0,-1dn,0-x+n+-d0,0+⟹∂-ω=-∑n≠0,-1dn,0-x+n-d0,0+⟹ω=-∑n≠0,-1dn,0-n+1x+n+1-d0,0-x++ω~~with [ω~~,x-]=0. Then, we can take ω as(96)ω=-∑n≠0,-1dn,0-n+1x+n+1+∑m≠0,-1d0,m-m+1x-m+1-d0,0-x+-d0,0-x-.
Now we need the following result, whose proof (see [24]) is similar in spirit to the one we just carried out.
Lemma  34.IfDis an outer derivation such thatD(x±)=0, thenDis a scalar multiple off0.
Lemmas  33 and  34 imply that Theorem32 holds in the case n=1. In order to perform the inductive step we use an appropriate version of the Künneth formula. Again we refer to [24].We use Theorem32 to classify central extensions of Hn: Theorem  1 of [23] tells us that Hn can be equipped with the bilinear form u,v=res(uv)=(uv)-ϑ and that this form is symmetric, nondegenerate, and invariant. Then, reasoning as in Theorem 27 we obtain that ci=res([log⁡ξ±i,A]B) and c0=res([f0,A]B) are Lie algebra cocycles of Hn. Now we use the key Corollary 11: we have that f0(A),A≠0 for A=x-ϑ and therefore f0 does not determine a central extension. On the other hand, in [23, page 135] the author shows that [log⁡ξ±i,A],A=0 for all A. We conclude that the space of central extensions of Hn is of dimension 2n. The case n=1 is discussed from a geometric point of view in [17, Remark  4.16].


## 3.1. Preliminaries

Our notation mainly follows [23]. We fix n∈Z+ and we set(62)Γn=α=α1,…,αi,…,αn:αi∈Z;Γn+=α∈Γn:αi∈Z+,1≤i≤n;εi=0,…,0,1,0,…,0;ϑ=∑i=1nεi∈Γn;α!=∏i=1nαi!,αi∈Γn+.Also, ξα=ξ1α1⋯ξnαn,  δγ(b)=δ1γ1∘⋯∘δnγn(b) for linear maps δ1,…,δn, and(63)αγ=∏iαiγi,in which the binomial coefficient is defined as in Section 2.LetA be an algebra on C and let δi, with i=1,…,n, be (commuting) derivations on A. The algebra of formal differential symbols in several variables DOn is, by definition, the algebra generated by A and symbols ξi with the relations(64)ξia=aξi+δiafor all a∈A and i=1,…,n. Elements of DOn are of the form A=∑α∈Γn+aαξα. Using (64), we can prove that(65)aξα·bξβ=∑γ∈Γn+αγaδγbξα+β-γ.We extend the algebraDOn to the algebra ΨnDO of formal pseudodifferential operators by introducing differentiations with negative exponents via(66)ξi-1a=∑j≥0-1jδijaξi-1-j,and we define a structure of Lie algebra on ΨnDO by the usual commutator(67)A,B=AB-BA=∑γ∈Γn+1γ!∂γ∂ξγAδγB-∂γ∂ξγBδγA,where ∂γ/∂ξγ and δγ are determined by linearity and the rules(68)∂γ∂ξγaξα=γ!αγaξα-γ,δγaξα=δγaξα.


## 3.2. The Logarithm of a Symbol and the Logarithmic 2-Cocycle

We definelog⁡ξi for all i=1,…,n. As in Section 2, the action of log⁡ξi on the algebra ΨnDO is via the commutator [log⁡ξi,·] considered in (67):(69)log⁡ξi,X=∑γ∈Γn+1γ!∂γ∂ξγlog⁡ξiδγX=∑γi≥01γi!∂iγ∂ξiγilog⁡ξiδγiXfor all X∈ΨDOn.Proposition 25.
The linear operator[log⁡ξi,·] defines a derivation of the (both associative and Lie) algebra ΨnDO for all i=1,…,n. The Lie algebra derivation [log⁡ξi,·] is outer.Proof.
One verifies, as in Section2, that for any two symbols A and B in ΨDO(70)log⁡ξi,AB=log⁡ξi,AB+Alog⁡ξi,B,so that [log⁡ξi,·] is a derivation of the associative algebra. Equation (67) implies that [log⁡ξi,·] is also a derivation of the Lie algebra structure. That [log⁡ξi,·] is an outer derivation is proven as in Proposition 13.Letτ be a δi-invariant trace, i=1,…,n, so that τ(δi(a))=0 for all a∈A. Then, as in Section 2, we have(71)τaδγb=-1γτbδγa.Proposition 26.
LetA be an algebra, δi derivations on A, and τ a δi-invariant trace, i=1,…,n. Then, the linear functional res:ΨnDO→CΨnDO→C defined by(72)res∑α∈Γnaαξα=τa-ϑis a trace on ΨnDO.Proof.
It suffices to prove that, for anya,b∈A and α,β∈Γn,(73)resaξαbξβ=resbξβaξα.Let α,β∈Γn such that for some αi, βj, we have αi,βj≥0 or αi,βj<0; then (73) holds by Proposition 3. Thus, without loss of generality, we assume that, for all i, αi≥0 and βi<0. If αi+βi<-1, then using Proposition 3 we have (73) again. Now suppose that αi+βi≥-1 and let ki=αi+βi. We have(74)resaξαbξβ=αk+ϑτaδk+ϑ.On the other hand, applying (12) n times and using (71), we obtain (75)resbξβaξα=τβk+ϑbδk+ϑa=k-αk+ϑτbδk+ϑa=-1k+ϑαk+ϑτbδk+ϑa=-1k+ϑ-1k+ϑαk+ϑτaδk+ϑb=αk+ϑτaδk+ϑb.As in Section2.1, we define the bilinear form A,B=res(AB) for A,B∈ΨnDO. 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 A.Theorem 27.
The 2-cochainsci(A,B)=res([log⁡ξi,A]B), i=1,…,n, are Lie algebra 2-cocycles of ΨnDO.Proof.
Skew-symmetry is proven as in Theorem14. Also, as before, a straightforward computation using that [log⁡ξi,·] is a Lie algebra derivation yields ci([A,B],C)+ci([C,A],B)+ci([B,C],A)=0.


## 3.3. Hierarchies of Centrally Extended Algebras of Pseudodifferential Symbols in Several Variables

We present two examples of hierarchies of subalgebras ofΨnDO and we prove that they admit nontrivial central extensions.For our first example, we setαi=m>0 for fixed 1≤i≤n, and(76)
Ψ
n
DO
α
i
=
∑
β
∈
Γ
n
b
β
ξ
β
=
∑
β
1
,
…
,
β
n
∈
Γ
n
a
β
ξ
1
β
1
⋯
ξ
n
β
n
∈
Ψ
n
DO
:
β
i
≥
α
i
.Theorem 28.
The cocycleci defined by log⁡ξi is a nontrivial cocycle for the subalgebra ΨnDOαi. Hence, it defines a nontrivial central extension of ΨnDOαi.Proof.
We identifyΨnDO∗ with ΨnDO making use of the nondegenerate bilinear form ,. We also make the identification ΨnDOαi∗≃∑θ∈Γn‍lθξθ:θi≤-αi-1. Let A,B∈ΨnDOαi and assume that ci is a coboundary. Then there exists L∈ΨnDOαi∗ such that for A,B∈ΨnDOαi we have (77)
c
A
,
B
=
r
e
s
A
,
B
L
=
r
e
s
∑
α
∈
Γ
n
a
α
ξ
α
,
∑
β
∈
Γ
n
b
β
ξ
β
∑
θ
∈
Γ
n
θ
i
≤
-
α
i
-
1
l
θ
ξ
θ
=
r
e
s
∑
coefficient
δ
φ
a
α
δ
ν
b
β
δ
μ
l
θ
·
ξ
α
+
β
+
θ
-
ν
+
μ
+
φ
=
τ
coefficient
-
ϑ
.The coefficient of ξ-ϑ verifies that(78)αi+βi+θi+νi+μi+φi=-1on the ith position. On the other hand, we have that (79)cA,B=reslog⁡ξi,AB=reslog⁡ξi,∑α∈Γnaαξα∑β∈Γnbβξβ=res∑coefficientδiψiaαδτibβξα+β-ψi+τ=τcoefficient-ϑ.The coefficient of ξ-ϑ satisfies that(80)αi+βi+τi+φi=-1on the ith position. Comparing (78) with (80) for arbitrary A,B we have that μi=θi, and this is a contradiction, because θi≤-αi-1 and μi≥0.For our second example, let us say thatα≥β if αi≥βi for all i. Fix α∈Γn+, and define ΨnDOα as the subalgebra of ΨnDO with elements of the form ∑β∈Γnbβξβ such that β≥α.Theorem 29.
The cocycle generated bylog⁡ξi for each i is a nontrivial cocycle in ΨnDOα. Hence, it defines a nontrivial central extension of ΨnDOα.Proof.
The proof is similar to the proof of Theorem28. We see that if the cocycle generated by log⁡ξi were trivial, then (78) and (80) would be true for all n components, and we would have μi=θi for all i, which is impossible.


## 3.4. The Dzhumadil’daev Classification Theorem

We sketch a new proof of the principal theorem of [23]. 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 [24]. We continue using the notation introduced in Section 3.1.LetPn+=∑α‍λαxα∣α∈Γn+ be the algebra of polynomials in variables x1,…,xn, and let Pn be the algebra of Laurent power series of the form ∑α∈Γn‍λαxα such that the number of α∈Γn+’s with nonzero λα is finite. The action of the derivation δi on this algebra is determined by δi(xα)=αixα-ϵi, with α∈Γn.LetU be another algebra with derivations δ~1,…,δ~n. The tensor product U⊗Pn becomes an associative algebra if we endow it with the multiplication rule(81)u⊗fv⊗g=∑α∈Γn+1α!uδ~αv⊗δαfg.Clearly this algebra contains U⊗Pn+ as a subalgebra and we have the following.Proposition 30.
There is an isomorphism between the associative algebra of formal pseudodifferential symbolsΨnDO and U⊗Pn determined by the correspondence (82)ΨnDO⟶U⊗Pn,uξα⟼u⊗xα.This isomorphism determines a Lie algebra isomorphism between ΨnDO and U⊗Pn.Now, following [23], we let Hn be the Lie algebra associated to U⊗Pn with U=Pn. Then, identifying x+αx-β≃xα⊗xβ, we see that an element of Hn is of the form(83)∑α,β∈Γnλα,βx+αx-βand that the Lie bracket on Hn is given by(84)x+αx-β,x+α¯x-β¯=∑γ∈Γn+1γ!∂-∧∂+x+αx-β,x+α¯x-β¯,where ∂+,∂- are derivations acting on Hn as(85)∂-γx+αx-β=γ!βγx+αx-β-γ,∂+γx+αx-β=γ!αγx+α-γx-β,and ∂-∧∂+=∂-⊗∂+-∂+⊗∂-.Definition 31.
We define a linear functionf0 on Hn via f0(x-ϑ)=1 and f0(xα)=0 if α≠-ϑ.It is not hard to check thatf0 is an outer derivation of the Lie algebra Hn. The following result is the main theorem of this section.Theorem 32.
The first group of cohomology of the Lie algebraHn with coefficients in Hn, H1Hn,Hn, is generated (as space of outer derivations, see Proposition 7) by f0,log⁡x+i,· and log⁡x-i,·, i=1,…,n.Proof.
We sketch an inductive proof of Theorem32. We consider first the one variable case; that is, the basic elements of the Lie algebra are x+=x+1 and x-=x-1. We write H instead of H1. Proposition 25 implies that [log⁡x+,·] and [log⁡x-,·] are outer derivations, and it is easy to check that the derivations log⁡x+,·, [log⁡x-,·], and f0 are linearly independent. Now we have the following.
Lemma  33.LetDbe a derivation onHwithD(x±)≠0. Then, there isω∈Hsuch that(86)D=λlog⁡x+,·+βlog⁡x-,·-ω,·for someλ,β.
Proof. We know that [u,1]=0 and then [D(1),u]=0 for all u∈H. This implies that D(1)∈Z(H)=C, and thus we can write D(1)=λ. Now, let D(x+)=∑‍dn,m+x+nx-m and D(x-)=∑‍dn,m-x+nx-m. We have(87)1=x-,x+⟹λ=Dx-,x++x-,Dx+⟹λ=∑dn,m-x+nx-m,x++∑dn,m+x-,x+nx-m⟹λ=∑dn,m-mx+nx-m-1+∑dn,m+nx+n-1x-m.Equation (87) implies that dn,m+=0 except when n=0. Thus d0,m≠0, and so D(x+)=∑‍d0,m+x-m. Analogously, we have that D(x-)=∑‍dn,0-x+n. Also, (87) implies that(88)dn,m-mx+nx-m-1=-dn,mnx+n-1x-mexcept when n=-1, m=-1, and (n,m)=(0,0). We rewrite (88) as(89)m+1dn,m+1-=-n+1dn+1,m.Then(90)Dx+=∑m≠0,-1-d0,m-x-m+d0,-1x--1+d0,0+,(91)Dx-=∑n≠0,-1dn,0-x+n+d-1,0x+-1+d0,0-.Now we write(92)Dx+=λlog⁡x+,x++βlog⁡x-,x+-ω,x+=βx--1-ω,x+for some ω to be determined. Comparing (90) and (92), we have that β=d0,-1+ and(93)ω,x+=∑m≠0,-1d0,m-x-m+d0,0+⟹∂-ω=∑m≠0,-1d0,m-x-m+d0,0+⟹ω=∑m≠0,-1d0,m-m+1x-m+1+d0,0+x-+ω~with [ω~,x+]=0. On the other hand,(94)Dx-=λlog⁡x+,x-+βlog⁡x-,x--ω,x-=λx+-1-ω,x-.Comparing (91) and (94) and using (93), we have that λ=d0,1- and(95)ω,x-=-∑n≠0,-1dn,0-x+n+-d0,0+⟹∂-ω=-∑n≠0,-1dn,0-x+n-d0,0+⟹ω=-∑n≠0,-1dn,0-n+1x+n+1-d0,0-x++ω~~with [ω~~,x-]=0. Then, we can take ω as(96)ω=-∑n≠0,-1dn,0-n+1x+n+1+∑m≠0,-1d0,m-m+1x-m+1-d0,0-x+-d0,0-x-.
Now we need the following result, whose proof (see [24]) is similar in spirit to the one we just carried out.
Lemma  34.IfDis an outer derivation such thatD(x±)=0, thenDis a scalar multiple off0.
Lemmas  33 and  34 imply that Theorem32 holds in the case n=1. In order to perform the inductive step we use an appropriate version of the Künneth formula. Again we refer to [24].We use Theorem32 to classify central extensions of Hn: Theorem  1 of [23] tells us that Hn can be equipped with the bilinear form u,v=res(uv)=(uv)-ϑ and that this form is symmetric, nondegenerate, and invariant. Then, reasoning as in Theorem 27 we obtain that ci=res([log⁡ξ±i,A]B) and c0=res([f0,A]B) are Lie algebra cocycles of Hn. Now we use the key Corollary 11: we have that f0(A),A≠0 for A=x-ϑ and therefore f0 does not determine a central extension. On the other hand, in [23, page 135] the author shows that [log⁡ξ±i,A],A=0 for all A. We conclude that the space of central extensions of Hn is of dimension 2n. The case n=1 is discussed from a geometric point of view in [17, Remark  4.16].


## 4. Applications

### 4.1. Manin Triples and Double Extensions of (Twisted) Pseudodifferential Symbols

Manin triples are ubiquitous in integrable systems; see, for instance, [17]. They were introduced by Drinfel’d in his seminal paper [15] on Hopf algebras and the quantum inverse scattering method.Definition 35.
Three Lie algebrasg¯, g-, and g+ form a Manin triple if the following conditions are satisfied.(1)
The Lie algebrasg- and g+ are Lie subalgebras of g¯ such that g¯=g-⊕g- as vector spaces.(2)
There exists a nondegenerate invariant bilinear form ong¯ such that g+ and g- are isotropic subspaces; that is, the restrictions of this form to both g+ and g- vanish.For example, if we assume that the bilinear formA,B=res(AB) on ΨDO is nondegenerate, then Proposition 6 tells us that the algebras ΨDO, DO, and INT form a Manin triple with respect to the bilinear form A,B. In the same way, Proposition 23 implies that the algebras ΨDOσ, DOσ, and INTσ form a Manin triple with respect to the twisted bilinear form A,Bσ.Let us discuss the several variables case in some detail. We say that the order of the pseudodifferential operatorL=∑α∈Γnaαξα is N, if there is α=(α1,…,αn) with αn=N such that aαn≠0 and for all β=(β1,…,βn) such that βn>N we have aβ=0. This definition is analogous to the notion of order used by Parshin in [22].The Lie algebraΨnDO has two natural subalgebras:(97)INTn=L∈ΨnDO∣ordL<0,DOn=L∈ΨnDO∣ordL≥0,and clearly, as a vector space, ΨnDO is a direct sum of these algebras: ΨnDO=INTn⊕DOn.Proposition 36.
If the bilinear formA,B=res(AB) is nondegenerate, the algebras (ΨnDO,INTn,DOn) form a Manin triple with respect to ,.Proof.
We reason as in the proof of Proposition6.We can also define order inΨnDO by using a fixed coordinate ai instead of αN, i≠N. Then, the Lie algebra ΨnDO has two natural subalgebras:(98)INTi=∑…,-∞,…∈Γn…,-1,…aαξα∣aα∈A,DOi=∑…,0,…∈Γn…,n,…aαξα∣aα∈A.As before we have that as a vector space, ΨnDO=INTi⊕DOi.Proposition 37.
The algebras(ΨnDO,INTi,DOi) form a Manin triple with respect to the bilinear form A,B=res(AB).Proof.
This is again analogous to the proof of Proposition6.Now we show how to construct further Manin triples with the help of central extensions using a general result by Bordemann, [25], which we now recall.LetA a nonassociative algebra over a field K and assume that f is a bilinear functional with the following properties:(1)
f(ab,c)=f(a,bc) for all a,b,c∈A.(2)
f(a,b)=0 for all b∈A implies a=0, and f(a,b)=0 for all a∈A implies b=0.The pair(A,f) is called apseudometrized algebra or if f is symmetric, ametrized algebra. Bordemann’s result is the following [25, Theorem  2.2].Theorem 38.
Let(A,f) be a metrized algebra over a field K, and let B be a Lie algebra over K. Suppose that there is a Lie homomorphism ϕ:B→Derf(A), where Derf(A) denotes the space of all f-antisymmetric derivations of A (i.e., the derivations d of A for which f(da,a~)+f(a,da~)=0 for all a,a~∈A).
LetB∗ denote the dual space of B. We define w:A×A→B∗ as the bilinear antisymmetric map (99)a,a~⟼b⟼fϕba,a~,and for b∈B and β∈B∗ we write b·β for the coadjoint representation; that is, (100)b·βb~=-βbb~.
We consider the vector spaceAB=B⊕A⊕B∗ and we define the following multiplication for b,b~∈B,  a,a~∈A, and β,β~∈B∗:(101)b+a+βb~+a~+β~=bb~+ϕba~-ϕb~a+aa~+wa,a~+b·β~-b~·β.Moreover, we define the following symmetric bilinear form fB on AB:(102)fBb+a+β,b~+a~+β~=βb~+β~b+fa,a~.The pair (AB,fB) is a metrized algebra over K called the double extension of A by (β,ψ).We sketch the proof of Theorem38 in the case of interest for us. We assume that A is a Lie algebra and that f(a,b)=(a,b) is a bilineal symmetric form on A which is invariant and nondegenerate. Then, it is in fact easy to check that the bilineal form (102) is symmetric, nondegenerate and invariant. On the other hand, that (101) defines a Lie bracket on AB follows from a straightforward computation using the identities(103)wa,a′,a~=wa′,a,a~-wa~,a,a′,b·wa′,a~=wa′,ϕb·a~+wϕb·a′,a~for a,a′,a~∈A and b∈B.As a first application we considerA=ΨDO, f(L1,L2)=L1,L2 as in Section 2, and B={b[log⁡ξ,·]:b∈R}, B∗=R, and we set ϕ:B→Derf(A) as the inclusion map. Then, w(L1,L2)=[log⁡ξ,L1],L2 and Theorem 38 gives us the double extension ΨDO~=B⊕ΨDO⊕B∗ with bracket(104)
b
1
,
L
1
,
β
1
,
b
2
,
L
2
,
β
2
=
0
,
b
1
log
⁡
ξ
,
L
2
-
b
2
log
⁡
ξ
,
L
1
+
L
1
,
L
2
,
log
⁡
ξ
,
L
1
,
L
2and invariant symmetric linear form(105)b1,L1,β1,b2,L2,β2=β1b2+β2b1+resL1L2.This extension is certainly known; see [17]. In an analogous way we obtain a double extension ΨDOσ~ of the Lie algebra of twisted pseudodifferential operators.Now, we note that the Lie algebraΨDO~ (and also ΨDOσ~) admits the (vector space) direct sum decomposition ΨDO~=DO^⊕INT~ (resp., ΨDOσ~=DOσ^⊕INTσ~), in which(106)DO^=αc+∑i=0naiξi∣ai∈A,α∈R,INT~=βlog⁡ξ+∑i=-∞-1aiξi∣ai∈A,β∈R,and analogous definitions for the twisted case. We have the following.Corollary 39.
If the bilinear form (102) is nondegenerate, then (107)ΨDO~,DO^,INT~,ΨσDO~,DOσ^,INTσ~are Manin triples.Now we consider the doubly extended algebraΨnDO~ of ΨnDO, that is, the semidirect product of ΨnDO^=ΨnDO⊕R·ci and the space of derivations βlog⁡ξi∣β∈R. We have the vector space decomposition(108)ΨnDO~=INTi~⊕DOi^,where(109)INTi~=βlog⁡ξi+∑…,-∞,…∈Γn…,-1,…aθξθ∣aθ∈A,DOi^=αci+∑…,0,…∈Γn…,n,…aθξθ∣aθ∈A.Theorem 38 yields the corollary.Corollary 40.
The algebras(ΨnDO~,INTi~,DOi^) form a Manin triple with respect to the bilinear form A+b1ci+α1log⁡ξi,B+b2ci+α2log⁡ξi=res(AB)+b2α1+b1α2 and Lie bracket [(A,b1,α1),(B,b2,α2)]  = ([A,B]+[α2log⁡ξi,A]-[α1log⁡ξi,B],ci(A,B),0).Theorem38 also allows us to define a double extension of ΨnDO~ by considering the n-central extension R·c1⊕⋯R·cn and the n symbols log⁡ξi. We have the vector space decomposition(110)ΨnDO~=ΨnDO⊕R·c1⊕⋯⊕R·cn⊕R·log⁡ξ1⊕⋯⊕R·log⁡ξn,with Lie bracket given by(111)
A
,
α
,
β
,
B
,
α
¯
,
β
¯
=
A
,
B
+
β
¯
·
log
⁡
ξ
,
A
-
β
·
log
⁡
ξ
,
B
,
c
A
,
B
,
0
,and bilinear form(112)A,B=resA,B+α·β¯+β·α¯,where α, β, α¯, and β¯ denote n-tuples, log⁡ξ=(log⁡ξ1,log⁡ξ2,…,log⁡ξn), 0=(0,…,0)∈Rn, and c(A,B)=(c1(A,B),…,cn(A,B)). The operation “·” is the usual dot product in Rn; for example, β·log⁡ξ=β1log⁡ξ+⋯+βnlog⁡ξn.


### 4.2. Hierarchies of Differential Equations

In this last section we apply our work to the construction of hierarchies of partial differential equations. We begin with some standard facts on hamiltonian systems modelled on (dual spaces of) Lie algebras (see, for instance, [17] or the recent paper [38]).Definition 41.
Ifg is a Lie algebra and g∗ is its dual space, the functional derivative of a function F:g∗→R at μ∈g∗ is the unique element δF/δμ of g determined by(113)ν,δFδμ=ddϵϵ=0Fμ+ϵνfor all ν∈g∗, in which , denotes a natural paring between g and g∗.Definition 42.
The Lie-Poisson bracket on the dual spaceg∗ is given by(114)F,Gμ=μ,δFδμ,δGδμ,in which F,G:g∗→R and μ∈g∗.The corresponding equations of motion for a HamiltonianH:g∗→R are(115)dμdt=μ,·,δHδμ.Definition 43.
Letg,, be a Lie algebra and R:g→g a linear operator on g. We define a bilinear antisymmetric bracket ,R on g by(116)P,QR=RP,Q+P,RQQ,P∈g.The operator R is called a classical R-matrix if the bracket ,R satisfies the Jacobi identity.Ifg=g-⊕g+ as vector space, we choose R=(1/2)π+-π-, where π± are the projection operators on g±. We write P±=π±(P) for P∈g. Then, [P,Q]R=[P+,Q+]-[P-,Q-] and(117)F,GR=μ,δFδμ,δGδμR.If F:g∗→R satisfies the infinitesimal Ad∗-invariance condition(118)μ,δFδμ,·=0for all μ∈g∗, bracket (117) becomes(119)F,GR=μ,δFδμ+,δGδμ.It follows that if H is a hamiltonian function on g∗ satisfying (118), the corresponding hamiltonian vector field is given by (120)XHμ·F=μ,-δHδμ+,δFδμand, if there exists a nondegenerate pairing , between g and g∗ such that the invariance condition (121)PQ,R=R,P,Qholds for P,Q,R∈g, then the corresponding equations of motion with respect to the Poisson structure (119) can be written as equations on g as(122)dPdt=δHδμ+,P.


#### 4.2.1. The KP Hierarchy

We apply the above remarks to the Lie algebraΨDO. Let us recall that we have a bilinear symmetric form on ΨDO which satisfies(123)P,Q,S=S,P,QS,P,Q∈ΨDO.We assume that this form is nondegenerate. The decomposition ΨDO=INT⊕DO=ΨDO-⊕ΨDO+ allows us to define a new structure of Lie algebra ,R on ΨDO using R-matrices as above. The following proposition is standard; see, for instance, [5] or the recent work [38].Proposition 44.
LetHk(L)=res(Lk), k=1,2,… for L∈ΨDO. Then, (124)δHkδL=kLk-1.Moreover, for any functional F on ΨDO∗, Hk,F(L)=0, and Hk,HlR(L)=0 for any R-matrix R.Equation (122) gives us the standard Kadomtsev-Petviashvili hierarchy as a Hamiltonian equation.Theorem 45.
LetHk(L)=res(Lk) for L∈ΨDO. The corresponding Hamiltonian equation of motion with respect to ·,·R is(125)∂L∂t=kLk-1+,L.


#### 4.2.2. The Twisted KP Hierarchy

We consider (59) and Proposition 23. The Manin triple ΨDOσ, DOσ, and INTσ allows us to consider a new Lie bracket ,R on ΨDOσ as in the previous subsection. Proposition 44 is also valid in the twisted context (see the proof appearing, for instance, in [38]), and therefore (122) yields the following result.Theorem 46.
LetHk(L)=res(Lk) for L∈ΨDOσ. The corresponding Hamiltonian equation of motion with respect to ·,·R is(126)∂L∂t=kLk-1+,L.A special case of (126) appears in [11]; see (7.4) therein.


#### 4.2.3. Then-KP Hierarchy

Now we consider Proposition36. As before, the Manin triple ΨnDO, DOn, and INTn allows us to induce a new Lie algebra structure on ΨnDO with bracket ,R. Once more we set Hk(L)=res(Lk) for L∈ΨnDO. Equation (122) yields the equation of motion(127)∂L∂t=kLk-1+,L.This is an n-variables generalization of KP of the kind considered by Parshin in [22].


#### 4.2.4. Zero Curvature Equations

Now we add a central extensionc to ΨDO and we study ΨDO^=ΨDO⊕R.Definition 47.
Let(A,α),(B,β)∈ΨDO^. We define a bilinear form on ΨDO^ by(128)A,α,B,β=A,B+αβ=resAB+αβ.We use this bilinear form to identify the dual space ofΨDO^ with itself.Proposition 48.
LetH:(ΨDO^)∗→R be a Hamiltonian function. The Hamiltonian equations of motion with respect to the bracket (114) are (129)∂L∂t,Q=∂H∂L,L,Q+lcQ,δHδL,(130)dldt=0,for any Q∈ΨDO, in which (L~,l~)=(L,l),(·,·)∈(ΨDO^)∗.Proof.
The Hamiltonian equations of motion on(ΨDO^)∗ with Hamiltonian H are(131)dL~,l~dt·Q,r=L~,l~,Q,r,δHδL~,l~=L~,l~,Q,r,δHδL~,δHδl~=L~,l~,Q,δHδL~,cQ,δHδL~.If we set (L~,l~)=(L,l),(·,·) for some (L,l)∈ΨDO^, the last equation becomes(132)∂L,l∂t,Q,r=L,l,Q,δHδL,cQ,δHδL=L,Q,δHδL+lcQ,δHδL.Using invariance of the bilinear form ,, we write this equation as(133)∂L∂t,Q+r∂l∂t=δHδL,L,Q+lcQ,δHδL.Since this equation is valid for any (Q,r), we conclude that dl/dt=0 and we obtain (129).In the context of current algebras, (129) and (130) yield the zero curvature equations obtained by Reyman and Semenov-Tian-Shansky in [28].Finally, we restrict our attention to the subalgebraVect={aξ:a∈A}. In this case, we can compute the cocycle c rather easily:(134)reslog⁡ξ,aξbξ=resbξlog⁡ξ,aξ=resbξ∑j=1∞-1j+1jδjaξ1-j=res∑j=0∞-1j+1jbδjaξ2-j+bδj+1aξ1-j=-13+13bδ3a+-12+12bδ3a=-16τbδ3a.We consider the subalgebraVect⊕R of ΨDO^, and we equip it with the bilinear form(135)aξ,α,bξ,β=τab+αβ.Then, if we set (L,l)=(uξ,l), Q=bξ, and δH/δL=h(u)ξ, and we use (129) and (130), which were derived without using the specific form of the inner product on ΨDO^, we obtain(136)dudtξ,bξ=huξ,uξ,bξ-16lτbδ3hu;that is, using that τ is nondegenerate,(137)dudt=huδu-uδhu-16lδ3hu,an abstract form of the ubiquitous Korteweg-de Vries equation.


## 4.1. Manin Triples and Double Extensions of (Twisted) Pseudodifferential Symbols

Manin triples are ubiquitous in integrable systems; see, for instance, [17]. They were introduced by Drinfel’d in his seminal paper [15] on Hopf algebras and the quantum inverse scattering method.Definition 35.
Three Lie algebrasg¯, g-, and g+ form a Manin triple if the following conditions are satisfied.(1)
The Lie algebrasg- and g+ are Lie subalgebras of g¯ such that g¯=g-⊕g- as vector spaces.(2)
There exists a nondegenerate invariant bilinear form ong¯ such that g+ and g- are isotropic subspaces; that is, the restrictions of this form to both g+ and g- vanish.For example, if we assume that the bilinear formA,B=res(AB) on ΨDO is nondegenerate, then Proposition 6 tells us that the algebras ΨDO, DO, and INT form a Manin triple with respect to the bilinear form A,B. In the same way, Proposition 23 implies that the algebras ΨDOσ, DOσ, and INTσ form a Manin triple with respect to the twisted bilinear form A,Bσ.Let us discuss the several variables case in some detail. We say that the order of the pseudodifferential operatorL=∑α∈Γnaαξα is N, if there is α=(α1,…,αn) with αn=N such that aαn≠0 and for all β=(β1,…,βn) such that βn>N we have aβ=0. This definition is analogous to the notion of order used by Parshin in [22].The Lie algebraΨnDO has two natural subalgebras:(97)INTn=L∈ΨnDO∣ordL<0,DOn=L∈ΨnDO∣ordL≥0,and clearly, as a vector space, ΨnDO is a direct sum of these algebras: ΨnDO=INTn⊕DOn.Proposition 36.
If the bilinear formA,B=res(AB) is nondegenerate, the algebras (ΨnDO,INTn,DOn) form a Manin triple with respect to ,.Proof.
We reason as in the proof of Proposition6.We can also define order inΨnDO by using a fixed coordinate ai instead of αN, i≠N. Then, the Lie algebra ΨnDO has two natural subalgebras:(98)INTi=∑…,-∞,…∈Γn…,-1,…aαξα∣aα∈A,DOi=∑…,0,…∈Γn…,n,…aαξα∣aα∈A.As before we have that as a vector space, ΨnDO=INTi⊕DOi.Proposition 37.
The algebras(ΨnDO,INTi,DOi) form a Manin triple with respect to the bilinear form A,B=res(AB).Proof.
This is again analogous to the proof of Proposition6.Now we show how to construct further Manin triples with the help of central extensions using a general result by Bordemann, [25], which we now recall.LetA a nonassociative algebra over a field K and assume that f is a bilinear functional with the following properties:(1)
f(ab,c)=f(a,bc) for all a,b,c∈A.(2)
f(a,b)=0 for all b∈A implies a=0, and f(a,b)=0 for all a∈A implies b=0.The pair(A,f) is called apseudometrized algebra or if f is symmetric, ametrized algebra. Bordemann’s result is the following [25, Theorem  2.2].Theorem 38.
Let(A,f) be a metrized algebra over a field K, and let B be a Lie algebra over K. Suppose that there is a Lie homomorphism ϕ:B→Derf(A), where Derf(A) denotes the space of all f-antisymmetric derivations of A (i.e., the derivations d of A for which f(da,a~)+f(a,da~)=0 for all a,a~∈A).
LetB∗ denote the dual space of B. We define w:A×A→B∗ as the bilinear antisymmetric map (99)a,a~⟼b⟼fϕba,a~,and for b∈B and β∈B∗ we write b·β for the coadjoint representation; that is, (100)b·βb~=-βbb~.
We consider the vector spaceAB=B⊕A⊕B∗ and we define the following multiplication for b,b~∈B,  a,a~∈A, and β,β~∈B∗:(101)b+a+βb~+a~+β~=bb~+ϕba~-ϕb~a+aa~+wa,a~+b·β~-b~·β.Moreover, we define the following symmetric bilinear form fB on AB:(102)fBb+a+β,b~+a~+β~=βb~+β~b+fa,a~.The pair (AB,fB) is a metrized algebra over K called the double extension of A by (β,ψ).We sketch the proof of Theorem38 in the case of interest for us. We assume that A is a Lie algebra and that f(a,b)=(a,b) is a bilineal symmetric form on A which is invariant and nondegenerate. Then, it is in fact easy to check that the bilineal form (102) is symmetric, nondegenerate and invariant. On the other hand, that (101) defines a Lie bracket on AB follows from a straightforward computation using the identities(103)wa,a′,a~=wa′,a,a~-wa~,a,a′,b·wa′,a~=wa′,ϕb·a~+wϕb·a′,a~for a,a′,a~∈A and b∈B.As a first application we considerA=ΨDO, f(L1,L2)=L1,L2 as in Section 2, and B={b[log⁡ξ,·]:b∈R}, B∗=R, and we set ϕ:B→Derf(A) as the inclusion map. Then, w(L1,L2)=[log⁡ξ,L1],L2 and Theorem 38 gives us the double extension ΨDO~=B⊕ΨDO⊕B∗ with bracket(104)
b
1
,
L
1
,
β
1
,
b
2
,
L
2
,
β
2
=
0
,
b
1
log
⁡
ξ
,
L
2
-
b
2
log
⁡
ξ
,
L
1
+
L
1
,
L
2
,
log
⁡
ξ
,
L
1
,
L
2and invariant symmetric linear form(105)b1,L1,β1,b2,L2,β2=β1b2+β2b1+resL1L2.This extension is certainly known; see [17]. In an analogous way we obtain a double extension ΨDOσ~ of the Lie algebra of twisted pseudodifferential operators.Now, we note that the Lie algebraΨDO~ (and also ΨDOσ~) admits the (vector space) direct sum decomposition ΨDO~=DO^⊕INT~ (resp., ΨDOσ~=DOσ^⊕INTσ~), in which(106)DO^=αc+∑i=0naiξi∣ai∈A,α∈R,INT~=βlog⁡ξ+∑i=-∞-1aiξi∣ai∈A,β∈R,and analogous definitions for the twisted case. We have the following.Corollary 39.
If the bilinear form (102) is nondegenerate, then (107)ΨDO~,DO^,INT~,ΨσDO~,DOσ^,INTσ~are Manin triples.Now we consider the doubly extended algebraΨnDO~ of ΨnDO, that is, the semidirect product of ΨnDO^=ΨnDO⊕R·ci and the space of derivations βlog⁡ξi∣β∈R. We have the vector space decomposition(108)ΨnDO~=INTi~⊕DOi^,where(109)INTi~=βlog⁡ξi+∑…,-∞,…∈Γn…,-1,…aθξθ∣aθ∈A,DOi^=αci+∑…,0,…∈Γn…,n,…aθξθ∣aθ∈A.Theorem 38 yields the corollary.Corollary 40.
The algebras(ΨnDO~,INTi~,DOi^) form a Manin triple with respect to the bilinear form A+b1ci+α1log⁡ξi,B+b2ci+α2log⁡ξi=res(AB)+b2α1+b1α2 and Lie bracket [(A,b1,α1),(B,b2,α2)]  = ([A,B]+[α2log⁡ξi,A]-[α1log⁡ξi,B],ci(A,B),0).Theorem38 also allows us to define a double extension of ΨnDO~ by considering the n-central extension R·c1⊕⋯R·cn and the n symbols log⁡ξi. We have the vector space decomposition(110)ΨnDO~=ΨnDO⊕R·c1⊕⋯⊕R·cn⊕R·log⁡ξ1⊕⋯⊕R·log⁡ξn,with Lie bracket given by(111)
A
,
α
,
β
,
B
,
α
¯
,
β
¯
=
A
,
B
+
β
¯
·
log
⁡
ξ
,
A
-
β
·
log
⁡
ξ
,
B
,
c
A
,
B
,
0
,and bilinear form(112)A,B=resA,B+α·β¯+β·α¯,where α, β, α¯, and β¯ denote n-tuples, log⁡ξ=(log⁡ξ1,log⁡ξ2,…,log⁡ξn), 0=(0,…,0)∈Rn, and c(A,B)=(c1(A,B),…,cn(A,B)). The operation “·” is the usual dot product in Rn; for example, β·log⁡ξ=β1log⁡ξ+⋯+βnlog⁡ξn.


## 4.2. Hierarchies of Differential Equations

In this last section we apply our work to the construction of hierarchies of partial differential equations. We begin with some standard facts on hamiltonian systems modelled on (dual spaces of) Lie algebras (see, for instance, [17] or the recent paper [38]).Definition 41.
Ifg is a Lie algebra and g∗ is its dual space, the functional derivative of a function F:g∗→R at μ∈g∗ is the unique element δF/δμ of g determined by(113)ν,δFδμ=ddϵϵ=0Fμ+ϵνfor all ν∈g∗, in which , denotes a natural paring between g and g∗.Definition 42.
The Lie-Poisson bracket on the dual spaceg∗ is given by(114)F,Gμ=μ,δFδμ,δGδμ,in which F,G:g∗→R and μ∈g∗.The corresponding equations of motion for a HamiltonianH:g∗→R are(115)dμdt=μ,·,δHδμ.Definition 43.
Letg,, be a Lie algebra and R:g→g a linear operator on g. We define a bilinear antisymmetric bracket ,R on g by(116)P,QR=RP,Q+P,RQQ,P∈g.The operator R is called a classical R-matrix if the bracket ,R satisfies the Jacobi identity.Ifg=g-⊕g+ as vector space, we choose R=(1/2)π+-π-, where π± are the projection operators on g±. We write P±=π±(P) for P∈g. Then, [P,Q]R=[P+,Q+]-[P-,Q-] and(117)F,GR=μ,δFδμ,δGδμR.If F:g∗→R satisfies the infinitesimal Ad∗-invariance condition(118)μ,δFδμ,·=0for all μ∈g∗, bracket (117) becomes(119)F,GR=μ,δFδμ+,δGδμ.It follows that if H is a hamiltonian function on g∗ satisfying (118), the corresponding hamiltonian vector field is given by (120)XHμ·F=μ,-δHδμ+,δFδμand, if there exists a nondegenerate pairing , between g and g∗ such that the invariance condition (121)PQ,R=R,P,Qholds for P,Q,R∈g, then the corresponding equations of motion with respect to the Poisson structure (119) can be written as equations on g as(122)dPdt=δHδμ+,P.


### 4.2.1. The KP Hierarchy

We apply the above remarks to the Lie algebraΨDO. Let us recall that we have a bilinear symmetric form on ΨDO which satisfies(123)P,Q,S=S,P,QS,P,Q∈ΨDO.We assume that this form is nondegenerate. The decomposition ΨDO=INT⊕DO=ΨDO-⊕ΨDO+ allows us to define a new structure of Lie algebra ,R on ΨDO using R-matrices as above. The following proposition is standard; see, for instance, [5] or the recent work [38].Proposition 44.
LetHk(L)=res(Lk), k=1,2,… for L∈ΨDO. Then, (124)δHkδL=kLk-1.Moreover, for any functional F on ΨDO∗, Hk,F(L)=0, and Hk,HlR(L)=0 for any R-matrix R.Equation (122) gives us the standard Kadomtsev-Petviashvili hierarchy as a Hamiltonian equation.Theorem 45.
LetHk(L)=res(Lk) for L∈ΨDO. The corresponding Hamiltonian equation of motion with respect to ·,·R is(125)∂L∂t=kLk-1+,L.


### 4.2.2. The Twisted KP Hierarchy

We consider (59) and Proposition 23. The Manin triple ΨDOσ, DOσ, and INTσ allows us to consider a new Lie bracket ,R on ΨDOσ as in the previous subsection. Proposition 44 is also valid in the twisted context (see the proof appearing, for instance, in [38]), and therefore (122) yields the following result.Theorem 46.
LetHk(L)=res(Lk) for L∈ΨDOσ. The corresponding Hamiltonian equation of motion with respect to ·,·R is(126)∂L∂t=kLk-1+,L.A special case of (126) appears in [11]; see (7.4) therein.


### 4.2.3. Then-KP Hierarchy

Now we consider Proposition36. As before, the Manin triple ΨnDO, DOn, and INTn allows us to induce a new Lie algebra structure on ΨnDO with bracket ,R. Once more we set Hk(L)=res(Lk) for L∈ΨnDO. Equation (122) yields the equation of motion(127)∂L∂t=kLk-1+,L.This is an n-variables generalization of KP of the kind considered by Parshin in [22].


### 4.2.4. Zero Curvature Equations

Now we add a central extensionc to ΨDO and we study ΨDO^=ΨDO⊕R.Definition 47.
Let(A,α),(B,β)∈ΨDO^. We define a bilinear form on ΨDO^ by(128)A,α,B,β=A,B+αβ=resAB+αβ.We use this bilinear form to identify the dual space ofΨDO^ with itself.Proposition 48.
LetH:(ΨDO^)∗→R be a Hamiltonian function. The Hamiltonian equations of motion with respect to the bracket (114) are (129)∂L∂t,Q=∂H∂L,L,Q+lcQ,δHδL,(130)dldt=0,for any Q∈ΨDO, in which (L~,l~)=(L,l),(·,·)∈(ΨDO^)∗.Proof.
The Hamiltonian equations of motion on(ΨDO^)∗ with Hamiltonian H are(131)dL~,l~dt·Q,r=L~,l~,Q,r,δHδL~,l~=L~,l~,Q,r,δHδL~,δHδl~=L~,l~,Q,δHδL~,cQ,δHδL~.If we set (L~,l~)=(L,l),(·,·) for some (L,l)∈ΨDO^, the last equation becomes(132)∂L,l∂t,Q,r=L,l,Q,δHδL,cQ,δHδL=L,Q,δHδL+lcQ,δHδL.Using invariance of the bilinear form ,, we write this equation as(133)∂L∂t,Q+r∂l∂t=δHδL,L,Q+lcQ,δHδL.Since this equation is valid for any (Q,r), we conclude that dl/dt=0 and we obtain (129).In the context of current algebras, (129) and (130) yield the zero curvature equations obtained by Reyman and Semenov-Tian-Shansky in [28].Finally, we restrict our attention to the subalgebraVect={aξ:a∈A}. In this case, we can compute the cocycle c rather easily:(134)reslog⁡ξ,aξbξ=resbξlog⁡ξ,aξ=resbξ∑j=1∞-1j+1jδjaξ1-j=res∑j=0∞-1j+1jbδjaξ2-j+bδj+1aξ1-j=-13+13bδ3a+-12+12bδ3a=-16τbδ3a.We consider the subalgebraVect⊕R of ΨDO^, and we equip it with the bilinear form(135)aξ,α,bξ,β=τab+αβ.Then, if we set (L,l)=(uξ,l), Q=bξ, and δH/δL=h(u)ξ, and we use (129) and (130), which were derived without using the specific form of the inner product on ΨDO^, we obtain(136)dudtξ,bξ=huξ,uξ,bξ-16lτbδ3hu;that is, using that τ is nondegenerate,(137)dudt=huδu-uδhu-16lδ3hu,an abstract form of the ubiquitous Korteweg-de Vries equation.


## 4.2.1. The KP Hierarchy

We apply the above remarks to the Lie algebraΨDO. Let us recall that we have a bilinear symmetric form on ΨDO which satisfies(123)P,Q,S=S,P,QS,P,Q∈ΨDO.We assume that this form is nondegenerate. The decomposition ΨDO=INT⊕DO=ΨDO-⊕ΨDO+ allows us to define a new structure of Lie algebra ,R on ΨDO using R-matrices as above. The following proposition is standard; see, for instance, [5] or the recent work [38].Proposition 44.
LetHk(L)=res(Lk), k=1,2,… for L∈ΨDO. Then, (124)δHkδL=kLk-1.Moreover, for any functional F on ΨDO∗, Hk,F(L)=0, and Hk,HlR(L)=0 for any R-matrix R.Equation (122) gives us the standard Kadomtsev-Petviashvili hierarchy as a Hamiltonian equation.Theorem 45.
LetHk(L)=res(Lk) for L∈ΨDO. The corresponding Hamiltonian equation of motion with respect to ·,·R is(125)∂L∂t=kLk-1+,L.


## 4.2.2. The Twisted KP Hierarchy

We consider (59) and Proposition 23. The Manin triple ΨDOσ, DOσ, and INTσ allows us to consider a new Lie bracket ,R on ΨDOσ as in the previous subsection. Proposition 44 is also valid in the twisted context (see the proof appearing, for instance, in [38]), and therefore (122) yields the following result.Theorem 46.
LetHk(L)=res(Lk) for L∈ΨDOσ. The corresponding Hamiltonian equation of motion with respect to ·,·R is(126)∂L∂t=kLk-1+,L.A special case of (126) appears in [11]; see (7.4) therein.


## 4.2.3. Then-KP Hierarchy

Now we consider Proposition36. As before, the Manin triple ΨnDO, DOn, and INTn allows us to induce a new Lie algebra structure on ΨnDO with bracket ,R. Once more we set Hk(L)=res(Lk) for L∈ΨnDO. Equation (122) yields the equation of motion(127)∂L∂t=kLk-1+,L.This is an n-variables generalization of KP of the kind considered by Parshin in [22].


## 4.2.4. Zero Curvature Equations

Now we add a central extensionc to ΨDO and we study ΨDO^=ΨDO⊕R.Definition 47.
Let(A,α),(B,β)∈ΨDO^. We define a bilinear form on ΨDO^ by(128)A,α,B,β=A,B+αβ=resAB+αβ.We use this bilinear form to identify the dual space ofΨDO^ with itself.Proposition 48.
LetH:(ΨDO^)∗→R be a Hamiltonian function. The Hamiltonian equations of motion with respect to the bracket (114) are (129)∂L∂t,Q=∂H∂L,L,Q+lcQ,δHδL,(130)dldt=0,for any Q∈ΨDO, in which (L~,l~)=(L,l),(·,·)∈(ΨDO^)∗.Proof.
The Hamiltonian equations of motion on(ΨDO^)∗ with Hamiltonian H are(131)dL~,l~dt·Q,r=L~,l~,Q,r,δHδL~,l~=L~,l~,Q,r,δHδL~,δHδl~=L~,l~,Q,δHδL~,cQ,δHδL~.If we set (L~,l~)=(L,l),(·,·) for some (L,l)∈ΨDO^, the last equation becomes(132)∂L,l∂t,Q,r=L,l,Q,δHδL,cQ,δHδL=L,Q,δHδL+lcQ,δHδL.Using invariance of the bilinear form ,, we write this equation as(133)∂L∂t,Q+r∂l∂t=δHδL,L,Q+lcQ,δHδL.Since this equation is valid for any (Q,r), we conclude that dl/dt=0 and we obtain (129).In the context of current algebras, (129) and (130) yield the zero curvature equations obtained by Reyman and Semenov-Tian-Shansky in [28].Finally, we restrict our attention to the subalgebraVect={aξ:a∈A}. In this case, we can compute the cocycle c rather easily:(134)reslog⁡ξ,aξbξ=resbξlog⁡ξ,aξ=resbξ∑j=1∞-1j+1jδjaξ1-j=res∑j=0∞-1j+1jbδjaξ2-j+bδj+1aξ1-j=-13+13bδ3a+-12+12bδ3a=-16τbδ3a.We consider the subalgebraVect⊕R of ΨDO^, and we equip it with the bilinear form(135)aξ,α,bξ,β=τab+αβ.Then, if we set (L,l)=(uξ,l), Q=bξ, and δH/δL=h(u)ξ, and we use (129) and (130), which were derived without using the specific form of the inner product on ΨDO^, we obtain(136)dudtξ,bξ=huξ,uξ,bξ-16lτbδ3hu;that is, using that τ is nondegenerate,(137)dudt=huδu-uδhu-16lδ3hu,an abstract form of the ubiquitous Korteweg-de Vries equation.



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*Source: 210346-2015-07-05.xml*