Multipoint Lax operator algebras. Almost-graded structure and central extensions
aa r X i v : . [ m a t h . QA ] A p r MULTIPOINT LAX OPERATOR ALGEBRAS.ALMOST-GRADED STRUCTUREAND CENTRAL EXTENSIONS
MARTIN SCHLICHENMAIER
Abstract.
Recently, Lax operator algebras appeared as a new class of higher genuscurrent type algebras. Based on I. Krichever’s theory of Lax operators on algebraic curvesthey were introduced by I. Krichever and O. Sheinman. These algebras are almost-gradedLie algebras of currents on Riemann surfaces with marked points (in-points, out-points,and Tyurin points). In a previous joint article of the author with Sheinman the localcocycles and associated almost-graded central extensions are classified in the case of onein-point and one out-point. It was shown that the almost-graded extension is essentiallyunique. In this article the general case of Lax operator algebras corresponding to severalin- and out-points is considered. In a first step it is shown that they are almost-graded.The grading is given by the splitting of the marked points which are non-Tyurin pointsinto in- and out-points. Next, classification results both for local and bounded cocyclesare shown. The uniqueness theorem for almost-graded central extensions follows. Forthis generalization additional techniques are needed which are presented in this article.
Contents
1. Introduction 22. The algebras 52.1. Lax operator algebras 52.2. Krichever-Novikov algebras of current type 73. The almost-graded structure 83.1. The statements 83.2. The function algebra A and the vector field algebra L A L D and the algebra D g Date : April 11, 2013.2000
Mathematics Subject Classification.
Key words and phrases. infinite-dimensional Lie algebras, current algebras, Krichever Novikov typealgebras, central extensions, Lie algebra cohomology, integrable systems.This work was partially supported by the IRP GEOMQ11 of the University of Luxembourg. L -invariant cocycles 245.3. Some remarks on the cocycles on D g L -invariant cocycles 327.1. The induction step 327.2. Simple Lie algebras g gl ( n ) 358. The simple case in general 36Appendix A. Example gl ( n ) 40References 411. Introduction
Lax operator algebras are a recently introduced new class of current type Lie algebras.In their full generality they were introduced by Krichever and Sheinman in [10]. Therethe concept of Lax operators on algebraic curves, as considered by Krichever in [5], wasgeneralized to g -valued Lax operators, where g is a classical complex Lie algebra. Krichever[5] extended the conventional Lax operator representation with a rational parameter tothe case of algebraic curves of arbitrary genus. Such generalizations of Lax operatorsappear in many fields. They are closely related to integrable systems (Krichever-Novikovequations on elliptic curves, elliptic Calogero-Moser systems, Baker-Akhieser functions),see [5], [6]. Another important application appears in the context of moduli spaces ofbundles. In particular, they are related to Tyurin’s result on the classification of framedsemi-stable holomorphic vector bundles on algebraic curves [30]. The classification uses Tyurin parameters of such bundles, consisting of points γ s ( s = 1 , . . . , ng ), and associatedelements α s ∈ P n − ( C ) (where g denotes the genus of the Riemann surface Σ, and n corresponds to the rank of the bundle). In the following I will not make any reference tothese applications. Beside the above mentioned work the reader might refer to Sheinman[27], [28] for more background in the case of integrable systems.Here I will concentrate on the mathematical structure of these algebras. Lax opera-tor algebras are infinite dimensional Lie algebras of geometric origin and are interestingmathematical objects. In contrast to the classical genus zero algebras, appearing in Con-formal Field Theory, they are not graded anymore. In this article we will introduce analmost-graded structure (see Definition 3.1) for them. Such an almost-grading will bean indispensable tool. A crucial task for such infinite dimensional Lie algebras is theconstruction and classification of central extensions. This is done in the article. We willconcentrate on such central extensions for which the almost-grading can be extended.In certain respect the Lax operator algebras can be considered as generalizations of thehigher-genus Krichever-Novikov type current and affine algebras, see [7], [24], [25], [22],[17], [19]. They themselves are generalizations of the classical affine Lie algebras as e.g.introduced by Kac [3], [4] and Moody [11]. ULTIPOINT LAX OPERATOR ALGEBRAS 3
This article extends the results on the two-point case (see the next paragraph for itsdefinition) to the multi-point case. As far as the almost-grading in the two-point case isconcerned, see Krichever and Sheinman [10]. For the central extensions in the two-pointcase see the joint work of the author with Sheinman [23].To describe the obtained results we first have to give a rough description of the setup.Full details will be given in Section 2. Let g be one of the classical Lie algebras gl ( n ) , sl ( n ) , so ( n ) , sp ( n ) over C and Σ a compact Riemann surface. Let A be a finiteset of points of Σ divided into two disjoint non-empty subsets I and O . Furthermore, let W be another finite set of points (called weak singular points). Our Lax operator algebraconsists of meromorphic functions Σ → g , holomorphic outside of W ∪ A with possiblypoles of order 1 (resp. of order 2 for sp ( n )) at the points in W and certain additionalconditions, depending on g , on the Laurent series expansion there (see e.g. (2.6)). It turnsout [10] that due to the additional condition this set of matrix-valued functions closes toa Lie algebra g under the point-wise commutator. In case that W = ∅ then g will bethe Krichever-Novikov type current algebra (associated to this special finite-dimensionalLie algebras). They were extensively studied by Krichever and Novikov, Sheinman, andSchlichenmaier see e.g. [7], [24], [25], [26], [17], [22], [19]. It has to be pointed out that theKrichever-Novikov type algebras can be defined for all finite-dimensional Lie algebras g .If furthermore, the genus of the Riemann surface is zero and A consists only of twopoints, which we might assume to be { } and {∞} , then the algebras will be the usualclassical current algebras. These classical algebras are graded algebras. Such a grading isused e.g. to introduce highest weight representations, Verma modules, Fock spaces, andto classify these representations. Unfortunately, the algebras which we consider here willnot be graded. But they admit an almost-grading, see Definition 3.1. As was realizedby Krichever and Novikov [7] for most applications it is a valuable replacement for thegrading. They also gave a method how to introduce it for the two-point algebras ofKrichever-Novikov type.For the multi-point case of the Krichever-Novikov type algebras such an almost-gradingwas given by the author [16], [15], [19], [20], [21], see also [12]. The crucial point is thatthe almost-grading will depend on the splitting of A into I and O . Different splittingswill give different almost-gradings. Hence, the multi-point case is more involved than hetwo-point case.For the Krichever-Novikov current algebra g the grading comes from the grading of thefunction algebra (to be found in the above cited works of the author). This is due to thefact, that they are tensor products. If W = ∅ the Lax operator algebras are not tensorproducts anymore and their almost-grading has to be constructed directly. This has beendone in the two point case by Krichever and Sheinman [10].Our first result in this article is to introduce an almost-grading of g for the arbitrarymulti-point case. As mentioned above, it will depend in an essential way on the splittingof A into I ∪ O . This is done in Section 3. The construction is much more involved thanin the two-point case. As far as G is concerned see the recent preprint of Sheinman [29], and the remark at the end of theintroduction. M. SCHLICHENMAIER
Our second goal is to study central extensions b g of the Lax operator algebras g . It iswell-known that central extensions are given by Lie algebra two-cocycles of g with valuesin the trivial module C . Equivalence classes of central extensions are in 1:1 correspondenceto the elements of the Lie algebra cohomology space H ( g , C ). Whereas for the classicalcurrent algebras associated to a finite-dimensional simple Lie algebra g the extension classwill be unique this is not the case anymore for higher genus and even for genus zero inthe multi-point case. But we are interested only in central extensions b g which allow usto extend the almost-grading of g . This reduces the possibilities. The condition for thecocycle defining the central extension will be that it is local (see (5.13)) with respect tothe almost-grading given by the splitting A = I ∪ O . Hence, which cocycles will be localwill depend on the splitting as well.If g is simple then the space of local cohomology classes for g will be one-dimensional.For gl ( n ) we have to add another natural property for the cocycle meaning that it isinvariant under the action of the vector field algebra L (see (5.3)). In this case the spaceof local and L -invariant cocycle classes will be two-dimensional.The action of the vector field algebra L on g is given in terms of a certain connection ∇ ( ω ) , see Section 4.2. With the help of the connection we can define geometric cocycles γ ,ω,C ( L, L ′ ) = 12 π i Z C tr( L · ∇ ( ω ) L ′ ) , (1.1) γ ,ω,C ( L, L ′ ) = 12 π i Z C tr( L ) · tr( ∇ ( ω ) L ′ ) , (1.2)where C is an arbitrary cycle on Σ avoiding the points of possible singularities. Thecocycle γ ,ω,C will only be different from zero in the gl ( n ) case.Special integration paths are circles C i around the points in I , resp. around the pointsin O , and a path C S separating the points in I form the points in O .Our main result is Theorem 6.7 about uniqueness of local cocycles classes and that thecocycles are given by integrating along C S . The proof presented in Section 6 is based onTheorem 6.4 which gives the classification of bounded (from above) cohomology classes(see (5.12)). The bounded cohomology classes constitute a subspace of dimension N ,(resp. 2 N for gl ( n )) where N = I and the integration is done over the C i , i = 1 , . . . , N .The proof of Theorem 6.4 is given in Section 7 and Section 8. We use recursive tech-niques as developed in [18] and [19]. Using the boundedness and L -invariance we showthat such a cocycle is given by its values at pairs of homogeneous elements for whichthe sum of their degrees is equal to zero. Furthermore, we show that an L -invariant andbounded cocycle will be uniquely fixed by a certain finite number of such cocycle values.A more detailed analysis shows that the cocycles are of the form claimed. In Section 8we show that in the simple Lie algebra case in each bounded cohomology class there isa representing cocycle which is L -invariant. For this we use the internal structure of theLie algebra g related to the root system of the underlying finite dimensional simple Liealgebra g , and the almost-gradedness of g . Recall that in the classical case g ⊗ C [ z, z − ]the algebra is graded. In this very special case the chain of arguments gets simpler and issimilar to the arguments of Garland [1]. ULTIPOINT LAX OPERATOR ALGEBRAS 5
As already mentioned above, in joint work with Sheinman [23] the two-point case wasconsidered. This article extends the result to the multi-point case. Unfortunately, it isnot an application of the results of the two-point case. In this more general context theproofs have to be done anew. (The two-point case will finally be a special case.) Only atfew places references to proofs in [23] can be made.I like to thank Oleg Sheinman for extensive discussions which were very helpful duringwriting this article. After I finished this work he succeeded [29] to give a definition of aLax operator algebra for the exceptional Lie algebra G in such a way, that all propertiesand statements presented here will also be true in this case. Hence, there is now anotherelement in the list of Lax operator algebra associated to simple Lie algebras.2. The algebras
Lax operator algebras.
Let g be one of the classical matrix algebras gl ( n ), sl ( n ), so ( n ), sp (2 n ), or s ( n ), wherethe latter denotes the algebra of scalar matrices. Our algebras will consist of certain g -valued meromorphic functions, forms, etc, defined on Riemann surfaces with additionalstructures (marked points, vectors associated to this points, ...).To become more precise, let Σ be a compact Riemann surface of genus g ( g arbitrary)and A a finite subset of points in Σ divided into two non-empty disjoint subsets(2.1) I := { P , P , . . . , P N } , O := { Q , Q , . . . , Q M } with A = N + M . The points in I are called incoming-points the points in O outgoing-points.To define Lax operator algebras we have to fix some additional data. Fix K ∈ N anda collection of points(2.2) W := { γ s ∈ Σ \ A | s = 1 , . . . , K } . We assign to every point γ s a vector α s ∈ C n (resp. from C n for sp (2 n )). The system(2.3) T := { ( γ s , α s ) ∈ Σ × C n | s = 1 , . . . , K } is called Tyurin data . We will be more general than in our earlier joint paper [23] withSheinman, not only in respect that we allow for A more than two points also that our K is not bound to be n · g . Even K = 0 is allowed. In the latter case the Tyurin data willbe empty. Remark.
For K = n · g and for generic values of ( γ s , α s ) with α s = 0 the tuples of pairs( γ s , [ α s ]) with [ α s ] ∈ P n − ( C ) parameterize framed semi-stable rank n and degree n g holomorphic vector bundles as shown by Tyurin [30]. Hence, the name Tyurin data.We fix local coordinates z l , l = 1 , . . . , N centered at the points P l ∈ I and w s centeredat γ s , s = 1 , . . . , K . In fact nothing will dependent on the choice of w s . This is essentiallyalso true for z l . Only its first jet will be used to normalize certain basis elements uniquely.We consider g -valued meromorphic functions(2.4) L : Σ → g , M. SCHLICHENMAIER which are holomorphic outside W ∪ A , have at most poles of order one (resp. of ordertwo for sp (2 n )) at the points in W , and fulfill certain conditions at W depending on T , A , and g . These conditions will be described in the following. The singularities at W arecalled weak singularities . These objects were introduced by Krichever [5] for gl ( n ) in thecontext of Lax operators for algebraic curves, and further generalized by Krichever andSheinman in [10]. The conditions are exactly the same as in [23]. But for the convenienceof the reader we recall them here.For gl ( n ) the conditions are as follows. For s = 1 , . . . , K we require that there exist β s ∈ C n and κ s ∈ C such that the function L has the following expansion at γ s ∈ W (2.5) L ( w s ) = L s, − w s + L s, + X k> L s,k w ks , with(2.6) L s, − = α s β ts , tr( L s, − ) = β ts α s = 0 , L s, α s = κ s α s . In particular, if L s, − is non-vanishing then it is a rank 1 matrix, and if α s = 0 then it isan eigenvector of L s, .The requirements (2.6) are independent of the chosen coordinates w s and the set of allsuch functions constitute an associative algebra under the point-wise matrix multiplication,see [10]. The proof transfers without changes to the multi-point case. For the convenienceof the reader and for illustration we will nevertheless recall the proof in an appendix to thisarticle. We denote this algebra by gl ( n ). Of course, it will depend on the Riemann surfaceΣ, the finite set of points A , and the Tyurin data T. As there should be no confusion,we prefer to avoid cumbersome notation and will just use gl ( n ). The same we do for theother Lie algebras.Note that if one of the α s = 0 then the conditions at the point γ s correspond to thefact, that L has to be holomorphic there. We can erase the point from the Tyurin data.Also if α s = 0 and λ ∈ C , λ = 0 then α and λα induce the same conditions at the point γ s . Hence only the projective vector [ α s ] ∈ P n − ( C ) plays a role.The splitting gl ( n ) = s ( n ) ⊕ sl ( n ) given by(2.7) X (cid:18) tr( X ) n I n , X − tr( X ) n I n (cid:19) , where I n is the n × n -unit matrix, induces a corresponding splitting for the Lax operatoralgebra gl ( n ):(2.8) gl ( n ) = s ( n ) ⊕ sl ( n ) . For sl ( n ) the only additional condition is that in (2.5) all matrices L s,k are trace-less. Theconditions (2.6) remain unchanged.For s ( n ) all matrices in (2.5) are scalar matrices. This implies that the corresponding L s, − vanish. In particular, the elements of s ( n ) are holomorphic at W . Also L s, , as ascalar matrix, has every α s as eigenvector. This means that beside the holomorphicitythere are no further conditions. And we get s ( n ) ∼ = A , where A be the (associative) ULTIPOINT LAX OPERATOR ALGEBRAS 7 algebra of meromorphic functions on Σ holomorphic outside of A . This is the (multi-point) Krichever-Novikov type function algebra. It will be discussed further down inSection 3.2.In the case of so ( n ) we require that all L s,k in (2.5) are skew-symmetric. In particular,they are trace-less. Following [10] the set-up has to be slightly modified. First only thoseTyurin parameters α s are allowed which satisfy α ts α s = 0. Then the first requirement in(2.6) is changed to obtain(2.9) L s, − = α s β ts − β s α ts , tr( L s, − ) = β ts α s = 0 , L s, α s = κ s α s . For sp (2 n ) we consider a symplectic form ˆ σ for C n given by a non-degenerate skew-symmetric matrix σ . The Lie algebra sp (2 n ) is the Lie algebra of matrices X such that X t σ + σX = 0. The condition tr( X ) = 0 will be automatic. At the weak singularities wehave the expansion(2.10) L ( z s ) = L s, − w s + L s, − w s + L s, + L s, w s + X k> L s,k w ks . The condition (2.6) is modified as follows (see [10]): there exist β s ∈ C n , ν s , κ s ∈ C suchthat(2.11) L s, − = ν s α s α ts σ, L s, − = ( α s β ts + β s α ts ) σ, β st σα s = 0 , L s, α s = κ s α s . Moreover, we require(2.12) α ts σL s, α s = 0 . Again under the point-wise matrix commutator the set of such maps constitute a Liealgebra.
Theorem 2.1.
Let g be the space of Lax operators associated to g , one of the aboveintroduced finite-dimensional classical Lie algebras. Then g is a Lie algebra under thepoint-wise matrix commutator. For g = gl ( n ) it is an associative algebra under point-wisematrix multiplication. The proof in [10] extends without problems to the multi-point situation (see the appen-dix for an example).These Lie algebras are called
Lax operator algebras .2.2.
Krichever-Novikov algebras of current type.
Let A be the (associative) algebra of meromorphic functions on Σ holomorphic outsideof A . Let g be an arbitrary finite-dimensional Lie algebra. On the tensor product g ⊗ A a Lie algebra structure is given by(2.13) [ x ⊗ f, y ⊗ g ] := [ x, y ] ⊗ ( f · g ) , x, y ∈ g , f, g ∈ A . The elements of this Lie algebra can be considered as the set of those meromorphic mapsΣ → g , which are holomorphic outside of A . These algebras are called (multi-point)Krichever Novikov algebras of current type , see [7], [8], [9], [24], [25], [17], [19].If the genus of the surface is zero and if A consists of two points, the Krichever-Novikovcurrent algebras are the classical current (or loop algebra) g ⊗ C [ z − , z ]. M. SCHLICHENMAIER
In the case that in the defining data of the Lax operator algebra there are no weaksingularities, resp. all α s = 0, then for the g -valued meromorphic functions the require-ments reduce to the condition that they are holomorphic outside of A . Hence, we obtain(for these g ) the Krichever-Novikov current type algebra. But note that not for all finite-dimensional g we have an extension of the notion Krichever-Novikov current to a Laxoperator algebra. 3. The almost-graded structure
The statements.
For the construction of certain important representations of infinite dimensional Liealgebras (Fock space representations, Verma modules, etc.) a graded structure is usuallyassumed and heavily used. The algebras we are considering for higher genus, or evenfor genus zero with many marked points were poles are allowed, cannot be nontriviallygraded. As realized by Krichever and Novikov [7] a weaker concept, an almost-grading,will be enough to allow to do the above mentioned constructions.
Definition 3.1.
A Lie algebra V will be called almost-graded (over Z ) if there existsfinite-dimensional subspaces V m and constants S , S ∈ Z such that(1) V = L m ∈ Z V m ,(2) dim V m < ∞ , ∀ m ∈ Z ,(3) [ V n , V m ] ⊆ P n + m + S h = n + m + S V h .If there exists an R such that dim V m ≤ R for all m it is called strongly almost-graded .Accordingly, an almost-grading can be defined for associative algebras and for modulesover almost-graded algebras.We will introduce for our multi-point Lax operator in the following such a (strong)almost-graded structure. The almost-grading will be induced by the splitting of our set A into I and O . Recall that I = { P , P , . . . , P N } . In the Krichever Novikov function,vector field, and current algebra case this was done by Krichever and Novikov [7] forthe two-point situation. In the two-point Lax operator algebra it was done by Kricheverand Sheinman [10]. In the two-point case there is only one splitting possible. This is incontrast to the multi-point case which turns out to be more difficult. The multi-pointKrichever-Novikov algebras of different types were done by Schlichenmaier [16],[15]. Wewill recall it in Section 3.2.In Section 3.3 we will single out for each m ∈ Z a subspace g m of g , called (quasi-)homogeneous subspace of degree m . The degree is essentially related to the order of theelements of g at the points in I . We will show Theorem 3.2.
Induced by the splitting A = I ∪ O the (multi-point) Lax operator algebra g becomes a (strongly) almost-graded Lie algebra (3.1) g = M m ∈ Z g m , dim g m = N · dim g [ g m , g n ] ⊆ n + m + S M h = n + m g h , ULTIPOINT LAX OPERATOR ALGEBRAS 9 with a constant S independent of n and m . In addition we will show
Proposition 3.3.
Let X be an element of g . For each ( m, s ) , m ∈ Z and s = 1 , . . . , N there is a unique element X m,s in g m such that locally in the neighbourhood of the point P p ∈ I we have (3.2) X m,s | ( z p ) = Xz mp · δ ps + O ( z m +1 p ) , ∀ p = 1 , . . . , N. Proposition 3.4.
Let { X u | u = 1 , . . . , dim g } be a basis of the finite dimensional Liealgebra g . Then (3.3) B m := { X um,p , u = 1 , . . . , dim g , p = 1 , . . . , N } is a basis of g m , and B = ∪ m ∈ Z B m is a basis of g .Proof. By (3.1) we know that dim g m = N · dim g . The elements in B m are pairwisedifferent. Hence, we have B m = N · dim g elements { X um,p } in g m . For being a basis itsuffices to show that they are linearly independent. Take P u P p α um,p X um,p = 0 a linearcombination of zero. We consider the local expansions at the point P s , for s = 1 , . . . , N .From (3.2) we obtain 0 = ( X u α um,s X u ) z ms + O ( z m +1 s ) . Hence 0 = P u α um,s X u . As the X u are a basis of g this implies that a um,s = 0 for all u, s .That B is a basis of the full g follows from the direct sum decomposition in (3.1). (cid:3) It is very convenient to introduce the associated filtration(3.4) g ( k ) := M m ≥ k g m , g ( k ) ⊆ g ( k ′ ) , k ≥ k ′ . Proposition 3.5. (a) g = S m ∈ Z g ( m ) ,(b) [ g ( k ) , g ( m ) ] ⊆ g ( k + m ) ,(c) g ( m ) / g ( m +1) ∼ = g m .(d) The equivalence classes of the elements of the set B m (see (3.3)) constitute a basis forthe quotient space g ( m ) / g ( m +1) .Proof. Equation (3.1) implies directly (a), (b), and (c). Part (d) follows from Proposi-tion 3.4. (cid:3)
There is another filtration.(3.5) g ′ ( m ) := { L ∈ g | ord P s ( L ) ≥ m, s = 1 , . . . , N } . Note that the elements L are meromorphic maps from Σ to g , hence it makes sense totalk about the orders of the component functions with respect to a basis. The minimumof these orders is meant in (3.5). The symbol δ ps denotes the Kronecker delta, which is equal to 1 if s = p , otherwise 0. Proposition 3.6. (a) g = S m ∈ Z g ′ ( m ) .(b) The two filtrations coincide, i.e. g ( m ) = g ′ ( m ) , ∀ m ∈ Z . Proof.
Let L ∈ g , then as g -valued meromorphic functions the pole orders of the componentfunctions at the points P s are individually bounded. As there are only finitely many, thereis a bound k for the pole order, hence L ∈ g ( − k ) . This shows (a) and consequently that( g ′ ( m ) ) is a filtration.By Proposition 3.4 we know that B is a basis of g . Let L ∈ g ′ ( m ) . Every element of L ∈ g will be a finite linear combination of the basis elements. The elements of B k have exactorder k and are linearly independent. Moreover, with respect to a fixed basis element ofthe finite dimensional Lie algebra we have N basis elements in B k with orders given by(3.2). Hence the individual orders at the points P s cannot increase with non-trivial linearcombinations. Hence only k ≥ m can appear in the combination. This shows L ∈ g ( m ) .Vice versa, obviously all elements from B k for k ≥ m lie in the set (3.4). Hence, we haveequality. (cid:3) The second description of the filtration has the big advantage, that it is very naturallydefined. The only data which enters is the splitting of the points A into I ∪ O . Hence,it is canonically given by I . In contrast, it will turn out that in the multi-point case if O > g m , like numbering the points in O ,resp. even some different rules for the points in O . But via Proposition 3.6 we know thatthe induced filtration (3.4) will not depend on any of these choices.Here we have to remark that we supplied above a proof of Proposition 3.6. But itwas based on results (i.e. Theorem 3.2 and Proposition 3.3) which we only will prove inSection 3.3. Our starting point there will be the filtration g ′ ( m ) , hence we cannot assumeequality from the very beginning.We have the very important fact Proposition 3.7.
Let X k,s and Y m,p be the elements in g k and g m corresponding to X, Y ∈ g respectively then (3.6) [ X k,s , Y m,p ] = [ X, Y ] k + m,s δ ps + L, with [ X, Y ] the bracket in g and L ∈ g ( k + m +1) .Proof. Using for X k,s and Y m,p the expression (3.2) we obtain[ X k,s , Y m,p ] | ( z t ) = [ X, Y ] z k + ms δ pt δ st + O ( z k + m +1 t ) , for every t . Hence, the element[ X k,s , Y m,p ] − ([ X, Y ]) k + m,s δ ps has at all points in I an order ≥ k + m + 1. With (3.5) and Proposition 3.6 we obtainthat it lies in g ( k + m +1) , which is the claim. (cid:3) ULTIPOINT LAX OPERATOR ALGEBRAS 11
The function algebra A and the vector field algebra L . Before we supply the proofs of the statements in Section 3.1 we want to introduce thoseKrichever-Novikov type algebras which are of relevance in the following. We start withthe Krichever-Novikov function algebra A and the Krichever-Novikov vector field algebra L . Both algebras are almost-graded algebras(3.7) A = M m ∈ Z A m , L = M m ∈ Z L m , where the almost-grading is induced by the same splitting of A into I ∪ O as used fordefining the Lax operator algebras. Recall that I = { P , . . . , P N } and O = { Q , . . . , Q M } .Let A , respectively L , be the space of meromorphic functions, respectively of meromor-phic vector fields on Σ, holomorphic on Σ \ A . In particular, they are holomorphic alsoat the points in W . Obviously, A is an associative algebra under the product of functionsand L is a Lie algebra under the Lie bracket of vector fields. In the two point case theiralmost-graded structure was introduced by Krichever and Novikov [7]. In the multi-pointcase they were given by Schlichenmaier [15], [16]. The results will be described in thefollowing.The homogeneous spaces A m have as basis the set of functions { A m,s , s = 1 , . . . , N } given by the conditions(3.8) ord P i ( A m,s ) = ( n + 1) − δ si , i = 1 , . . . , N, and certain compensating conditions at the points in O to make it unique up to multipli-cation with a scalar. For example, in case that O = M = 1 and the genus is either 0, or ≥
2, and the points are in generic position, then the condition is (with the exception forfinitely many m )(3.9) ord Q M ( A m,s ) = − N · ( n + 1) − g + 1 . To make it unique we require for the local expansion at the P s (with respect to the chosenlocal coordinate z s )(3.10) A n,s | ( z s ) = z ns + O ( z n +1 s ) . For the vector field algebra L m we have the basis { e m,s | s = 1 , . . . , N } , where theelements e m,s are given by the condition(3.11) ord P i ( e m,s ) = ( n + 2) − δ si , i = 1 , . . . , N, and corresponding compensating conditions at the points in O to make it unique up tomultiplication with a scalar. In exactly the same special situation as above the conditionis(3.12) ord Q M ( e m,s ) = − N · ( n + 2) − g − . The local expansion at P s is(3.13) e n,s | ( z s ) = ( z n +1 s + O ( z n +2 s )) ddz s . There are constants S and S (not depending on m, n ) such that(3.14) A k · A m ⊆ k + m + S M h = k + m A h , [ L k , L m ] ⊆ k + m + S M h = k + m L h . This says that we have almost-gradedness. In what follows we will need the fine structureof the almost-grading A k,s · A m,t = A k + m,s δ ts + Y, Y ∈ k + m + S X h = k + m +1 A h , (3.15) [ e k,s , e m,t ] = ( m − k ) e k + m,s δ ts + Z, Z ∈ k + m + S X h = k + m +1 L h . (3.16)Again we have the induced filtrations A ( m ) and L ( m ) .The elements of the Lie algebra L act on A as derivations. This makes the space A analmost-graded module over L . In particular, we have(3.17) e k,s .A m,r = mA k + m δ rs + U, U ∈ k + m + S X h = k + m +1 A h , with a constant S not depending on k and m .Induced by the almost-grading of A = ⊕ m A m we get an almost-grading for the Krichever-Novikov type algebra of current type by setting(3.18) g ⊗ A = M m ∈ Z ( g ⊗ A ) m with ( g ⊗ A ) m := g ⊗ A m , ∀ m ∈ Z . The proofs.
Readers being in a hurry, or readers only interested in the results may skip this rathertechnical section (involving Riemann-Roch type arguments) during a first reading andjump directly to Section 4.Recall the definition(3.19) g ′ ( m ) := { L ∈ g | ord P s ( L ) ≥ m, s = 1 , . . . , N } of the filtration. We will only deal with this filtration in this section, hence for notationalreason we will drop the ′ in the following. Finally, the primed and unprimed will coincide. Proposition 3.8.
Given X ∈ g , X = 0 , s = 1 , . . . , N , m ∈ Z then there exists at leastone X m,s such that (3.20) X m,s | ( z p ) = Xz ms δ sp + O ( z m +1 p ) . The proof is based on the theorem of Riemann-Roch. The technique will be used all-over in this section. Hence, we will introduce some notation, before we proceed with theproof. For any m ∈ Z we will consider certain divisors(3.21) D m = ( D m ) I + D W + ( D m ) O . ULTIPOINT LAX OPERATOR ALGEBRAS 13
Where(3.22) ( D m ) I = − m N X s =1 P s , ( D m ) O = M X s =1 a s,m Q s , a s,m ∈ Z D W = ǫ K X s =1 γ s , ǫ = 1 , for gl ( n ) , sl ( n ) , so ( n ) , ǫ = 2 , for sp ( n ) . Recall that the genus of Σ is g . Denote by K a canonical divisor. Set L ( D ) the spaceconsisting of meromorphic functions u on Σ for which we have for their divisors ( u ) ≥ − D .Riemann-Roch says(3.23) dim L ( D ) − dim L ( K − D ) = deg D − g + 1 . In particular, we have(3.24) dim L ( D ) ≥ deg D − g + 1 . We have several cases which we will need in the following(1) If deg D ≥ g − D is a generic divisor then also for g ≤ deg D ≤ g − D ≥ D is generic we have dim L ( D ) = 1 for 0 ≤ deg D ≤ g − D D with negativemultiplicity) and D is generic we have dim L ( D ) = 0 for 0 ≤ deg D ≤ g − g = 0 every divisor is generic and we have equality in (3.24) as long as theright hand side is ≥
0, i.e. dim L ( D ) = max(0 , deg D + 1).See e.g. [13] for informations on divisors, Riemann-Roch and their applications, see also[2].In case that u = ( u , u , . . . , u r ) is a vector valued function we define L ( D ) to be thevector space of vector valued functions with ( u ) ≥ − D . This means that ( u i ) ≥ − D forall i = 1 , . . . , r . Now all dimension formulas have to be multiplied by r :(3.25) dim L ( D ) ≥ r (deg D − g + 1) . We apply this to our Lax operator algebra g by considering the component functions u i , i = 1 , . . . , r = dim g with respect to a fixed basis. We set(3.26) L ′ ( D ) := { u ∈ L ( D ) | u gives an element of g } ⊆ L ( D ) . In L ′ ( D m ) we have to take into account that at the weak singular points γ s we have H additional linear conditions for the elements of the solution space L ( D m ) to be fulfilled.They are formulated in terms of the corresponding α s for some finite part of the Laurentseries. In total this are finitely many conditions. In case that the α s are generic they willexactly compensate for the possible poles at γ s [10]. But for the moment we still considerthem to be arbitrary. By the very definition of the filtration we always have(3.27) g ( m ) = L ′ (( D m ) I ) and g ( m ) ≥ L ′ ( D m ) . Proof. (Proposition 3.8) We start with a divisor D m by choosing the part ( D m ) O = T suchthat the degree of the divisors D m and D m − P P i is still big enough such that for both thecase (1) of the Riemann-Roch equality (3.25) is true and that dim L ( D ) = l ≥ r ( N +1)+ H .Hence, after applying the H linear conditions we have dim L ′ ( D ) ≥ r ( N + 1). Let P s bea fixed point from I . We consider(3.28) D ′ m = D m − N X i =1 P i , D ′′ m = D ′ m + P s . This yields(3.29) dim L ′ ( D ′ m ) = l − rN, dim L ′ ( D ′′ m ) = l − rN + r. The element in L ′ ( D ′ m ) have orders ≥ ( m + 1) at all points in I . The elements in L ′ ( D ′′ m )have orders ≥ ( m + 1) at all points P i , i = s and orders ≥ m at P s . From the dimensionformula (3.29) we conclude that there exists r elements which have exact order m at P s and orders ≥ ( m + 1) at the other points in I . This says that there is for every basiselement X u in the Lie algebra g an element X um,s ∈ g which has exact order m at thepoint P s and order higher than m at the other points in I and can be written there asrequired in (3.20). By linearity we get the statement for all X ∈ g . (cid:3) Remark. By modifying the divisor T in its degree we can even show that there existselements such that the orders of X m,s at the points P p , p = s are equal to m + 1. We remark that for this proof no genericity arguments, neither with respect to thepoints A and W , nor with respect to the parameter α s were used. Hence, the statementis true for all situations. In the very definition of X m,s the local coordinate z s enters. In fact it only depends onthe first order jet of the coordinate, two different elements will just differ by a rescaling. The elements X m,s are highly non-unique. For introducing the almost-grading we willhave to make them essentially unique by trying to find a divisor T as small as possiblebut such that the statement is still true. Further down, we will come back to this. Proposition 3.9.
Let X u , u = 1 , . . . , dim g be a basis of g and (3.30) X um,s , u = 1 , . . . , dim g , s = 1 , . . . , N, m ∈ Z any fixed set of elements chosen according to Proposition 3.8 then(a) These elements are linearly independent.(b) The set of classes [ X um,s ] , u = 1 , . . . , dim g , s = 1 , . . . , N will constitute a basis of thequotient g ( m ) / g ( m +1) .(c) dim g ( m ) / g ( m +1) = N · dim g .(d) The classes of the elements X um,s will not depend on the elements chosen. ULTIPOINT LAX OPERATOR ALGEBRAS 15
Proof.
By the local expansion it follows like in the proof of Proposition 3.4 that theelements (3.30) are linearly independent, hence (a). Furthermore, by ignoring higherorders, i.e. elements from g ( m +1) they stay linearly independent. Hence (b), and (c)follows. Part (d) is true by the very definition of the elements. (cid:3) Given X ∈ g we will denote for the moment by X m,s any element fulfilling the conditionsin Proposition 3.8.As the proof of Proposition 3.7 stays also valid for these elements we have Proposition 3.10.
The algebra g is a filtered algebra with respect to the introduced filtra-tion ( g ( m ) ) i.e. (3.31) [ g ( m ) , g ( k ) ] ⊆ g ( m + k ) . Moreover, (3.32) [ X k,s , Y m,p ] = [ X, Y ] k + m,s δ ps + L, L ∈ g ( m + k +1) . Our next goal is to introduce the homogeneous subspaces g m . A too naive methodwould be to take the linear span of a fixed set of elements (3.30) for g m . The condition ofalmost-gradedness with respect to the lower bound would be fulfilled by m + k , but notnecessarily for the upper bound. To fix this we have to place more strict conditions on thepole orders at O , and we have to specify the divisor ( D m ) O in a coherent manner (withrespect to m ). By our recipe the elements will become essentially unique in the genericsituation at least for nearly all m . For non-generic α s it might be necessary to modify theprescription for individual component functions. But all these modifications will changeonly the upper bound by a constant. Remark.
Before we advance we recall that for A and for the usual Krichever-Novikovcurrent algebra g ⊗ A we have an almost-graded structure. As explained in Section 2 we have the direct sum decomposition (2.8). Moreover, s ( n ) ∼ = A . Hence the scalar part is almost-graded and fulfills Theorem 3.2 and Proposition 3.3.If we show the statements for sl ( n ) then it will follow for gl ( n ). Hence, it is enough toconsider in the following the case of g simple. Moreover, if the Tyurin data is empty (or all α s = 0) then our Lax operator algebrasreduce to the Krichever-Novikov current algebras. For those we have the statements.Hence, it is enough to consider Lax operator algebras with non-empty Tyurin data. Thereader might ask why we make such a different treatment. In fact, for non-empty Tyurindata the proof will need less case distinctions.We will now give the general description for the generic situation for g simple, andproof the claim about almost-gradedness in detail. For the non-generic situation we willshow where things have to be modified.Recall that for the divisor D m we had the decomposition (3.21). The terms ( D m ) I and D W stay as above. For ( D m ) O we require(3.33) ( D m ) O = M X i =1 ( a i m + b m,i ) Q i , with a i , b m,i ∈ Q such that a i m + b m,i ∈ Z , a i > B such that | b m,i | < B, ∀ m ∈ Z , i = 1 , . . . , M . Furthermore,(3.34) M X i =1 a i = N, M X i =1 b m,i = N + g − , ( D m +1 ) O > ( D m ) O . For the degrees we calculate(3.35) deg(( D m ) O ) = m · N + ( N + g − , deg(( D m +1 ) O ) = deg(( D m ) O ) + N. Example. For M = 1 we have the unique solution(3.36) ( D m ) O = ( N · m + ( N + g − Q M . For N ≥ M the prescription(3.37) ( D m ) O = ( m + 1) M − X j =1 Q j + (cid:0) ( N − M + 1)( m + 1) + g − (cid:1) Q M will do. Apart from the D W the corresponding divisor D m was introduced in [15] wherethe almost-grading in case of multi-point Krichever-Novikov algebras and tensors has beenconsidered for the first time (see also [12]). In [15] also prescriptions for the case
N < M were given. We will not reproduce ithere.Hence in all cases we can find such divisors.Now we set(3.38) g m := { L ∈ g | ( L ) ≥ − D m } . Proposition 3.11. (a) dim g m = N dim g .(b) A basis of g m is given by elements X um,s , u = 1 , . . . , dim g , s = 1 , . . . , N fulfilling theconditions (3.39) X um,s | ( z p ) = X u z ms δ sp + O ( z m +1 p ) . Proof.
We set r := dim g . First we deal with the generic situation. As explained above atthe weak singular points we have exactly as much relations as we get parameters by thepoles. Hence for the calculation of dim L ′ ( D ) the contribution of the degree of D W (whichis ǫ · K ) will be canceled by the relations (which are r · ǫ · K ). Here ǫ is equal to 1 or 2,depending on g . For the degree of D m we calculate(3.40) deg D m = g + ( N −
1) + ǫK ≥ g. We stay in the region where equality for (3.25) is true and calculate(3.41) dim L ′ ( D m ) = dim L ( D m ) − ǫrK = rN + rǫK − ǫrK = rN. As by definition g m = L ′ ( D m ) we get (a).Next we consider D ′ m = D m − P Ni =1 P i . For its degree we calculate deg( D m − P Ni =1 P i ) = ULTIPOINT LAX OPERATOR ALGEBRAS 17 g − ǫK . As K ≥ L ′ ( D ′ m ) = 0. Now for D ′′ m = D ′ m + P s we calculate dim L ′ ( D ′′ m ) = r .This shows that for every basis element X u of g there exists up to multiplication with ascalar a unique element X um,s ∈ g m which has the local expansion(3.42) X um,s | ( z p ) = X u δ ps z p + O ( z m +1 p ) . Hence, (b).In the non-generic case we have to change the pole orders in the definition of the divisorpart ( D m ) O in a minimal way by adding or subtracting finitely many points to reach thesituation such that we obtain exactly the dimension formula and existence of the basis ofthe required type. We have to take care that the number of changes maximally needed willbe bounded independent of m . In fact this number is bounded by the number of points Q from O needed to add to the divisor D m of the generic situation (which is of degree N + g − ǫK ) to reach a divisor D ′ m with deg D ′ m ≥ g − H , where H is the numberof relations for the α s . (cid:3) Proposition 3.12. (3.43) g = M m ∈ Z g m . Proof.
The elements X um,s introduced as the basis elements in g m are elements of thetype of Proposition 3.8 with respect to the grading. By Proposition 3.9 they stay linearlyindependent even if we considered all m ’s together, as their classes are linearly independent.Hence, the sum on the r.h.s. of (3.43) is a direct sum.To avoid to take care of special adjustments to be done for the non-generic situations weconsider m ≫ E m := − ( D m ) I + D W + ( D m ) O = m N X i =1 P i + D W + ( D m ) O , where ( D m ) O is the divisor used for fixing the basis elements in g m , see (3.33). For itsdegree we have(3.45) deg E m = 2 mN + ( N + g −
1) + ǫK.
For m ≫ L ′ ( E m ) = dim g · ((2 m + 1) N ) . The basis elements(3.47) X uk,s , u = 1 , . . . , dim g , s = 1 , . . . , N, − m ≤ k ≤ m are in L ′ ( E m ). This is shown by considering the orders at I and O . For I it is obvious. For O we have to use from (3.34) the fact that ( D ( k +1) ) O > ( D k ) O . Hence, − ( D m ) O is a lower bound for the O -part of the divisors for the element (3.47). But these are (2 m +1) · N · dim g linearly independent elements. Hence,(3.48) L ′ ( E m ) = m M k = − m g k . An arbitrary element L ∈ g has only finite pole orders at the points in I and O . Hence,there exists an m such that L ∈ L ′ ( E m ). This is again obvious for the points in I . Forthe points in O we use that by the conditions for ( D m ) O , see (3.33) for all i = 1 , . . . , M we have that a i >
0. Hence every pole order at O will be superseded by a ( D m ) O with m suitably big. This shows the claim. (cid:3) Proposition 3.13.
There exist a constant S independent of n and m such that (3.49) [ g m , g k ] ⊆ m + k + S M h = m + k g h . Proof.
We will give the proof for the generic case (and g simple) first and then point outthe modification needed for the general situation. Let L ∈ [ g m , g k ] then(3.50) ( L ) ≥ − ( D m + D k ) I − D W − ( D m + D k ) O (observe that D W does not redouble here). We consider the divisors D h . Recall theformula (3.33). As all a i > h such that ∀ h ≥ h we have(3.51) ( D h ) O ≥ ( D m + D k ) O Hence, there exists also a smallest h ∈ Z such that (3.51) is still true. We call this h max .Again by (3.34) h max ≥ m + k . Now we consider the divisor(3.52) E m = ( D m + D k ) I + D W + ( D h max ) O . From (3.35) we calculate(3.53) deg(( D hmax ) O ) = deg(( D m + D k ) O ) + ( h max − ( m + k )) N. Hence,(3.54) deg( E m ) = − ( m + k ) N + ǫK + h max · N + ( N + g − . As deg( E m ) ≥ g and under the assumption of genericity we stay in the region where(3.55) dim L ′ ( E m ) = deg g · ( h max − ( m + k ) + 1) N. As in the proof of Proposition 3.12 we get that the elements (3.47) for m + k ≤ h ≤ h max lie in L ′ ( E m ). They are linearly independent, hence(3.56) L ′ ( E m ) = h max M h = n + m g h . By (3.50) the L , we started with, lies also in L ′ ( E m ) and consequently also on the righthand side of (3.56). ULTIPOINT LAX OPERATOR ALGEBRAS 19
To show almost-grading we have to show that there exists an S (independent of m and k such that h max = m + k + S . The relation (3.51) can be rewritten as(3.57) a i h + b h,i ≥ a i ( m + k ) + b m,i + b k,i , ∀ i = 1 , . . . , M. This rewrites to(3.58) h ≥ ( m + k ) + b m,i + b k,i − b h,i a i , ∀ i = 1 , . . . , M. The minimal h for which this is true is(3.59) h max = ( m + k ) + min i =1 ,...,M ⌈ b m,i + b k,i − b h,i a i ⌉ , where for any real number x the ⌈ x ⌉ denotes the smallest integer ≥ x . As our | b m,i | arebounded uniformly by B the 3.term in (3.59) will be uniformly bounded by a constant S too. Hence, we get almost-grading. In the case of non-generic points and α s ’s the divisorsat O have to be modified by finitely many modifications. Hence the constant S has to beadapted by adding a finite constant to it. But still everything remains almost-graded. (cid:3) From the proof we can even calculate h max if needed. As an example we give Corollary 3.14.
In the generic simple Lie algebra case for N ≥ M with the standardprescription (3.37) we have h max = n + m + S with (3.60) S = , g = 0 , N = M = 1 , , g = 0 , M > , , g = 11 + ⌈ g − N − M +1 ⌉ , g ≥ . Proof.
For the standard prescription we have(3.61) a i = 1 , i = 1 , . . . , M − , a M = N − M + 1 ,b i = b m,i = 1 , i = 1 , . . . , M − , b M = b m,M = N − M + g. Hence,(3.62) S = max i =1 ,...,M ⌈ b i a i ⌉ . which yields the result. (cid:3) Now we are ready to collect the results of Propositions 3.11, 3.12 and 3.13. The state-ments are exactly the statements both of Theorem 3.2 and Proposition 3.3. All statementsof Section 3.1 are now shown to be true. In particular, now we know that both filtrations(3.5) and (3.4) coincide. Hence, (3.4) is also canonically defined by the splitting of A into I and O .A Lie algebra V is called perfect if V = [ V , V ]. Simple Lie algebras are of course perfect.The usual Krichever-Novikov current algebras g for g simple are perfect too [19, Prop.3.2]. Lax operator algebras are not necessarily perfect (at least we do not have a proof ofit). Lemma 3.15 below might be considered as a weak analog of that property. Lemma 3.15.
Let g be simple and y ∈ g then for every m ∈ Z there exists finitely manyelements y ( s, , y ( s, ∈ g , i = 1 , . . . , l = l ( m ) such that (3.63) y − l X s =1 [ y ( s, , y ( s, ] ∈ g m . Proof.
Let y be an element of g . Hence there exists a k such that y ∈ g ( k ) , but y = g ( k +1) .In particular there exists for every point P i elements X ik,i such that(3.64) y − N X i =1 X ik,i ∈ g ( k +1) , where X ik,i = ( X i ) k,i is the element corresponding to X i ∈ g . As g is perfect we have X i = [ Y i , Z i ] with elements Y i , Z i ∈ g . We calculate(3.65) X ik,i = [ Y i ,i , Z ik,i ] + y i , y ( i ) ∈ g ( k +1) . In total(3.66) y ( k ) = y − N X i =1 [ Y i ,i , Z ik,i ] ∈ g ( k +1) . Using the same again for y ( k ) etc., we can approximate y to every finite order by sums ofcommutators. (cid:3) Module structure
Lax operator algebras as modules over A . The space g is an A -module with respect to the point-wise multiplication. Obviously,the relations (2.5), (2.6), (2.9), (2.11), are not disturbed. Proposition 4.1. (a) The Lax operator algebra g is an almost-graded module over A , i.e. there exists aconstant S (not depending on k and m ) such that (4.1) A k · g m ⊆ k + m + S M h = k + m g h . (b) For X ∈ g (4.2) A m,s · X n,p = X m + n,s δ sp + L, L ∈ g ( m + n +1) . Proof.
We consider the orders of the elements in I and O . As in the proof of Proposi-tion 3.13 the existence of a constant S follows so that (4.1) is true. Hence (a).We study the lowest order term of A m,s · X n,r at the points P i ∈ I . Using (3.20), (3.8),(3.10)we see that if s = r then A m,s · X n,r ∈ g ( m + n +1) as all orders are ≥ n + m + 1. The sameis true for s = r for the element A m,s · X n,s − X m + n,s . Hence the claim. (cid:3) ULTIPOINT LAX OPERATOR ALGEBRAS 21
Warning: in general we do not have A m,s · X ,s = X m,s as the orders at O do notcoincide. Also, A m,s · X does not necessarily belong to g .4.2. Lax operator algebras as modules over L . Next we introduce an action of L on g . This is done with the help of a certain connection ∇ ( ω ) following the lines of [5], [6], [10] with the modification made in [23]. The connectionform ω is a g -valued meromorphic 1-form, holomorphic outside I , O and W , and has acertain prescribed behavior at the points in W . For γ s ∈ W with α s = 0 the requirementis that ω is also regular there. For the points γ s with α s = 0 it is required that it has anexpansion of the form(4.3) ω ( z s ) = ω s, − z s + ω s, + ω s, + X k> ω s,k z ks ! dz s . For gl ( n ): there exist ˜ β s ∈ C n and ˜ κ s ∈ C such that(4.4) ω s, − = α s ˜ β ts , ω s, α s = ˜ κ s α s , tr( ω s, − ) = ˜ β ts α s = 1 . For so ( n ): there exist ˜ β s ∈ C n and ˜ κ s ∈ C such that(4.5) ω s, − = α s ˜ β ts − ˜ β s α ts , ω s, α s = ˜ κ s α s , ˜ β ts α s = 1 . For sp (2 n ): there exist ˜ β s ∈ C n , ˜ κ s ∈ C such that(4.6) ω s, − = ( α s ˜ β ts + ˜ β s α ts ) σ, ω s, α s = ˜ κ s α s , α ts σω s, α s = 0 , ˜ β ts σα s = 1 . The existence of nontrivial connection forms fulfilling the listed conditions is proved byRiemann-Roch type argument as Proposition 3.8. We might even require, and actuallyalways will do so, that the connection form is holomorphic at I . Note also that if all α s = 0 we could take ω = 0.The connection form ω induces the following connection ∇ ( ω ) on g (4.7) ∇ ( ω ) = d + [ ω, . ] . Let e ∈ L be a vector field. In a local coordinate z the connection form and the vectorfield are represented as ω = ˜ ωdz and e = ˜ e ddz with a local function ˜ e and a local matrixvalued function ˜ ω . The covariant derivative in direction of e is given by(4.8) ∇ ( ω ) e = dz ( e ) ddz + [ ω ( e ) , . ] = e. + [ ˜ ω ˜ e , . ] = ˜ e · (cid:0) ddz + [ ˜ ω , . ] (cid:1) . Here the first term ( e. ) corresponds to taking the usual derivative of functions in eachmatrix element separately, whereas ˜ e · means multiplication with the local function ˜ e .Using the last description we obtain for L ∈ g , g ∈ A , e, f ∈ L (4.9) ∇ ( ω ) e ( g · L ) = ( e.g ) · L + g · ∇ ( ω ) e L, ∇ ( ω ) g · e L = g · ∇ ( ω ) e L, and(4.10) ∇ ( ω )[ e,f ] = [ ∇ ( ω ) e , ∇ ( ω ) f ] . The proofs of the following statements are completely the same as the proofs for thetwo-point case presented in [23]. Hence, they are here omitted.
Proposition 4.2. (a) ∇ ( ω ) e acts as a derivation on the Lie algebra g , i.e. (4.11) ∇ ( ω ) e [ L, L ′ ] = [ ∇ ( ω ) e L, L ′ ] + [ L, ∇ ( ω ) e L ′ ] . (b) The covariant derivative makes g to a Lie module over L .(c) The decomposition gl ( n ) = s ( n ) ⊕ sl ( n ) is a decomposition into L -modules, i.e. (4.12) ∇ ( ω ) e : s ( n ) → s ( n ) , ∇ ( ω ) e : sl ( n ) → sl ( n ) . Moreover, the L -module s ( n ) is equivalent to the L -module A . Proposition 4.3. (a) g is an almost-graded L -module.(b) For the corresponding L -action we have (4.13) ∇ ( ω ) e k,s X m,r = m · X k + m,s δ rs + L, L ∈ g ( k + m +1) . Proof. (a) By Proposition 4.2 g is an L -module. It remains to show that there is an upperbound for the order of the elements of the type n + m + S , with S independent of n and m (but may depend on ω ). We write (4.8) for homogeneous elements and obtain(4.14) ∇ ( ω ) e k,s X m,r = e k,s .X m,r + [ ˜ ω ˜ e k,s , X m,r ] . The form ω has fixed orders at I and at O , the action of L on A is almost-graded, andthe bracket corresponds to the commutator in the almost-graded g . By considering thecorresponding bounds for the order of poles at I and O we get such an universal bound.(b) Locally at P i , i = 1 , . . . , N we have(4.15) X m,r | ( z i ) = Xz mi δ ri + O ( z m +1 i ) , e k | ( z i ) = z k +1 i δ ki ddz + O ( z k +2 i ) . This implies(4.16) e k,s .X m,r ( z i ) = mXz k + mi δ ri δ si + O ( z k + m +1 i ) , ˜ ω ˜ e k ( z i ) = Bz k +1 i + O ( z k +2 i ) , with B ∈ gl ( n ). Hence(4.17) [ ˜ ω ˜ e k , X m ] = O ( z k + m +1 i ) , ∀ i, and the second term will only contribute to higher order. It remains the first term in(4.14). If r = s then e k,s .X m,r ( z i ) ∈ O ( z k + m +1 i ) for all i . If r = s then ( e k,s .X m,s − mX m + k,s )( z i )) ∈ O ( z k + m +1 i ). Hence, (4.13) follows. (cid:3) Module structure over D and the algebra D g . The Lie algebra D of meromorphic differential operators on Σ of degree ≤ I ∪ O is defined as the semi-direct sum of A and L with the commutatorbetween them given by the action of L on A . It is the vector space direct sum D = A ⊕ L with Lie bracket(4.18) [( g, e ) , ( h, f )] := ( e.h − f .g, [ e, f ]) . ULTIPOINT LAX OPERATOR ALGEBRAS 23
In particular(4.19) [ e, h ] = e.h.
It is an almost-graded Lie algebra [18].
Proposition 4.4.
The Lax operator algebras g are almost-graded Lie modules over D via (4.20) e.L := ∇ ( ω ) e L, h.L := h · L. Proof. As g is an almost-graded A - and L -module it is enough to show that the relation(4.19) is satisfied. For e ∈ L , h ∈ A , L ∈ g using (4.8) we get e. ( h.L ) − h. ( e.L ) = ∇ ( ω ) e ( hL ) − h ∇ ( ω ) e ( L ) =˜ e (cid:18) d ( hL ) dz + [˜ ω, hL ] (cid:19) − h ˜ e (cid:18) dLdz + [˜ ω, L ] (cid:19) = (cid:18) ˜ e dhdz (cid:19) L = ( e.h ) L = [ e, h ] .L. (cid:3) The Lax operator algebra g is a module over the Lie algebra L which acts on g byderivations (according to Proposition 4.2). Proposition 4.2 says that this action of L on g is an action by derivations. Hence as above we can consider the semi-direct sum D g = g ⊕L with Lie product given by(4.21) [ e, L ] := e.L = ∇ ( ω ) e L, for the mixed pairs. See [19] for the corresponding construction for the classical Krichever-Novikov algebras of affine type. 5. Cocycles
In this section we will study 2-cocycles for the Lie algebra g with values in C . It is well-known that the corresponding cohomology space H ( g , C ) classifies equivalence classes of(one-dimensional) central extensions of g .For the convenience of the reader we recall that a 2-cocycle for g is a bilinear form γ : g × g → C which is (1) antisymmetric and (2) fulfills the condition(5.1) γ ([ L, L ′ ] , L ′′ ) + γ ([ L ′ , L ′′ ] , L ) + γ ([ L ′′ , L ] , L ′ ) = 0 , L, L ′ , L ′′ ∈ g . A 2-cocycle γ is a coboundary if there exists a linear form φ on g such that(5.2) γ ( L, L ′ ) = φ ([ L, L ′ ]) , L, L ′ ∈ g . Given a 2-cocycle γ for g , the associated central extension b g γ is given as vector spacedirect sum b g γ = g ⊕ C · t with Lie product(5.3) [ b L, b L ′ ] = \ [ L, L ′ ] + γ ( L, L ′ ) · t, [ b L, t ] = 0 , L, L ′ ∈ g . Here we used b L := ( L,
0) and t := (0 , −−−−→ C i −−−−→ b g p −−−−→ g −−−−→ , defines a 2-cocycle γ : g → C by choosing a section s : g → b g .Two central extensions b g γ and b g γ ′ are equivalent if and only if the defining cocycles γ and γ ′ are cohomologous. Geometric cocycles.
Next we introduce geometric 2-cocycles. Let ω be a connection form as introduced inSection 4.2 for defining the connection (4.7). Furthermore, let C be a (not necessarilyconnected ) differentiable cycle on Σ not meeting the sets A = I ∪ O and W .As in the two point situation considered in [23] we define the following bilinear formson g :(5.5) γ ,ω,C ( L, L ′ ) = 12 π i Z C tr( L · ∇ ( ω ) L ′ ) , L, L ′ ∈ g , and(5.6) γ ,ω,C ( L, L ′ ) = 12 π i Z C tr( L ) · tr( ∇ ( ω ) L ′ ) , L, L ′ ∈ g . The following propositions and their proofs remain the same as in [23] (of course now tobe interpreted in this more general context), and we will not repeat them.
Proposition 5.1.
The bilinear forms γ ,ω,C and γ ,ω,C are cocycles. Proposition 5.2. (a) The cocycle γ ,ω,C does not depend on the choice of the connection form ω .(b) The cohomology class [ γ ,ω,C ] does not depend on the choice of the connection form ω .More precisely (5.7) γ ,ω,C ( L, L ′ ) − γ ,ω ′ ,C ( L, L ′ ) = 12 π i Z C tr (cid:0) ( ω − ω ′ )[ L, L ′ ] (cid:1) . As γ ,ω,C does not depend on ω we will drop ω in the notation. Note that γ ,C vanisheson g for g = sl ( n ) , so ( n ) , sp (2 n ). But it does not vanish on s ( n ), hence not on gl ( n ).5.2. L -invariant cocycles. As explained in Section 4.2 after fixing a connection form ω ′ the vector field algebra L operates on g via the covariant derivative e
7→ ∇ ( ω ′ ) e . Definition 5.3.
A cocycle γ for g is called L -invariant (with respect to ω ′ ) if(5.8) γ ( ∇ ( ω ′ ) e L, L ′ ) + γ ( L, ∇ ( ω ′ ) e L ′ ) = 0 , ∀ e ∈ L , ∀ L, L ′ ∈ g . Proposition 5.4. (a) The cocycle γ ,C is L -invariant.(b) If ω = ω ′ then the cocycle γ ,ω,C is L -invariant. The proof is the same as presented in [23] for the two point case.We call a cohomology class L -invariant if it has a representing cocycle which is L -invariant. The reader should be warned that this does not mean that all representingcocycles are L -invariant. On the contrary, see Corollary 6.5. Clearly, the L -invariantclasses constitute a subspace of H ( g , C ) which we denote by H L ( g , C ). ULTIPOINT LAX OPERATOR ALGEBRAS 25
Some remarks on the cocycles on D g . In the following let ω = ω ′ . The property of L -invariance of a cocycle has a deepermeaning. In Section 4.3 we introduced the algebra D g . The Lax operator algebra g is asubalgebra of D g . Given a 2-cocycle γ for g we might extend it to D g as a bilinear formby setting ( L, L ′ ∈ g , e, f ∈ L )(5.9) ˜ γ ( L, L ′ ) = γ ( L, L ′ ) , ˜ γ ( e, L ) = ˜ γ ( L, e ) = 0 , ˜ γ ( e, f ) = 0 . Proposition 5.5.
The extended bilinear form ˜ γ is a cocycle for D g if and only if γ is L -invariant.Proof. The conditions defining a cocycle are obviously fulfilled for the triples of elementsconsisting either of currents or of vector fields. The only condition which does not followautomatically from (5.9) for ˜ γ is(5.10) ˜ γ ([ L, L ′ ] , e ) + ˜ γ ([ L ′ , e ] , L ) + ˜ γ ([ e, L ] , L ′ ) = 0 . Using (4.21) we get that (5.10) is true if an only if(5.11) γ ( ∇ ( ω ) e L, L ′ ) + γ ( L, ∇ ( ω ) e L ′ ) = 0 , which is L -invariance. (cid:3) Bounded and local cocycles.Definition 5.6.
Given an almost-graded Lie algebra V = L m ∈ Z V m . A cocycle γ is called bounded (from above) if there exists a constant R ∈ Z such that(5.12) γ ( V n , V m ) = 0 = ⇒ n + m ≤ R . Similarly bounded from below is defined.A cocycle is called local if and only if it is bounded from above and below. Equivalently,there exist R , R ∈ Z such(5.13) γ ( V n , V m ) = 0 = ⇒ R ≤ n + m ≤ R . The almost-grading of V can be extended from V to the corresponding central extension b V γ (5.3) by assigning to the central element t a certain degree (e.g. the degree 0) if andonly if the defining cocycle for the central extension is local.We call a cohomology class bounded (resp. local) if it contains a bounded (resp. local)representing cocycle. Again, not every representing cocycle of a bounded (resp. local)class is bounded (resp. local). The set of bounded cohomology classes is a subspaceof H ( g , C ) which we denote by H b ( g , C ). It contains the subspace of local cohomologyclasses denoted by H loc ( g , C ). This space classifies the almost-graded central extensions of g up to equivalence. Both spaces admit subspaces consisting of those cohomology classesadmitting a representing cocycle which is both bounded (resp. local) and L -invariant.The subspaces are denoted by H b, L ( g , C ), resp. H loc, L ( g , C ).If we consider our geometric cocycles γ ,C and γ ,ω,C obtained by integrating over anarbitrary cycle then they will neither be bounded, nor local, nor will they define a boundedor local cohomology class. Next we will consider special integration paths. Let C i be positively oriented (deformed)circles around the points P i in I , i = 1 , . . . , N and C ∗ j positively oriented ones around thepoints Q j in O , j = 1 , . . . , M . The cocycle values of γ if integrated over such cycles canbe calculated via residues, e.g.(5.14) γ ,ω,C i ( L, L ′ ) = res P i (tr( L · ∇ ( ω ) L ′ )) , i = 1 , . . . , N . Proposition 5.7. (1) The 1-form tr( L · ∇ ( ω ) L ′ ) has no poles outside of A = I ∪ O .(2) The 1-form tr( L ) · tr( dL ′ ) has no poles outside of A = I ∪ O .Proof. For (1) see [10]. For (2) see [23]. (cid:3)
A cycle C S is called a separating cycle if it is smooth, positively oriented of multiplicityone, it separates the points in I from the points in O , and it does not meet A or W . Itmight have multiple components. For our cocycles (5.5), (5.6) we integrate the forms ofProposition 5.7 over closed curves C . By this proposition the integrals will yield the sameresults if [ C ] = [ C ′ ] in H (Σ \ A, Z ). Note that the weak singular points will not show upin this context. In this sense we can write for every separating cycle(5.15) [ C S ] = K X i =1 [ C i ] = − M X j =1 [ C ∗ j ] . The minus sign appears due to the opposite orientation. In particular the cocycle valuesobtained by integrating over a C S can be obtained by calculating residues either over thepoints in I or the points in O . Theorem 5.8.
Let ω coincides with the connection form ω ′ associated to the L -actionthen(a) For i = 1 , . . . , N the cocycles γ ,ω,C i and γ ,C i with C i a circle around P i will bebounded from above and L -invariant.(b) For j = 1 , . . . , M the cocycles γ ,ω,C ∗ j and γ ,C ∗ j with C ∗ j a circle around Q j will bebounded from below and L -invariant.(c) The cocycles γ ,ω,C S and γ ,C S with C S a separating cycle will be local and L -invariant.(d) In case (a) and (c) the upper bound will be zero.Proof. The statement about L -invariance follows from Proposition 5.4. In fact only forthis ω = ω ′ is needed.As explained above the cocycle calculation if integrated over C i (or over C ∗ j ) reduces tothe calculation of residues. Let L ∈ g n , L ′ ∈ g m then ord P i ( L ) ≥ n and ord P i ( L ′ ) ≥ m .As ω is holomorphic at P i we obtainord P i ( dL ′ ) ≥ m − , ord P i ( ∇ ( ω ) L ′ ) ≥ m − . Hence, if n + m > I and consequently no residues. This shows (a).For (b) we have to consider the orders at the points in O of the basis elements of g m . Bythe prescriptions (3.33) and (3.34) and taking into account possible poles of ω at O wefind an R such that if n + m ≤ R the integrands will not have poles anymore. This ULTIPOINT LAX OPERATOR ALGEBRAS 27 shows (b).Using (5.15) we can obtain the values of the cocycles integrated over C S either by addingup the values obtained by integration either over I or over O . Hence boundedness frombelow and from above. Hence, locality.That zero is an upper bound followed already during the proof. (cid:3) Classification Results
Recall that we are in the multi-point situation A = I ∪ O with I = N . The C i , C ∗ j ,and C S are the special cycles introduced in Section 5.4. If we will use the word bounded for a cocycle we always mean bounded from above if nothing else is said. Proposition 6.1.
The cocycles γ ,ω,C i , i = 1 , . . . , N (and γ ,C i , i = 1 , . . . , N for gl ( n ) )are linearly independent.Proof. Assume that there is a linear relation(6.1) 0 = N X i =1 α i γ ,ω,C i + N X i =1 β i γ ,C i , α i , β i ∈ C . The last sum will not appear in the simple algebra case. Recall that for a pair
L, L ′ ∈ g the above cocycles can be calculated by taking residues(6.2) 0 = N X i =1 α i res P i (tr( L · ∇ ( ω ) L ′ )) + N X i =1 β i res P i (tr( L ) · tr( ∇ ( ω ) L ′ )) . In the first sum the Cartan-Killing form is present which is non-degenerated. Hence thereexist
X, Y ∈ g such that tr( XY ) = 0 and tr( X ) = tr( Y ) = 0. For k = 1 , . . . , N , using thealmost-graded structure and following Proposition 3.3 we take L = X ,k and L ′ = Y − ,k .In the neighbourhood of the point P l , l = 1 , . . . , N we have(6.3) L ( z l ) = Xz l δ kl + O ( z l ) , L ′ ( z l ) = Y z − l δ kl + O ( z l ) , ∇ ( ω ) L ′ ( z l ) = − Y z − l δ kl + O ( z − l ) , as ∇ ( ω ) L ′ = dL ′ + [ ω, L ′ ]. Hence,(6.4) res P l (tr( L · ∇ ( ω ) L ′ )) = − tr( XY ) δ kl . As tr( X ) = 0 the second sum will vanish anyway and we conclude α k = 0, for all k =1 , . . . , N . For the second sum we take X = Y a nonvanishing scalar matrix and chose L = X ,k and L ′ = X − ,k . We obtain β k = 0 for all k = 1 , . . . , N . (cid:3) Proposition 6.2. ( g = gl ( n ) ) Let γ = P Ni =1 β i γ ,C i be a nontrivial linear combination,then it is not a coboundary.Proof. Recall from (2.8) that s ( n ) is an abelian subalgebra of gl ( n ). Hence, every cobound-ary restricted to it will be identically zero. If we take again as in the previous proof elements X ,k and X − ,k from the scalar subalgebra we obtain as above β k = 0. (cid:3) Proposition 6.3.
Let γ = P Ni =1 α i γ ,ω,C i be a non-trivial linear combination then it isnot a coboundary.Proof. Assume that γ is a coboundary. This means that there exists a linear form φ : g → C such that ∀ L, L ′ ∈ g (6.5) γ ( L, L ′ ) = N X i =1 α i res P i tr( L · ∇ ( ω ) L ′ ) = φ ([ L, L ′ ]) . Assume that γ = 0, hence one of the coefficients α k will be non-zero. Take H ∈ h with κ ( H, H ) = 0, where h is the Cartan subalgebra of the simple part of g and κ itsCartan-Killing form. Let H ,k ∈ g be the element defined by (3.2). In particular, we have H ,k = H + O ( z k ). We set H ( n,k ) := H ,k · A n,k ∈ g and hence H ( n,k ) = H · A n,k + O ( z n +1 k )in the neighbourhood of the point P k . Recall that from the local forms (3.2) and (3.8) ofour basis elements we have in the neighbourhood of points P l with l = k (6.6) H n,k = O ( z n +1 l ) , A n,k = O ( z n +1 l ) , H ( n,k ) = O ( z n +1 l ) . In the following, let n = 0. We have(6.7) ∇ ( ω ) H ( n,k ) = ∇ ( ω ) ( H ,k · A n,k ) = ∇ ( ω ) ( H ,k ) · A n,k + H ,k dA n,k . The expression ∇ ( ω ) H ,k is of nonnegative order, A n,k is of order n , H ,k of order 0 and dA n,k of order n − P k . Hence(6.8) ∇ ( ω ) H ( n,k ) = H ,k dA n,k + O ( z nk ) dz k . Now we compute(6.9) γ ( H ( − ,k ) , H (1 ,k ) ) = N X i =1 α i res P i tr( H ( − ,k ) · ∇ ( ω ) H (1 ,k ) ) = α k res P k tr( H ( − ,k ) · ∇ ( ω ) H (1 ,k ) ) . The last equality follows from the fact that by (6.6) we do not have any poles at the points P l for l = k . From the above it follows(6.10) ( α k ) − γ ( H ( − ,k ) , H (1 ,k ) ) = res P k tr( H ,k A − ,k H ,k dA ,k ) = res P k tr( H ,k dz k z k ) . As H ,k = H + O ( z k ) we obtain(6.11) ( α k ) − γ ( H ( − ,k ) , H (1 ,k ) ) = res P k (tr( H ) dz k z k ) = tr( H ) = β · κ ( H, H ) = 0 , with a non-vanishing constant β relating the trace form with the Cartan-Killing form. But(6.12) [ H ( − ,k ) , H (1 ,k ) ] = [ H ,k A − ,k , H ,k A ,k ] = [ H ,k , H ,k ] A − ,k A ,k = 0 . The relations (6.11) and (6.12) are in contradiction to (6.5). (cid:3)
Now we are able to formulate the basic theorem. Notice that H ( n,k ) and H n,k , in general, are different but coincide up to higher order. ULTIPOINT LAX OPERATOR ALGEBRAS 29
Theorem 6.4. (a) If g is simple (i.e. g = sl ( n ) , so ( n ) , sp (2 n ) ) then the space of bounded cohomologyclasses is N -dimensional. If we fix any connection form ω then this space has as basisthe classes of γ ,ω,C i , i = 1 , . . . , N . Every L -invariant (with respect to the connection ω )bounded cocycle is a linear combination of the γ ,ω,C i .(b) For g = gl ( n ) the space of local cohomology classes which are L -invariant havingbeen restricted to the scalar subalgebra is 2N-dimensional. If we fix any connection form ω then the space has as basis the classes of the cocycles γ ,ω,C i and γ ,C i , i = 1 , . . . , N .Every L -invariant local cocycle is a linear combination of the γ ,ω,C i and γ ,C i .Proof of the theorem. Here we only outline the proof. The technicalities are postponeduntil Sections 7 and 8.By Propositions 7.8 and 7.10 it follows that L -invariant and bounded cocycles arenecessarily linear combinations of the claimed form. This proves the theorem for thecohomology space H b, L ( g , C ). For the scalar subalgebra we are done since we includedthe L -invariance into the conditions of the theorem. For semi-simple algebras we haveto show that there is an L -invariant representative in each local cohomology class. Butby Theorem 8.1 the space H b ( g , C ) is at most N-dimensional. As by Proposition 6.3 nonon-trivial linear combination of the cocycles γ ,ω,C i is a coboundary, this space is exactlyN-dimensional and [ γ ,ω,C i ] for i = 1 , . . . , N constitute a basis. (cid:3) We conclude the following.
Corollary 6.5.
Let g be a simple classical Lie algebra and g the associated Lax operatoralgebra. Let ω be a fixed connection form. Then in each [ γ ] ∈ H b ( g , C ) there exists aunique representative γ ′ which is bounded and L -invariant (with respect to ω ). Moreover, γ ′ = P Ni =1 a i γ ,ω,C i , with a i ∈ C . Proposition 6.6. (a) Let γ be a bounded and L -invariant cocycle which is a coboundary, then γ = 0 .(b) Let g be simple, then the cocycle γ ,ω ′ ,C i is L -invariant with respect to ω , if and onlyif ω = ω ′ .Proof. (a) By Theorem 6.4 we get γ = P Ni =1 ( α i γ ,ω,C i + β i γ ,C i ), with all β i = 0 for thecase g is simple. The summands constitute a basis of the cohomology. Hence, γ can onlybe a coboundary if all coefficients vanish.(b) As γ ,ω,C i and γ ,ω ′ ,C i are local and L -invariant with respect to ω their difference γ ,ω,C i − γ ,ω ′ ,C i is also local and L -invariant. By Proposition 5.2 it is a coboundary.Hence by part (a) γ ,ω,C i − γ ,ω ′ ,C i = 0. The relation (5.7) gives the explicit expression forthe left hand side. Assume ω = ω ′ . Let m be the order of the element(6.13) θ = ω − ω ′ = ( θ m z mi + O ( z mi )) dz i at the point P i . As g is simple the trace form tr( A · B ) is nondegenerate and we find(6.14) ˆ θ = ˆ θ − m − z − m − i + O ( z − mi ) , such that β = tr( θ m · ˆ θ − m − ) = 0. By Lemma 3.15 we get ˆ θ = [ L, L ′ ]+ L ′′ with ord P i ( L ′′ ) ≥− m . Hence,(6.15) 0 = β = tr( θ m · ˆ θ − m − ) = 12 π i Z C i tr (cid:0) ( ω − ω ′ ) · ([ L, L ′ ] + L ′′ ) (cid:1) = 12 π i Z C i tr (cid:0) ( ω − ω ′ ) · [ L, L ′ ] (cid:1) = γ ,ω,C i ( L, L ′ ) − γ ,ω ′ ,C i ( L, L ′ ) = 0which is a contradiction. (cid:3) After these results which are valid for bounded cocycles we will deduce the correspondingclassification theorem for local cocycles. In some sense this is the main theorem of thisarticle. It will show for example that for Lax operator algebras associated to simple Liealgebras there is up to rescaling and equivalence only one non-trivial almost-graded centralextension.Recall the relation for the separating cycle(6.16) [ C S ] = N X i =1 [ C i ] = − M X j =1 [ C ∗ j ] , and the corresponding relation for the cocycle obtained by integration. Theorem 6.7. (a) If g is simple (i.e. g = sl ( n ) , so ( n ) , sp (2 n ) ) then the space of local cohomology classesis one-dimensional. If we fix any connection form ω then this space will be generated bythe class of γ ,ω,C S . Every L -invariant (with respect to the connection ω ) local cocycle isa scalar multiple of γ ,ω,C S .(b) For g = gl ( n ) the space of local cohomology classes which are L -invariant havingbeen restricted to the scalar subalgebra is two-dimensional. If we fix any connection form ω then the space will be generated by the classes of the cocycles γ ,ω,C S and γ ,C S . Every L -invariant local cocycle is a linear combination of γ ,ω,C S and γ ,C S .Proof. Let γ be a local cocycle. This says it is bounded from above and from below. Forsimplicity we abbreviate in this proof(6.17) γ ,i := γ ,ω,C i γ ,i := γ ,C i , γ ∗ ,j := γ ,ω,C ∗ j , γ ∗ ,j := γ ,C ∗ j . If we switch the role of I and O we get an inverted almost-grading. Every bounded frombelow cocycle of the original grading, will get bounded from above with respect to theinverted grading. Hence we can employ Theorem 6.4 in both directions and obtain for thesame cocycle two representations(6.18) γ = N X i =1 a i γ ,i + N X i =1 b i γ ,i = − M X j =1 a ∗ j γ ∗ ,j − M X j =1 b ∗ j γ ∗ ,j , with a i , a ∗ j , b i , b ∗ j ∈ C . If either N = 1 or M = 1 then via (6.16) the cocycle is obtained via integration over aseparating cycle. Hence the statement. ULTIPOINT LAX OPERATOR ALGEBRAS 31
Otherwise both
N, M >
1. By Proposition 6.1 the type (1) and type (2) cocyclesare linearly independent, hence can be treated independently also in this context. Firstconsider type (1). Note that from (6.16) we get the relation that(6.19) N X i =1 γ ,i = − M X j =1 γ ∗ ,j . Hence,(6.20) γ ∗ , = − N X i =1 γ ,i − M X j =2 γ ∗ ,j . From (6.18) we get(6.21) 0 = N X i =1 a i γ ,i + M X j =1 a ∗ j γ ∗ ,j . If we plug (6.20) into this relation we obtain(6.22) 0 = ( a − a ∗ ) N X i =1 γ ,i + N X i =2 ( a i − a ) γ ,i + M X j =2 ( a ∗ k − a ∗ ) γ ∗ ,j . Fix a k . We take X, Y ∈ g such that tr( XY ) = 0. By Riemann-Roch there exists L ′ , L ∈ g such that around the point P k we have(6.23) L ( z k ) = Xz k + O ( z k ) , L ′ ( z k ) = Y z − k + O ( z k ) , both holomorphic at the points in O and at P l , l = 1 , k . The elements might have poleorders of sufficiently high degree at P to guarantee existence. The weak singularities willnot disturb. By construction(6.24) γ ,k ( L, L ′ ) = 0 , γ ,l ( L, L ′ ) = 0 , l = 2 , . . . , N, l = k,γ ∗ ,j ( L, L ′ ) = 0 , j = 1 , . . . , M. Hence(6.25) N X i =1 γ ,i ( L, L ′ ) = − M X j =1 γ ∗ ,j ( L, L ′ ) = 0 . If we plug (
L, L ′ ) into (6.22), all terms in (6.22) will vanish, with the only exception(6.26) 0 = ( a k − a ) γ ,k ( L, L ′ ) . This shows a k − a for all k . (In a similar way we get a ∗ j − a ∗ for all j .) In particular,our cocycle we started with (resp. the γ part of it) is a multiple of the cocycle obtainedby integration over the separating cycle. This was the claim. The proof for the γ partworks completely the same if we take X = Y a nonzero scalar matrix. (cid:3) As in the bounded case we obtain also for the local case the following corollary.
Corollary 6.8.
Let g be a simple classical Lie algebra and g the associated Lax operatoralgebra. Let ω be a fixed connection form. Then in each [ γ ] ∈ H loc ( g , C ) there exists aunique representative γ ′ which is local and L -invariant (with respect to ω ). Moreover, γ ′ = aγ ,ω , with a ∈ C . Uniqueness of L -invariant cocycles The induction step.
Recall from Section 3 the almost-graded structure of the Lax operator algebra g andin particular the decomposition g = ⊕ n ∈ Z g n into subspaces of homogeneous elements ofdegree n . Also there the basis { L un,p | u = 1 , . . . , dim g , p = 1 , . . . , N } of the subspace g n was introduced (see (3.3)).Let γ be an L -invariant cocycle for the algebra g which is bounded from above, i.e.there exists an R (independent of n and m ) such that γ ( g n , g m ) = 0 implies n + m ≤ R .Furthermore, we recall that our connection ω needed to define the action of L on g ischosen to be holomorphic at the points in I .For a pair ( L un,k , L vm,t ) of homogeneous elements we call n + m the level of the pair.We apply the technique developed in [18]. We will consider cocycle values γ ( L un,k , L vm,t )on pairs of level l = n + m and will make induction over the level. By the boundednessfrom above, the cocycle values will vanish at all pairs of sufficiently high level. It will turnout that everything will be fixed by the values of the cocycle at level zero. Finally, wewill show that the cocycle is a linear combination of the N (resp. 2 N ) basic cocycles asclaimed in Theorem 6.4.For a cocycle γ evaluated for pairs of elements of level l we will use the symbol ≡ todenote that the expressions are the same on both sides of an equation involving cocyclevalues up to values of γ at higher level. This has to be understood in the following strongsense:(7.1) X α ( n,p,t )( u,v ) γ ( L un,p , L vl − n,t ) ≡ , α ( n,p,t )( u,v ) ∈ C means a congruence modulo a linear combination of values of γ at pairs of basis elementsof level l ′ > l . The coefficients of that linear combination, as well as the α ( n,p,t )( u,v ) , dependonly on the structure of the Lie algebra g and do not depend on γ .We will also use the same symbol ≡ for equalities in g which are true modulo terms ofhigher degree compared to the terms under consideration.By the L -invariance we have(7.2) γ ( ∇ ( ω ) e k,r L um,p , L vn,s ) + γ ( L um,p , ∇ ( ω ) e k,r L vn,s ) = 0 . Using the almost-graded structure (4.13) we obtain (up to order > ( k + m + n ))(7.3) mγ ( L uk + m,p , L vn,s ) δ pr + nγ ( L vm,p , L vn + k,s ) δ sr ≡ , valid for all n, m, k ∈ Z .If in (7.3) all three indices r, p and s are different then the term on the left hand sidevanishes. If r = p = s then we obtain(7.4) mγ ( L uk + m,p , L vn,s ) ≡ . ULTIPOINT LAX OPERATOR ALGEBRAS 33 which is true for every m . Hence(7.5) γ ( L ul,p , L vn,s ) ≡ , for p = s . It remains r = p = s and this yields(7.6) mγ ( L uk + m,s , L vn,s ) + nγ ( L vm,s , L vn + k,s ) ≡ . Proposition 7.1.
Let m + n = 0 then at level m + n we have (7.7) γ ( g m , g n ) ≡ . Proof.
From (7.5) we conclude that only elements with the same second index could con-tribute in level m + n . We put k = 0 in (7.6) and obtain(7.8) ( m + n ) γ ( L um,s , L vn,s ) ≡ , ∀ u, v. Hence if ( m + n ) = 0 the claim follows. (cid:3) Proposition 7.2. (7.9) γ ( g m , g ) ≡ , ∀ m ∈ Z . Proof.
We evaluate (7.6) for the values m = 1 and n = 0 and obtain the result. (cid:3) Proposition 7.3. (a) We have γ ( g n , g m ) = 0 if n + m > , i.e. the cocycle is boundedfrom above by zero.(b) If γ ( g n , g − n ) = 0 then the cocycle γ vanishes identically.Proof. The proof stays word by word the same as in [23]. But as it is one of the centralarguments and for the convenience of the reader we repeat the arguments. If γ = 0 thereis nothing to prove. Assume γ = 0. As γ is bounded from above, there will be a minimalupper bound l , such that above l all cocycle values will vanish. Assume that l >
0, then byProposition 7.1 the values at level l are expressions of levels bigger than l . But the cocyclevanishes there. Hence it vanishes at level l too. This is a contradiction which proves (a).By induction, using again Proposition 7.1 we obtain that if the cocycle vanishes at level0, it vanishes everywhere. This proves (b). (cid:3) Combining Propositions 7.2 and 7.3 we obtain
Corollary 7.4. (7.10) γ ( g m , g ) = 0 , ∀ m ≥ . Proposition 7.5. (7.11) γ ( L un,r , L v − n,s ) = n · γ ( L u ,r , L v − ,r ) δ sr , (7.12) γ ( L u ,r , L v − ,s ) = γ ( L v ,s , L u − ,r ) Proof.
Assume s = r then all expressions are of positive level and vanishes by Proposi-tion 7.3, hence the statement is true. For r = s we take in (7.6) the values n = − p , m = 1and k = p −
1. This yields the expression (7.11) up to higher level terms. But as the levelis zero, the higher level terms vanish. Setting n = − (cid:3) Independently of the structure of the Lie algebra g , we obtained the following resultsfor every L -invariant and bounded cocycle γ :(1) The cocycle is bounded from above by zero.(2) The cocycle is uniquely given by its values at level zero.(3) At level zero the cocycle is uniquely fixed by its values γ ( L u ,s , L v − ,s ), for u, v =1 , . . . , dim g and s = 1 , . . . , N .(4) The other cocycle values at level zero are given by γ ( L un,s , L v − n,r ) = 0 if s = r , γ ( L u ,s , L v ,s ) = 0 and γ ( L un,s , L v − n,s ) given by (7.11).Let X ∈ g then we denote as always by X n,s , s = 1 , . . . , N the element in g with leadingterm Xz ns at P s and higher orders at the other points in I . We define for s = 1 , . . . , N themaps(7.13) ψ γ,s : g × g → C ψ γ,s ( X, Y ) := γ ( X ,s , Y − ,s ) . Obviously, ψ γ,s is a bilinear form on g . Proposition 7.6. (a) ψ γ,s is symmetric, i.e. ψ γ,s ( X, Y ) = ψ γ,s ( Y, X ) .(b) ψ γ,s is invariant, i.e. (7.14) ψ γ,s ([ X, Y ] , Z ) = ψ γ,s ( X, [ Y, Z ]) . Proof.
First we have by (7.12) ψ γ,s ( X, Y ) = γ ( X ,s , Y − ,s ) = γ ( Y ,s , X − ,s ) = ψ γ ( Y, X ) . This is the symmetry. Furthermore, using [ X ,s , Y ,s ] ≡ [ X, Y ] ,s , the fact that the cocyclevanishes for positive level, and by the cocycle condition we obtain ψ γ,s ([ X, Y ] , Z ) = γ ([ X, Y ] ,s , Z − ,s ) = γ ([ X ,s , Y ,s ] , Z − ,s ) = − γ ([ Y ,s , Z − ,s ] , X ,s ) − γ ([ Z − ,s , X ,s ] , Y ,s ) . The last term vanishes due to Corollary 7.4. Hence ψ γ,s ([ X, Y ] , Z ) = γ ( X ,s , [ Y ,s , Z − ,s ]) = γ ( X ,s , [ Y, Z ] − ,s ) = ψ γ,s ( X, [ Y, Z ]) . (cid:3) As the cocycle γ is fixed by the values γ ( L u ,s , L v − ,s ), s = 1 , . . . , N and they are fixedby the bilinear maps ψ γ,s we proved: Theorem 7.7.
Let γ be an L -invariant cocycle for g which is bounded from above. Then γ it is bounded from above by zero and is completely fixed by the associated symmetric andinvariant bilinear forms ψ γ,s , s = 1 , . . . , N on g defined via (7.13). Simple Lie algebras g . By Theorem 7.7 the L -invariant cocycle γ is completely given by fixing the N -tuple( ψ γ, , ψ γ, , . . . , ψ γ,N ) of symmetric invariant bilinear forms ψ γ,s . For a finite-dimensionalsimple Lie algebra every such form is a multiple of the Cartan-Killing form κ . Hencethe space of bounded cocycles is at most N -dimensional. Our geometric cocycles γ ,ω,C i ,see (5.14), for i = 1 , . . . , N are L -invariant and bounded cocycles. They are linearlyindependent, see Proposition 6.1. Hence, we obtain that every bounded and L -invariant ULTIPOINT LAX OPERATOR ALGEBRAS 35 cocycle is a linear combination of the γ ,ω,C i . Moreover, they are a basis of the space of L -invariant and bounded cocycles. By Proposition 6.3 they stay linearly independent afterpassing to cohomology and we obtain Proposition 7.8.
Let g be simple, then (7.15) dim H b, L ( g , C ) = N, and this cohomology space is generated by the classes of γ ,ω,C i , i = 1 , . . . , N . The case of gl ( n ) . We have the direct decomposition, as Lie algebras, gl ( n ) = s ( n ) ⊕ sl ( n ). Let γ be acocycle of gl ( n ) and denote by γ ′ and γ ′′ its restriction to s ( n ) and sl ( n ) respectively. Asin [23] we obtain using Lemma 3.15 Proposition 7.9. (7.16) γ ( x, y ) = 0 , ∀ x ∈ s ( n ) , y ∈ sl ( n ) . Hence we can decompose the cocycle as γ = γ ′ ⊕ γ ′′ . If γ is bounded/local and/or L -invariant the same is true for γ ′ and γ ′′ .First we consider the algebra s ( n ). It is isomorphic to A , the isomorphism is given by(7.17) L n tr( L ) . In [18, Thm. 5.7] it was shown that the space of L -invariant cocycles for A bounded fromabove is N -dimensional and a basis is given by(7.18) γ i ( f, g ) = 12 π i Z C i f dg = res P i ( f dg ) , i = 1 , . . . , N. Note that as A is abelian there do not exist non-trivial coboundaries. We obtain(7.19) γ ′ ( L, M ) = N X i =1 α i res P i (tr( L ) · tr( dM )) = N X i =1 α i γ ,C i ( L, M ) , by Definition (5.6).For the cocycle γ ′′ of sl ( n ) we use Proposition 7.8 and obtain γ ′′ = P Ni =1 β i γ ,ω,C i .Altogether we showed Proposition 7.10. (7.20) dim H b, L ( gl ( n ) , C ) = 2 N. A basis is given by the classes of γ ,ω,C i and γ ,C i , i = 1 , . . . , N . In this section we showed those parts of Theorem 6.4 which deal with L -invariantcocycles. In fact we showed the complete theorem under the additional assumption thatour cohomology classes are L -invariant. For the scalar part this is the best what could beexpected. Without L -invariance there will be much more non-trivial cohomology classesfor the scalar algebra, see [18] for more information. In the next section we will present away how to get rid of this condition for the simple Lie algebras. The simple case in general
In this section the Lax operator algebra g is always based on a finite simple classicalLie algebra. As explained in the previous section if we put L -invariance in the assumptionthen Theorem 6.4 would have been proved. One way to complete the general proof isto to show that after cohomological changes every bounded cocycle has also a bounded L -invariant representing it. In fact, we will do this. But unfortunately, we do not havea direct proof. Instead, by a quite different approach we will show that for the simpleLie algebra case the space of bounded cohomology classes (of the Lax operator algebras)is at most N -dimensional without assuming L -invariance a priori. Combining this resultwith the result of the last section that the space of L -invariant bounded cohomologyclasses is N -dimensional we see that in the simple case each bounded cohomology classis automatically L -invariant. Moreover, we showed there that it has a unique L -invariantrepresenting cocycle which is given as linear combination of γ ,ω,C i , i = 1 , . . . , N .The theorem we are heading for is Theorem 8.1.
Let g be a simple classical Lie algebra over C and g the associated Laxoperator algebra with its almost-grading. Every bounded cocycle on g is cohomologous toa distinguished cocycle which is bounded from above by zero. The space of distinguishedcocycles is at most N -dimensional.Remark. What we will show is the following. Every cocycle bounded from above is co-homologous to a cocycle which is fixed by its value at N special pair of elements in g (namely by γ ( H α ,s , H α − ,s ) for one fixed simple root α , see below for the notation). Besidesthe structure of g we only use the almost-gradedness of g with leading terms given in (8.4).The presentation is quite similar to [23]. Those proofs which are completely of the samestructure will not be repeated here.First we need to recall some facts about the Chevalley generators of g . Choose a rootspace decomposition g = h ⊕ α ∈ ∆ g α . As usual ∆ denotes the set of all roots α ∈ h ∗ .Furthermore, let { α , α , . . . , α p } be a set of simple roots ( p = dim h ). With respect tothis basis, the root system splits into positive and negative roots, ∆ + and ∆ − respectively.With α a positive root, − α is a negative root and vice versa. For α ∈ ∆ we have dim g α = 1.Certain elements E α ∈ g α and H α ∈ h , α ∈ ∆ can be fixed so that for every positive root α (8.1) [ E α , E − α ] = H α , [ H α , E α ] = 2 E α , [ H α , E − α ] = − E − α . We use also H i := H α i , i = 1 , . . . , p for the elements assigned to the simple roots. Avector space basis, the Chevalley basis, of g is given by { E α , α ∈ ∆; H i , ≤ i ≤ p } . ULTIPOINT LAX OPERATOR ALGEBRAS 37
We denote by ( , ) the inner product on h ∗ induced by the Cartan-Killing form of g .The following relations hold(8.2) [ H α , H β ] = 0 , [ H α , E ± β ] = ± β, α )( β, β ) E ± α , [ H, E α ] = α ( H ) E α , H ∈ h , [ E α , E β ] = H α , α ∈ ∆ + , β = − α, − H α , α ∈ ∆ − , β = − α, ± ( r + 1) E α + β , α, β, α + β ∈ ∆ , , otherwise.Here r is the largest nonnegative integer such that α − rβ still is a root.As in the other parts of this article, we denote by E αn,s , H αn,s the unique elements in g n (i.e. of degree n ) for which the expansions at P s start with E α z ns and H α z ns respectively,and at the Points P i ∈ I , i = s it is of higher order.The following elements form a basis of g :(8.3) { E αn,s , α ∈ ∆; H in,s , ≤ i ≤ p | n ∈ Z , s = 1 , . . . , N } . The structure equations, up to higher degree terms, are(8.4) [ H αn,s , H βm,r ] ≡ , [ H αn,s , E ± βm,r ] ≡ ± β, α )( β, β ) E ± βn + m,r δ sr , [ H n,s , E αm,r ] ≡ α ( H ) E αn + m,r δ sr , H ∈ h , [ E αn,s , E βm,r ] ≡ H αn + m,s δ sr , α ∈ ∆ + , β = − α, − H αn + m,s δ sr , α ∈ ∆ − , β = − α, ± ( r + 1) E α + βn + m,s δ sr , , α, β, α + β ∈ ∆ , , otherwise.Recall that the symbol ≡ denotes equality up to elements of degree higher than the sumof the degrees of the elements under consideration. Here, the elements not written downare elements of degree > n + m . Also recall that by the almost-gradedness there exists a S , independent of n and m , such that only elements of degree ≤ n + m + S appear.Let γ ′ be a cocycle for g which is bounded from above. For the elements in g we get(8.5) E ± α = ± / H α , E ± α ] , H i = [ E α i , E − α i ] , i = 1 , . . . , p. Consequently, for g we obtain(8.6) E ± αn,s = ± / H α ,s , E ± αn,s ] + Y ( n, s, α ) ,H in,s = [ E α i ,s , E − α i n,s ] + Z ( n, s, i ) , i = 1 , . . . , p. where Y ( n, s, α ) and Z ( n, s, i ) are sums of elements of degree between n + 1 and n + S .Fix a number M ∈ Z such that the cocycle γ ′ vanishes for all levels ≥ M . We define alinear map Φ : g → C by (descending) induction on the degree of the basis elements (8.3).First(8.7) Φ( E αn,s ) := Φ( H in,s ) := 0 , α ∈ ∆ , i = 1 , . . . , p, s = 1 , . . . , N n ≥ M. Next we define inductively ( α ∈ ∆ + , s = 1 , . . . , N )(8.8) Φ( E ± αn,s ) := ± / γ ′ ( H α ,s , E ± an,s ) + Φ( Y ( n, s, ± α )) , Φ( H in,s ) := γ ′ ( E α i ,s , E − α i n,s ) + Φ( Z ( n, s, i )) . The cocycle γ = γ ′ − δ Φ is cohomologous to the original cocycle γ ′ . As γ ′ is bounded fromabove, and, by definition, Φ is also bounded from above, the cocycle γ is bounded fromabove too.By the construction of Φ we have Φ([ H α ,s , E ± αn,s ] = γ ′ ( H α ,s , E ± αn,s ) and Φ([ E α i ,s , E − α i n,s ]) = γ ′ ( E α i ,s , E − α i n,s ). Hence Proposition 8.2. (8.9) γ ( H α ,s , E ± αn,s ) = 0 , γ ( E α i ,s , E − α i n,s ) = 0 ,α ∈ ∆ + , i = 1 , . . . , p, s = 1 , . . . , N, n ∈ Z . Definition 8.3.
A cocycle γ is called normalized if it fulfills (8.9).By the above construction we showed that every cocycle bounded from above is coho-mologous to a normalized one, which is also bounded from above. In the following weassume that our cocycle is already normalized. Proposition 8.4.
Let α be a fixed simple root, α and β arbitrary roots and γ a normalizedcocycle, then for all s, r = 1 , . . . , N , n, m ∈ Z we have (8.10) γ ( E αm,s , H n,r ) ≡ , H ∈ h , α ∈ ∆ γ ( E αm,s , E βn,r ) ≡ , α, β ∈ ∆ , β = − α,γ ( E αm,r , E − αn,s ) ≡ uγ ( H α m,r , H α n,r ) δ sr , α ∈ ∆ ,γ ( H αm,r , H βn,s ) ≡ tγ ( H α m,r , H α n,r ) δ sr , , α, β ∈ ∆ + , with u, t ∈ C . (8.11) γ ( H α n,r , H α ,s ) ≡ , (8.12) γ ( H α n +1 ,s , H α l − ( n +1) ,r ) ≡ (cid:16) γ ( H α n − ,s , H α l − ( n − ,s ) + 2 γ ( H α ,s , H α ) l − ,s (cid:17) δ sr . For a simple root α and for a level l = 0 we have (8.13) γ ( H α n,r , H α l − n,s ) ≡ . ULTIPOINT LAX OPERATOR ALGEBRAS 39
Proof.
In the two point case the statement of the proposition consists of a chain of indi-vidual statement which were proved in [23]. In fact, the proofs presented there remainvalid if one just adds in all relations there for the Lie algebra elements Y n the secondindex to obtain Y n,s . By the almost-graded structure, resp. its fine structure (3.6) for theexpressions [ Y n,s , Z m,r ] in the relations only terms involving s = r will contribute on thelevel under considerations. If s = r they will contribute only to higher level. Hence, allrelations there can be read with respect to all the second indices the same up to higherlevel. Hence, the proof is completely analogous. (cid:3) Proposition 8.5.
Let γ be a normalized cocycle. Then(a) it vanishes for levels greater than zero, i.e. (8.14) γ ( g n , g n ) = 0 , for n + m > . (b) All levels l < are fixed by the level zero.Proof. By the propositions above we showed that the expressions at level l of the cocyclecan be reduced to expressions of levels > l and values γ ( H αn,r , H αl − n,r ). As long as the levelis = 0, by (8.13) also these values can be expressed by higher level. Hence by induction,starting with the upper bound of the cocycle, we obtain that the upper bound for the levelof the cocycle values is equal to zero. Also it follows that the values at levels l < (cid:3) Hence it remains to consider the level zero.
Proposition 8.6.
Let α be a simple root. At level l = 0 the cocycle values for s = 1 , . . . , N are given by the relations (8.15) γ ( H αn,s , H α − n,r ) = n · γ ( H α ,s , H α − ,s ) δ rs , γ ( H α ,r , H α ,s ) = 0 . Proof.
If we set the value l = 0 in (8.12) we obtain the relation(8.16) γ ( H αn +1 ,s , H α − ( n +1) ,r ) ≡ (cid:16) γ ( H αn − ,s , H α − ( n − ,s ) + 2 γ ( H α ,s , H α − ,s ) (cid:17) · δ sr . As all cocycle values of level l > ≡ by =. Now the claimedexpression follows. (cid:3) Proof of Theorem 8.1.
After adding a suitable coboundary we might replace the given γ by a normalized one. Using Proposition 8.2, 8.4, and 8.6 everything depends only on thevalues γ ( H α ,s , H α − ,s ), s = 1 , . . . , N for one (fixed) simple root. This proves that there areat most N linearly independent normalized cocycles. (cid:3) Proposition 8.7.
If a normalized cocycle γ is a coboundary then it vanishes identically.Proof. As explained above, a normalized cocycle is fixed by the values γ ( H α ,s , H α − ,s ). Weset(8.17) H α (1 ,s ) := H α ,s A ,s ≡ H α ,s , and H α ( − ,s ) := H α ,s A − ,s ≡ H α − ,s . Hence(8.18) [ H α (1 ,s ) , H α ( − ,s ) ] = [ H α ,s , H α ,s ] A ,s A − ,s = 0 . As the cocycle vanishes for positive levels, and as γ = δφ is assumed to be a coboundarywe get(8.19) γ ( H α ,s , H α − ,s ) = γ ( H α (1 ,s ) , H α ( − ,s ) ) = φ ([ H α (1 ,s ) , H α ( − ,s ) ]) = φ (0) = 0 . Hence, all cocycle values are zero, as claimed. (cid:3)
Appendix A. Example gl ( n )In this appendix we will reproduce as an illustration for the reader the proof that theproduct of two Lax operators for the algebra gl ( n ) is again a Lax operator. This meansthat the equations (2.6) are fulfilled for their product. Hence, gl ( n ) will be closed undercommutator too. This result is due to Krichever and Sheinman [10]. In a similar mannerthe other cases are treated (but now only for the commutators). Furthermore, it is shownthat the connection operators ∇ ( ω ) e act indeed on g . The original proofs (involving partlytedious calculations) can be found in [10], [23], [28].The singularities at the points in A are not bounded. Hence, they will not createproblems and the proofs need only to consider the weak singular points. Consequently,the statements are also true in the multi-point case.We start with two elements L ′ and L ′′ with corresponding expansions (2.5) and examinetheir product L = L ′ L ′′ . For this we have to consider each point γ s (with local coordinate w s ) of the weak singularities with α s = 0 separately. Taking into account only those partswhich might contribute we obtain for L (A.1) L = L ′ s, − L ′′ s, − w s + L ′ s, − L ′′ s, + L ′ s, L ′′ s, − w s + (cid:0) L ′ s, − L ′′ s, + L ′ s, L ′′ s, + L ′ s, L ′′ s, − (cid:1) + O ( w s ) . By expanding the first numerator we get(A.2) L ′ s, − L ′′ s, − = α s β ′ st α s β ′′ s t = 0as β ′ st α s = 0 by (2.6). Hence, there is no pole of order two appearing.Next we consider the expression which comes with pole order one.(A.3) L s, − = L ′ s, − L ′′ s, + L ′ s, L ′′ s, − = α s β ′ st L ′′ s, + L ′ s, α s β ′′ s t . As by the conditions L ′ s, α s = κ ′ s α s we can write(A.4) L s, − = α s β ts , with β ts = β ′ st L ′′ s, + κ ′ s β ′′ s t . For the trace condition we obtain(A.5) tr( L s, − ) = ( β ′ st L ′′ s, + κ ′ s β ′′ s t ) α s = κ ′′ s β ′ st α s + κ ′ s β ′′ s t α s = 0 . Hence, we have the required form.Finally we have to verify that α s is an eigenvector of L s, . First we note that L ′′ s, − α s = 0and L ′ s, L ′′ s, α s = κ ′ s κ ′′ s α s . Also(A.6) L ′ s, − L ′′ s, α s = α s ( β ′ st L ′′ s, α s ) = ( β ′ st L ′′ s, α s ) α s . ULTIPOINT LAX OPERATOR ALGEBRAS 41
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