aa r X i v : . [ m a t h . QA ] M a y DOUBLE YANGIAN AND THE UNIVERSAL R -MATRIX MAXIM NAZAROV
Abstract.
We describe the double Yangian of the general linear Lie algebra gl N by following a general scheme of Drinfeld. This description is based on theconstruction of the universal R -matrix for the Yangian. To make exposition selfcontained, we include the proofs of all necessary facts about the Yangian itself.In particular, we describe the centre of the Yangian by using its Hopf algebrastructure, and provide a proof of the analogue of the Poincar´e–Birkhoff–Witttheorem for the Yangian based on its representation theory. This proof extendsto the double Yangian, thus giving a description of its underlying vector space. Contents
Introduction 11. Definition of the Yangian 32. Matrix form of the definition 43. Automorphisms and anti-automorphisms 64. Hopf algebra structure 95. Two filtrations on the Yangian 116. Vector and covector representations 127. Evaluation representations 138. Poincar´e–Birkhoff–Witt theorem 159. Centre of the Yangian 1710. Dual Yangian 1911. Canonical pairing 2212. Non-degeneracy of the pairing 2413. Universal R -matrix 2814. Double Yangian 3215. Filtration on the double Yangian 35References 39 Introduction
The main subject of this article is a Hopf algebra that appeared in the frameworkof quantum inverse scattering method introduced by L. D. Faddeev, E. K. Sklyaninand their collaborators, see for instance [9, 15, 26, 27, 28, 30]. This algebra thenbecame a part of a family of examples in the theory of quantum groups createdby V. G. Drinfeld [3, 4, 5]. He gave to this family the name
Yangians in honour ofC. N. Yang, the author of a seminal work [32]. The Yangian that we consider herecorresponds to the general linear Lie algebra gl N . It is a canonical deformation ofthe universal enveloping algebra of the polynomial current Lie algebra gl N [ z ] . MAXIM NAZAROV
The general notion of a quantum double was also introduced in [5]. Howeverthe Yangians were not discussed there in the context of this notion. Here we definethe double Yangian of the Lie algebra gl N similarly to [10]. Yet many details andproofs are also missing in the latter work. In the present article we fill these gaps.We denote by Y( gl N ) the Yangian of gl N , and by DY( gl N ) its quantum double.There are several equivalent definitions of the Hopf algebra Y( gl N ) available [20].In this article we use the definition that appeared first, see for instance [16, 17, 31].Details of this definition are given in our Sections 1,2 and 4 by closely following [21].Sections 3,5 and 6 describe basic properties of the Yangian Y( gl N ) that we will use.We will also use an analogue of the classical Poincar´e–Birkhoff–Witt theorem [2]for the algebra Y( gl N ) . The first proof of this analogue was given by V. G. Drinfeldbut not published. Other proofs were given later in [18, 24]. In Section 8 we giveyet another proof of this analogue by using the representation theory of current Liealgebras. The fact from the theory that we use is established in Section 7. It is thisproof that will be extended to the double Yangian DY( gl N ) in the present article.This method was used in [23] to prove analogues of the Poincar´e–Birkhoff–Witttheorem for the Yangian of the queer Lie superalgebra q N and its quantum double.For the algebra dual to the coalgebra Y( gl N ) the same method was used in [7].The structure of a Hopf algebra includes a canonical anti-automorphism relativeto both multiplication and comultiplication, called the antipodal map. In generalthis map is not involutive. In Section 3 we also compute the square of this map forthe Yangian Y( gl N ) , by following [22] where the Yangian of the general linear Liesuperalgebra gl M | N was considered. This yields a family of central elements of thealgebra Y( gl N ) , see also [6]. In Section 9 we prove that these elements generate thewhole centre. Our proof uses another general fact from the theory of current Liealgebras, which we establish in the beginning of the section. The idea of reducingthe proof to that fact belongs to V. G. Drinfeld, as acknowledged in [21].In Section 10 we introduce the bialgebra Y ∗ ( gl N ) dual to Y( gl N ) . First we defineit in terms generators and relations similarly to Y( gl N ) . However Y ∗ ( gl N ) is not aHopf algebra. The antipodal map is defined only on a certain completion Y ◦ ( gl N )of Y ∗ ( gl N ) described at the end of that section. In Section 11 we define a bialgebrapairing between Y( gl N ) and Y ∗ ( gl N ) . This definition goes back to [25] where thequantized universal enveloping algebras of simple Lie algebras were considered. InSection 12 we prove that this pairing is non-degenerate. Details of this proof firstappeared in [23] where instead of gl N , the Lie superalgebra q N was considered.In Section 13 we define the universal R -matrix for Y( gl N ). This is the canonicalelement of a suitable completion of the tensor product Y ∗ ( gl N ) ⊗ Y( gl N ) , whichcorresponds to the bialgebra pairing. There we also describe the basic properties ofthis element relative to the Hopf algebra structures on both Y( gl N ) and Y ◦ ( gl N ) .In Section 14 we define the double Yangian DY( gl N ) as a bialgebra generated byY( gl N ) and Y ∗ ( gl N ) . Following [5, 25] the cross relations between the elements ofY( gl N ) and Y ∗ ( gl N ) are introduced by means of the universal R -matrix. Then weprovide a more explicit description of the algebra DY( gl N ) . Using this descriptionone can define a central extension of DY( gl N ) , see for instance [8, 11].Finally, in Section 15 we introduce a filtration on the algebra DY( gl N ) and showthat the corresponding graded algebra is isomorphic to the universal envelopingalgebra of the current Lie algebra gl N [ z, z − ] . This implies our analogue of thePoincar´e–Birkhoff–Witt theorem for DY( gl N ) . This also implies that the defininghomomorphisms of the algebras Y( gl N ) and Y ∗ ( gl N ) to DY( gl N ) are embeddings. OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 3 The purpose of the present article is to provide the basic facts about the doubleYangian DY( gl N ) with their detailed proofs. We do not not aim to review all workswhich involve this remarkable object. Still let us mention here the pioneering works[1, 19, 29] where the double Yangian of the special linear Lie algebra sl was studied.The double Yangians of all simple Lie algebras were studied in [13, 14] by using thedefinition of the underlying Yangians from [4]. This approach to double Yangiansis different from ours. Recently some of the results on DY( gl N ) presented here havebeen extended to the double Yangians of the other classical Lie algebras [12].1. Definition of the Yangian
The
Yangian of the general linear Lie algebra gl N is a unital associative algebraY( gl N ) over the complex field C with countably many generators T (1) ij , T (2) ij , . . . where i, j = 1 , . . . , N . The defining relations of the algebra Y( gl N ) are(1.1) [ T ( r +1) ij , T ( s ) kl ] − [ T ( r ) ij , T ( s +1) kl ] = T ( r ) kj T ( s ) il − T ( s ) kj T ( r ) il where r, s = 0 , , . . . and T (0) ij = δ ij . By introducing the formal generating series(1.2) T ij ( u ) = δ ij + T (1) ij u − + T (2) ij u − + . . . ∈ Y( gl N )[[ u − ]]we can write (1.1) in the form(1.3) ( u − v ) [ T ij ( u ) , T kl ( v ) ] = T kj ( u ) T il ( v ) − T kj ( v ) T il ( u ) . Here the indeterminates u and v are considered to be commuting with each otherand with the elements of the Yangian. The following is an equivalent form of (1.1). Proposition 1.1.
The system of relations (1.1) is equivalent to the system (1.4) [ T ( r ) ij , T ( s ) kl ] = min( r,s ) X a =1 (cid:16) T ( a − kj T ( r + s − a ) il − T ( r + s − a ) kj T ( a − il (cid:17) . Proof.
Observe that the multiplication of both sides of (1.3) by the formal series P ∞ p =0 u − p − v p yields an equivalent relation[ T ij ( u ) , T kl ( v ) ] = (cid:16) T kj ( u ) T il ( v ) − T kj ( v ) T il ( u ) (cid:17) ∞ X p =0 u − p − v p . Taking the coefficients of u − r v − s on both sides gives[ T ( r ) ij , T ( s ) kl ] = r X a =1 (cid:16) T ( a − kj T ( r + s − a ) il − T ( r + s − a ) kj T ( a − il (cid:17) . This agrees with (1.4) in the case r s . Finally, if r > s observe that r X a = s +1 (cid:16) T ( a − kj T ( r + s − a ) il − T ( r + s − a ) kj T ( a − il (cid:17) = 0 . (cid:3) We shall be often using formal series to define or describe maps between variousalgebras. If A ( u ) and B ( u ) are formal series in u with coefficients in certain algebrasthen assignments of the type A ( u ) B ( u ) are understood in the sense that everycoefficient of A ( u ) is mapped to the corresponding coefficient of B ( u ). MAXIM NAZAROV
Many applications of Y( gl N ) are based on the following observation. Let E ij bethe standard generators of the Lie algebra gl N so that(1.5) [ E ij , E kl ] = δ jk E il − δ li E kj . Proposition 1.2.
The assignment (1.6) T ij ( u ) δ ij + E ij u − defines a surjective homomorphism Y( gl N ) → U( gl N ) . The assignment (1.7) E ij T (1) ij defines an embedding U( gl N ) → Y( gl N ) .Proof. By the definition (1.3) we need to verify the equality( u − v ) [ E ij , E kl ] u − v − =( δ kj + E kj u − )( δ il + E il v − ) − ( δ kj + E kj v − )( δ il + E il u − ) . But this clearly holds by the commutation relations (1.5) in gl N , which proves thefirst part of the proposition. In order to prove the second part, put r = s = 1 in(1.4). This gives [ T (1) ij , T (1) kl ] = δ kj T (1) il − δ il T (1) kj . Thus (1.7) is an algebra homomorphism. Its injectivity follows from the observationthat by applying (1.7) and then (1.6), we get the identity map on U( gl N ) . (cid:3) The homomorphism (1.6) is called the evaluation homomorphism . By its virtueany representation of the Lie algebra gl N can be regarded as representation of theY( gl N ). Any irreducible representation of gl N remains irreducible over Y( gl N ) dueto surjectivity of this homomorphism. We will also be using its composition withthe automorphism E ij
7→ − E ji of the algebra U( gl N ). The composition maps(1.8) T ij ( u ) δ ij − E ji u − . The reason for using it rather than (1.6) will be explained in Section 6.2.
Matrix form of the definition
Introduce the N × N matrix T ( u ) whose ij -th entry is the series T ij ( u ) . Onecan regard T ( u ) as an element of the algebra End C N ⊗ Y( gl N )[[ u − ]] . Then(2.1) T ( u ) = N X i,j =1 e ij ⊗ T ij ( u )where e ij ∈ End C N are the standard matrix units. If e , . . . , e N are the standardbasis vectors of C N , then T ( u ) e j is interpreted as the linear combination T ( u ) e j = N X i =1 e i ⊗ T ij ( u ) ∈ C N ⊗ Y( gl N )[[ u − ]] . For any positive integer m we shall be using algebras of the form(2.2) (End C N ) ⊗ m ⊗ Y( gl N ) . OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 5 For any a = 1 , . . . , m we denote by T a ( u ) the matrix T ( u ) which corresponds tothe a -th copy of the algebra End C N in the tensor product (2.2). That is, T a ( u ) isa formal power series in u − with the coefficients from the algebra (2.2), T a ( u ) = N X i,j =1 ⊗ ( a − ⊗ e ij ⊗ ⊗ ( m − a ) ⊗ T ij ( u )where e ij belongs to the a -th copy of End C N and 1 is the identity matrix. If C is anelement of the tensor square (End C N ) ⊗ then for a, b = 1 , . . . , m with a < b we willdenote by C ab the image of C under this embedding (End C N ) ⊗ → (End C N ) ⊗ m : e ij ⊗ e kl ⊗ ( a − ⊗ e ij ⊗ ⊗ ( b − a − ⊗ e kl ⊗ ⊗ ( m − b ) . Here the tensor factors e ij and e kl belong to the a -th and b -th copies of End C N respectively. The element C ab can be identified with the element C ab ⊗ t : End C N → End C N : e ij e ji is the matrix transposition, then for any a = 1 , . . . , m we shall denote by t a thecorresponding partial transposition on the algebra (2.2). It acts as t on the a -thcopy of End C N and as the identity map on all the other tensor factors.Consider now the permutation operator(2.3) P = N X i,j =1 e ij ⊗ e ji ∈ End C N ⊗ End C N . The rational function(2.4) R ( u ) = 1 − P u − with values in End C N ⊗ End C N is called the Yang R - matrix . Here and below wewrite 1 instead of 1 ⊗ R ( u ) R ( − u ) = 1 − u − . We will also work with the rational function R t ( u ) = 1 − Q u − where Q = N X i,j =1 e ij ⊗ e ij = P t = P t . We should write either R t ( u ) or R t ( u ) instead of R t ( u ) but we will not do so.Note that Q is a one-dimensional operator on C N ⊗ C N such that Q = N Q . Hence(2.5) R t ( u ) − = 1 + Q ( u − N ) − . Proposition 2.1.
In the algebra (End C N ) ⊗ ( u, v ) we have the identity (2.6) R ( u ) R ( u + v ) R ( v ) = R ( v ) R ( u + v ) R ( u ) . Proof.
Multiplying both sides of the relation (2.6) by uv ( u + v ) we come to verify(2.7) ( u + P )( u + v + P )( v + P ) = ( v + P )( u + v + P )( u + P ) . Each operator P ij is the image of the corresponding transposition ( ij ) ∈ S underthe natural action of the symmetric group S on ( C N ) ⊗ by permutations of thetensor factors. So (2.7) follows from the relations in the group algebra C [ S ]. (cid:3) MAXIM NAZAROV
The relation (2.6) is known as the
Yang–Baxter equation . The Yang R -matrix isits simplest nontrivial solution. Below we regard T ( u ) and T ( v ) as formal powerseries with the coefficients from the algebra (2.2) where m = 2 . We also identify R ( u − v ) with the rational function R ( u − v ) ⊗ Proposition 2.2.
The defining relations of the algebra Y( gl N ) can be written as (2.8) R ( u − v ) T ( u ) T ( v ) = T ( v ) T ( u ) R ( u − v ) . Proof.
Let us apply both sides of (2.8) to an any basis vector e j ⊗ e l ∈ C N ⊗ C N as explained in the beginning of this section. For the left hand side we get X i,k T ij ( u ) T kl ( v ) ⊗ e i ⊗ e k − u − v X i,k T ij ( u ) T kl ( v ) ⊗ e k ⊗ e i , while the right hand side gives X i,k T kl ( v ) T ij ( u ) ⊗ e i ⊗ e k − u − v X i,k T kj ( v ) T il ( u ) ⊗ e i ⊗ e k . Multiplying by u − v and equating the coefficients of e i ⊗ e k we recover (1.3). (cid:3) Automorphisms and anti-automorphisms
In this section, we will use the N × N matrix T ( u ) to define several distinguishedautomorphisms and anti-automorphisms of the associative unital algebra Y( gl N ).For each of them, we will describe the N × N matrix whose ij -entry is the formalpower series in u − with the coefficients being the images of the correspondingcoefficients of the series T ij ( u ). For example, the assignment (3.2) below meansthat for all indices r = 1 , , . . . and i, j = 1 , . . . , NT ( r ) ij ( − r T ( r ) ij . Proposition 3.1.
For any c ∈ C an automorphism of Y( gl N ) can be defined by (3.1) T ( u ) T ( u − c ) . Proof.
The image of T ( u ) relative to (3.1) clearly satisfies the defining relation (2.8).Further, the mapping (3.1) is obviously invertible which completes the proof. (cid:3) We may regard the element T ( u ) defined by (2.1) as a formal power series in u − whose coefficients are matrices with the entries from the algebra Y( gl N ). Since theleading term of this series is the identity matrix, the element T ( u ) is invertible. Wedenote by T − ( u ) the inverse element. Further, denote by T t ( u ) the transposedmatrix for T ( u ). Then T t ( u ) = N X i,j =1 e ij ⊗ T ji ( u ) . Proposition 3.2.
Each of the assignments T ( u ) T ( − u ) , (3.2) T ( u ) T t ( u ) , (3.3) S : T ( u ) T − ( u )(3.4) defines an anti-automorphism of Y( gl N ) . OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 7 Proof.
The images T ′ ij ( u ) of the series T ij ( u ) under any anti-automorphism of thealgebra Y( gl N ) must satisfy the relations (1.3) with the opposite multiplication:( u − v ) [ T ′ ij ( u ) , T ′ kl ( v ) ] = T ′ il ( u ) T ′ kj ( v ) − T ′ il ( v ) T ′ kj ( u ) . Exactly as in the proof of Proposition 2.2, one can show that these relations canbe equivalently written in the following matrix form R ( u − v ) T ′ ( v ) T ′ ( u ) = T ′ ( u ) T ′ ( v ) R ( u − v )where T ′ ( u ) is the N × N matrix whose ij -th entry is T ′ ij ( u ). But the relation R ( u − v ) T ( − v ) T ( − u ) = T ( − u ) T ( − v ) R ( u − v )follows from (2.8) if we conjugate both sides by P and replace ( u, v ) by ( − v, − u ) .This shows that (3.2) defines an anti-homomorphism. Furthermore, the applicationof the partial transposition t to both sides of the relation (2.8) yields(3.5) T t ( u ) R t ( u − v ) T ( v ) = T ( v ) R t ( u − v ) T t ( u ) . Since R ( u − v ) is fixed by the composition of t with t , applying t to (3.5) yields T t ( u ) T t ( v ) R ( u − v ) = R ( u − v ) T t ( v ) T t ( u ) . Hence (3.3) is an anti-homomorphism. Finally, for (3.4) observe that the relation R ( u − v ) T − ( v ) T − ( u ) = T − ( u ) T − ( v ) R ( u − v )is equivalent to (2.8). Note now that the mappings (3.2) and (3.3) are involutiveand so these two anti-homomorphisms are bijective.The bijectivity of the anti-homomorphism S of Y( gl N ) defined by (3.4) followsfrom the bijectivity of its square S which is computed at the end of this section. (cid:3) The anti-automorphisms (3.2) and (3.3) are involutive and commute with eachother. Their composition is an involutive automorphism of Y( gl N ) such that(3.6) T ( u ) T t ( − u ) . This automorphism of the algebra Y( gl N ) will play an important role in Section 6.However, the anti-automorphism (3.4) is not involutive unless N = 1 . This is theantipodal map S of the Hopf algebra Y( gl N ), see Section 4 below.To compute the square of the anti-homomorphism (3.4) consider N × N matrixobtained from T − ( u ) by transposition. Let us denote this new matrix by T ♯ ( u ) .Accordingly, the ij -th entry of this matrix will be denoted by T ♯ij ( u ) . This entry is aformal power series in u − with coefficients from the algebra Y( gl N ) . By definition,(3.7) S : T ij ( u ) T ♯ji ( u ) . Our computation of the image of T ij ( u ) relative to S is based on the next lemma. Lemma 3.3.
There is a formal power series Z ( u ) in u − with the coefficients fromthe centre of the algebra Y( gl N ) and with the leading term such that for all i and j (3.8) N X k =1 T ki ( u + N ) T ♯kj ( u ) = δ ij Z ( u ) . MAXIM NAZAROV
Proof.
Let us multiply both sides of the relation (2.8) by T − ( v ) on the left andright and then apply transposition relative to the second copy of End C N . We get R t ( u − v ) T ♯ ( v ) T ( u ) = T ( u ) T ♯ ( v ) R t ( u − v ) . Multiplying both sides of this result on the left and right R t ( u − v ) − we get(3.9) R t ( u − v ) − T ( u ) T ♯ ( v ) = T ♯ ( v ) T ( u ) R t ( u − v ) − . Multiplying the latter equality by u − v − N and then setting u = v + N we get(3.10) Q T ( v + N ) T ♯ ( v ) = T ♯ ( v ) T ( v + N ) Q , see (2.5). Because the operator Q is one-dimensional, either side of (3.10) must beequal to Q times a certain power series in v − with the coefficients from Y( gl N ) .Denote this series by Z ( v ) . By applying the left hand side of (3.10) to the basisvector e i ⊗ e j we obtain the required equality (3.8).It is immediate from (1.2) and (3.8) that the leading term of series Z ( v ) is 1 .Let us prove that all the coefficients of this series are central in Y( gl N ) . We willwork with the algebra (2.2) where m = 3 . By using the relations (2.8) and (3.9), R t ( u − v ) − R ( u − v − N ) T ( u ) T ( v + N ) T ♯ ( v ) = R t ( u − v ) − T ( v + N ) T ( u ) T ♯ ( v ) R ( u − v − N ) = T ( v + N ) T ♯ ( v ) T ( u ) R t ( u − v ) − R ( u − v − N ) . Note that by using the expressions (2.4) and (2.5) we obtain the equality Q R t ( u − v ) − R ( u − v − N ) = Q ( 1 − ( u − v − N ) − ) . So multiplying the first and third lines of previous display by Q on the left gives( 1 − ( u − v − N ) − ) T ( u ) Z ( v ) Q = Q Z ( v ) T ( u ) ( 1 − ( u − v − N ) − )where we also used (3.10). The last display shows that any generator T ( r ) ij commuteswith every coefficient of the series Z ( v ) . (cid:3) It follows from (1.2) and (3.8) that the coefficient of the series Z ( u ) at u − iszero. In Section 9 we will show that the coefficients of Z ( u ) at u − , u − , . . . are freegenerators of the centre of the algebra Y( gl N ) . Hence we will again use Lemma 3.3. Proposition 3.4.
The square of the map S is the automorphism of Y( gl N ) given by S : T ( u ) Z ( u ) − T ( u + N ) . Proof.
Let us apply the anti-homomorphism S to both sides of the identity N X k =1 T jk ( u ) T ♯ik ( u ) = δ ij . Using (3.7) we get N X k =1 S ( T ki ( u )) T ♯kj ( u ) = δ ij . Comparing this with (3.8) we conclude that S ( T ki ( u )) = Z ( u ) − T ki ( u + N ) . (cid:3) OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 9 Hopf algebra structure A coalgebra over the field C is a complex vector space A equipped with a linearmap ∆ : A → A ⊗ A called the comultiplication , and another linear map ε : A → C called the counit , such that the following three diagrams are commutative:A ∆ −−−−→ A ⊗ A ∆ y y ∆ ⊗ id A ⊗ A −−−−→ id ⊗ ∆ A ⊗ A ⊗ Awhich gives the coassociativity axiom of the comultiplication ∆ , andA ∆ −−−−→ A ⊗ A id y y ε ⊗ id A −−−−→ ∼ = C ⊗ A A ∆ −−−−→ A ⊗ A id y y id ⊗ ε A −−−−→ ∼ = A ⊗ C A bialgebra over C is a complex associative unital algebra A equipped with acoalgebra structure, such that ∆ and ε are algebra homomorphisms. In particular,then ∆(1) = 1 ⊗ ε (1) = 1. A bialgebra A is called a Hopf algebra , if it isalso equiped with an anti-automorphism S : A → A called the antipode , such thatanother two diagrams are commutative:A δ ε −−−−→ A ∆ y x µ A ⊗ A −−−−→ S ⊗ id A ⊗ A A δ ε −−−−→ A ∆ y x µ A ⊗ A −−−−→ id ⊗ S A ⊗ AHere µ : A ⊗ A → A is the algebra multiplication and δ : C → A is the unit map ofthe algebra A, that is δ ( c ) = c · c ∈ C . Proposition 4.1.
The Yangian Y( gl N ) is a Hopf algebra with comultiplication (4.1) ∆ : T ij ( u ) N X k =1 T ik ( u ) ⊗ T kj ( u ) , the antipode (3.4) and the counit ε : T ( u ) .Proof. We start by verifying the axiom that ∆ : Y( gl N ) → Y( gl N ) ⊗ Y( gl N ) is analgebra homomorphism. We shall slightly generalize the notation used in Section 2.Let m and n be positive integers. Introduce the algebra(4.2) (End C N ) ⊗ m ⊗ Y( gl N ) ⊗ n . For all a ∈ { , . . . , m } and b ∈ { , . . . , n } consider the formal power series in u − with the coefficients in this algebra, T a [ b ] ( u ) = N X i,j =1 ⊗ ( a − ⊗ e ij ⊗ ⊗ ( m − a ) ⊗ ⊗ ( b − ⊗ T ij ( u ) ⊗ ⊗ ( n − b ) . The definition of ∆ can now be written in a matrix form,(4.3) ∆ : T ( u ) T [1] ( u ) T [2] ( u )where T [ b ] ( u ) is an abbreviation for the series T b ] ( u ) with the coefficients from thealgebra (4.2) where m = 1 and n = 2. We need to show that ∆( T ( u )) obeys (2.8): R ( u − v ) T ( u ) T ( u ) T ( v ) T ( v ) = T ( v ) T ( v ) T ( u ) T ( u ) R ( u − v ) . Here m = n = 2, and R ( u − v ) is identified with R ( u − v ) ⊗ ⊗
1. But this relationis implied by the relation (2.8), and by the observation that the elements T ( u )and T ( v ) commute, as well as the elements T ( u ) and T ( v ) do.Our S is an anti-automorphism relative to multiplication due to Proposition 3.2.Since ∆ is a homomorphism of algebras, the definition (4.3) implies that∆ : T − ( u ) T − ( u ) T − ( u ) . Therefore S is also an anti-automorphism relative to comultiplication. The other twoaxioms involving S are readily verified since(S ⊗ id) ∆ : T ( u ) T − ( u ) T [2] ( u )and (id ⊗ S) ∆ : T ( u ) T [1] ( u ) T − ( u )so that subsequent application of µ yields the identity matrix in both the cases. (cid:3) We have ε (cid:0) T ( r ) ij (cid:1) = 0 for r > u − in(4.1) we obtain a more explicit definition of the comultiplication ∆ on Y( gl N ) ,(4.4) ∆ (cid:0) T ( r ) ij (cid:1) = T ( r ) ij ⊗ ⊗ T ( r ) ij + N X k =1 r − X s =1 T ( s ) ik ⊗ T ( r − s ) kj . Hence this comultiplication is not cocommutative unless N = 1 . Proposition 4.2.
For the series Z ( u ) defined above we have ∆ : Z ( u ) Z ( u ) ⊗ Z ( u ) . Proof.
The square S of of the antipodal map is a coalgebra automorphism. Hencethe images of T ( u ) relative to the compositions ∆ S and (S ⊗ S ) ∆ are the same.By Proposition 3.4 these images are respectively equal to∆ ( Z ( u ) − T ( u + N )) = ∆ ( Z ( u ) − ) ( T ( u + N ) ⊗ T ( u + N ))and (S ⊗ S )( T ( u ) ⊗ T ( u )) = ( Z ( u ) − T ( u + N )) ⊗ ( Z ( u ) − T ( u + N )) . Here we identify Z ( u ) − with the series 1 ⊗ Z ( u ) − which takes its coefficients fromEnd C N ⊗ Y( gl N ) and use the homomorphism property of ∆ . By dividing the righthand sides of above two equalities by T ( u + N ) ⊗ T ( u + N ) and equating the results∆ : Z ( u ) − Z ( u ) − ⊗ Z ( u ) − . (cid:3) OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 11 Two filtrations on the Yangian
There are two natural ascending filtrations on the associative algebra Y( gl N ) .The first one is defined by deg T ( r ) ij = r . For any r > b T ( r ) ij the image of the generator T ( r ) ij in the degree r component of the corresponding graded algebra gr Y( gl N ) . It is immediate fromthe defining relations (1.4) that all these images pairwise commute. In Section 8we will prove that these images are also algebraically independent.Now introduce another filtration on Y( gl N ) by setting for r > ′ T ( r ) ij = r − . Let gr ′ Y( gl N ) be the corresponding graded algebra. Let e T ( r ) ij be the image of T ( r ) ij in the component of gr ′ Y( gl N ) of the degree r − ′ Y( gl N ) inherits from Y( gl N ) the Hopf algebra structure.Namely, by using (4.4) for any r > (cid:0) e T ( r ) ij (cid:1) = e T ( r ) ij ⊗ ⊗ e T ( r ) ij , (5.2) ε (cid:0) e T ( r ) ij (cid:1) = 0 and S (cid:0) e T ( r ) ij (cid:1) = − e T ( r ) ij . (5.3)For any Lie algebra g over the field C consider the universal enveloping algebraU( g ). There is a natural Hopf algebra structure on U( g ). The comultiplication ∆,the counit ε and the antipode S on U( g ) are defined by setting for X ∈ g ∆( X ) = X ⊗ ⊗ X , (5.4) ε ( X ) = 0 and S( X ) = − X . (5.5)In the next proposition g is the polynomial current Lie algebra gl N [ z ] ∼ = gl N ⊗ C [ z ] .The latter Lie algebra is naturally graded by degrees of the indeterminate z . Proposition 5.1.
The graded Hopf algebra gr ′ Y( gl N ) is isomorphic to U( gl N [ z ]) .Proof. Using the defining relations (1.4) we get[ e T ( r ) ij , e T ( s ) kl ] = δ kj e T ( r + s − il − δ il e T ( r + s − kj . Hence the assignments(5.6) E ij z r − e T ( r ) ij for r > gl N [ z ]) → gr ′ Y( gl N )of graded associative algebras. At the end of Section 8 we will show that the kernelof this homomorphism is trivial. Hence comparing the definitions (5.2),(5.3) withthe general definitions (5.4),(5.5) completes the proof of Proposition 5.1. (cid:3) Vector and covector representations
We shall often use the matrix T ( u ) to describe homomorphisms from Y( gl N ) toother algebras. Namely, let A be any unital associative algebra over the field C . Let X ( u ) be the N × N matrix whose ij -entry is any formal power series X ij ( u ) in u − with the leading term δ ij and all coefficients from the algebra A . If α : Y( gl N ) → Ais any homomorphism, then the assignment(6.1) α : T ( u ) X ( u )means that every coefficient of the series T ij ( u ) gets mapped to the correspondingcoefficient of the series X ij ( u ) for all indices i, j = 1 , . . . , N . If we regard T ( u ) asa series in u with the coefficients from the algebra End C N ⊗ Y( gl N ) then, moreformally, we may write id ⊗ α : T ( u ) X ( u )instead of (6.1). Here X ( u ) = N X i,j =1 e ij ⊗ X ij ( u ) , is regarded as a series in u with coefficients from the algebra End C N ⊗ A; cf. (2.1).Setting A = End C N and X ( u ) = R ( u ) above, we can define a homomorphismY( gl N ) → End C N by the assignment T ( u ) R ( u ). To prove the homomorphismproperty by using the matrix form (2.8) of the defining relations of the algebraY( gl N ), we have to check the equality of rational functions in u and v with valuesin the algebra (End C N ) ⊗ , R ( u − v ) R ( u ) R ( v ) = R ( v ) R ( u ) R ( u − v ) . But this equality is just another form of (2.6). In other words, the assignment T ( u ) R ( u ) defines a representation of Y( gl N ) on the vector space C N . Here T ij ( u ) δ ij − e ji u − by (2.3) and (2.4). Note that this representation of the algebra Y( gl N ) can also beobtained by pulling the defining representation E ij e ij of the Lie algebra gl N back through the homomorphism (1.8). This remark justifies the definition (1.8).By pulling the defining representation E ij e ij of the Lie algebra gl N backthrough the homomorphism (1.6), we get the representation of Y( gl N ) such that T ij ( u ) δ ij + e ij u − . Hence this representation can be described by the assignment T ( u ) R t ( − u ) .Observe that the representations T ( u ) R ( u ) and T ( u ) R t ( − u ) differ by theinvolutive automorphism (3.6) of the algebra Y( gl N ).By pulling the representation T ( u ) R ( u ) back through the automorphism(3.1) of Y( gl N ) for any c ∈ C , we get the representation of Y( gl N ) on the vectorspace C N , such that T ( u ) R ( u − c ). It is called a vector representation of Y( gl N ),and is denoted by ρ c . Thus ρ c : T ij ( u ) δ ij − e ji ( u − c ) − or equvalently,(6.2) ρ c : T ( r ) ij
7→ − c r − e ji for any r > . OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 13 By pulling the representation T ( u ) R t ( − u ) back through the automorphism(3.1), we get the representation of Y( gl N ) on C N , such that T ( u ) R t ( c − u ). Itis called a covector representation of Y( gl N ), and is denoted by σ c . Thus σ c : T ij ( u ) δ ij + e ij ( u − c ) − or equvalently,(6.3) σ c : T ( r ) ij c r − e ij for any r > . In Section 5 we introduced an ascending filtration on algebra Y( gl N ) such thatany generator T ( r ) ij of Y( gl N ) has the degree r − ′ Y( gl N ) and defined a surjective homomorphism (5.7) by (5.6).Under this homomorphism the element T ( r ) ij of Y( gl N ), or rather its image e T ( r ) ij in gr ′ Y( gl N ) , corresponds to the generator E ij z r − of U( gl N [ z ]). One can definea representation e σ c of the algebra U( gl N [ z ]) on the vector space C N by(6.4) e σ c : E ij z r − c r − e ij for any r > , so that e σ c ( E ij z r − ) = σ c ( T ( r ) ij ) . The representation e σ c is an example of an evaluation representation of U( gl N [ z ]),see the general definition in Section 7 below.7. Evaluation representations
For any Lie algebra a over C consider the corresponding polynomial current Liealgebra a [ z ] = a ⊗ C [ z ] . Let θ be any representation of a on the vector space C N ,and c be any complex number. Then one can define a representation of a [ z ] by X z s c s θ ( X ) for any s > . This is the evaluation representation of the Lie algebra a [ z ], corresponding to θ at the point z = c of the complex plane C . When a = gl N and θ is the definingrepresentation of the Lie algebra gl N on C N , we obtain e σ c in this way.We will need the following general property of evaluation representations. Forany c , . . . , c n ∈ C let us denote by θ c ...c n the tensor product of the evaluationrepresentations of the Lie algebra a [ z ] corresponding to θ at the points c , . . . , c n .We extend the representation θ c ...c n to the universal enveloping algebra U( a [ z ]). Lemma 7.1.
Suppose that the Lie algebra a is finite-dimensional, and θ is itsfaithful representation. Let the parameters c , . . . , c n and integer n > vary. Thenthe intersection in U( a [ z ]) of the kernels of all representations θ c ...c n is trivial.Proof. Using the faithful representation θ of the Lie algebra a , we can identify a [ z ]with a subalgebra of the Lie algebra gl N [ z ]. Hence it suffices to consider the casewhen a is the Lie algebra gl N , and θ : gl N → End C N is the defining representation.Choose any basis X , . . . , X N in gl N such that one of the basis vectors is I = E + · · · + E NN . To distinguish between the algebras U( gl N ) and End C N , the operators on C N corresponding to the elements X , . . . , X N ∈ gl N will be denoted by x , . . . , x N respectively. Note that one of these operators is the identity operator 1.Take any finite non-zero linear combination C of the products(7.1) ( X a z s ) . . . ( X a m z s m ) ∈ U( gl N [ z ]) where the indices a , . . . , a m and s , . . . , s m > m offactors in (7.1) may also vary. Let S m be the symmetric group acting on the set { , . . . , m } . For each fixed m >
0, we will suppose that the elements(7.2) X q ∈ S m ( X a q (1) z s q (1) ) ⊗ . . . ⊗ ( X a q ( m ) z s q (1) ) ∈ ( gl N [ z ]) ⊗ m corresponding to the products (7.1) which have exactly m factors and appear in C with non-zero coefficients, are linearly independent. We may suppose so withoutloss of generality, due to the commutation relations in the algebra U( gl N [ z ]). Usingthe natural identification of vector spaces( gl N [ z ]) ⊗ m = gl ⊗ mN [ z , . . . , z m ] , the sum (7.2) may be also regarded as a polynomial function in m independentcomplex variables z , . . . , z m . This function takes values in the vector space gl ⊗ mN .For every element (7.1) appearing in the linear combination C , suppose that X a , . . . , X a l = I and X a l +1 = · · · = X a m = I for a certain number l >
0. Then x a , . . . , x a l = 1 and x a l +1 = . . . = x a m = 1. Wemay suppose so without loss of generality, as the elements I, Iz, Iz , . . . ∈ gl N [ z ]are central. Further, suppose that s l +1 > · · · > s m . Of course, the number l heremay depend on the given element (7.1) ; let l be the maximum of these numbers.Let us consider two cases. First suppose that l = 0 , so that l = 0 for everyelement (7.1) appearing in the linear combination C . The image of (7.1) under therepresentation θ c ...c n of U( gl N [ z ]) is then the operator of multiplication by(7.3) m Y k =1 ( c s k + · · · + c s k n ) ∈ C . Since s > · · · > s m here, any non-trivial linear combination of the scalars (7.3)corresponding to different sequences s , . . . , s m cannot vanish identically for all n > c , . . . , c n ∈ C . This proves the claim when l = 0.Second, suppose that l >
0. For any element (7.1) appearing in the linearcombination C , take its image under the representation θ c ...c n with n > l . Thisimage belongs to the algebra End ( C N ) ⊗ n , which we will identify with (End C N ) ⊗ n .Let V l be the subspace in (End C N ) ⊗ n spanned by the elements x b ⊗ . . . ⊗ x b n whereat least one of the first l tensor factors x b , . . . , x b l is 1. The indices b , . . . , b n hererange over 1 , . . . , N . Modulo V l the image of (7.1) in (End C N ) ⊗ n equals the sum(7.4) X p ∈ S l ( c s p (1) x a p (1) ) ⊗ . . . ⊗ ( c s p ( l ) l x a p ( l ) ) ⊗ ⊗ ( n − l ) multiplied by(7.5) m Y k = l +1 ( c s k + · · · + c s k n ) . Notice that the sum (7.4) does not belong to V l unless this sum is zero or l = 0 .In the linear combination C , take the terms where l = l . Let D ∈ (End C N ) ⊗ n be the image of the sum of these terms under the representation θ c ...c n . We assumethat here n > l . The images of the terms with l < l under the representation θ c ...c n belong to the subspace V l . We will show that D = 0 for some n > l and c , . . . , c n ∈ C . Then D / ∈ V l , and Lemma 7.1 will follow. OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 15 Observe that the sum (7.4) does not depend on the parameters c l +1 , . . . , c n while the product (7.5) can depend on these parameters. Due to this observation,we may now assume that for all terms in the linear combination C with l = l ,the sequences s l +1 , . . . , s m are the same; see the case l = 0 considered above.Moreover, we may assume that m = l for all these terms. But under the latterassumption, the inequality D = 0 for some c , . . . , c l ∈ C follows from the linearindependence of the elements (7.2) with m = l . (cid:3) Poincar´e–Birkhoff–Witt theorem
Let us now make use of the bialgebra structure on Y( gl N ). For any c , . . . , c n ∈ C take the tensor product of the vector representations ρ c , . . . , ρ c n of Y( gl N ). Weget a representation ρ c ...c n : Y( gl N ) → (End C N ) ⊗ n . If n = 0, the representation ρ c ...c n is understood as the counit homomorphism ε :Y( gl N ) → C . Using the matrix form (4.3) of the definition of the comultiplicationon Y( gl N ), we see thatid ⊗ ρ c ...c n : T ( u ) R ( u − c ) . . . R ,n +1 ( u − c n ) . Here we apply the convention made in the beginning of Section 6 to the algebraA = (End C N ) ⊗ n and to the homomorphism α = ρ c ...c n .The tensor product of the covector representations σ c , . . . , σ c n will be denotedby σ c ...c n . By using the matrix form (4.3) of the definition of the comultiplicationon Y( gl N ) again, we see thatid ⊗ σ c ...c n : T ( u ) R t ( c − u ) . . . R t ,n +1 ( c n − u ) . By using Lemma 7.1, we will now prove the following proposition.
Proposition 8.1.
Let the parameters c , . . . , c n ∈ C and the integer n > vary.Then the intersection of the kernels of all representations σ c ...c n is trivial.Proof. Take any finite linear combination A of the products T ( r ) i j . . . T ( r m ) i m j m ∈ Y( gl N )with certain coefficients A r ...r m i j ...i m j m ∈ C where the indices r , . . . , r m > m > i , j , . . . , i m , j m . Suppose that A = 0 as an element of Y( gl N ).The algebra Y( gl N ) comes with an ascending filtration such that T ( r ) ij has thedegree r − d be the degree of A rtelative to this filtration. Let B be the imageof A in the degree d component of the graded algebra gr ′ Y( gl N ). Then B = 0.We can also assume that A r ...r m i j ...i m j m = 0 if r + · · · + r m > d + m. Let C be the sum of the elements of the algebra U( gl N [ z ]), X r + ··· + r m = d + m A r ...r m i j ...i m j m ( E i j z r − ) . . . ( E i m j m z r m − ) . The image of C under the homomorphism (5.7) equals B . In particular, C = 0. Consider the image of A under the representation σ c ...c n . This image dependson c , . . . , c n polynomially. The degree of this polynomial does not exceed d by thedefinition (6.3). Let D be the sum of the terms of degree d of this polynomial.Now equip the tensor product Y( gl N ) ⊗ n with the ascending filtration where thedegree is the sum of the degrees on the tensor factors. Then under the n -foldcomultiplication Y( gl N ) → Y( gl N ) ⊗ n T ( r ) ij n X b =1 ⊗ ( b − ⊗ T ( r ) ij ⊗ ⊗ ( n − b ) plus terms of degree less than r − , see (4.4). But under the n -fold comultiplication U( gl N [ z ]) → U( gl N [ z ]) ⊗ n , E ij z r − n X b =1 ⊗ ( b − ⊗ ( E ij z r − ) ⊗ ⊗ ( n − b ) . The definitions (6.3) and (6.4) now imply that the sum D ∈ (End C N ) ⊗ n coincideswith the image of the sum C ∈ U( gl N [ z ]) under the tensor product of the evaluationrepresentations e σ c , . . . , e σ c n . Since C = 0, using Lemma 7.1 we can choose n and c , . . . , c n so that D = 0 . Then σ c ...c n ( A ) = 0 by the definition of D . (cid:3) Proposition 8.2.
Let the parameters c , . . . , c n ∈ C and the integer n > vary.Then the intersection of the kernels of all representations ρ c ...c n is trivial. The proof of Proposition 8.2 is similar to that of Proposition 8.1 and is omitted.We will now prove the injectivity of homomorphism (5.7) by modifying the logic ofour proof of Proposition 8.1. Take any finite linear combination C of the products( E i j z r − ) . . . ( E i m j m z r m − ) ∈ U( gl N [ z ])with certain coefficients C r ...r m i j ...i m j m ∈ C where the indices r , . . . , r m > m > i , j , . . . , i m , j m . Suppose that C = 0 as an element of U( gl N [ z ]).The algebra U( gl N [ z ]) is graded so that for any integer s > E ij z s has the degree s . The homomorphism (5.7) preserves the degree. Withoutloss of generality suppose that the element C is homogeneous of degree d , that is C r ...r m i j ...i m j m = 0 if r + · · · + r m = d + m. Now define the element A ∈ Y( gl N ) as the sum X r + ··· + r m = d + m C r ...r m i j ...i m j m T ( r ) i j . . . T ( r m ) i m j m . Let B be the image of A in the d -th component of the graded algebra gr ′ Y( gl N ).The element B coincides with the image of C under the homomorhism (5.7).Now let D ∈ (End C N ) ⊗ n be the image of C under the tensor product of theevaluation representations e σ c , . . . , e σ c n . The image of A under the representation σ c ...c n depends on c , . . . , c n polynomially. The degree of this polynomial does notexceed d by (6.3). The sum of the terms of degree d of this polynomial equals D , seethe proof of Proposition 8.1. Since C = 0, using Lemma 7.1 we can choose n and c , . . . c n so that D = 0 . Then deg ′ A = d . Indeed, if deg ′ A < d then the degree ofthe polynomial σ c ,...,c n ( A ) would be also less then d . This would contradict to thenon-vanishing of D . By the definition of the element B ∈ gr ′ Y( gl N ), the equalitydeg ′ A = d means that B = 0. So the homomorphism (5.7) is injective. OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 17 Let us now invoke the classical Poincar´e–Birkhoff–Witt theorem for the universalenveloping algebras of Lie algebras [2, Section 2.1]. By applying this theorem tothe Lie algebra gl N [ z ] we now obtain its analogue for the Yangian Y( gl N ) . Theorem 8.3.
Given an arbitrary linear ordering of the set of generators T ( r ) ij with r > , any element of the algebra Y( gl N ) can be uniquely written as a linearcombination of ordered monomials in these generators. Corollary 8.4.
The graded algebra gr Y( gl N ) is the algebra of polynomials in thegenerators b T ( r ) ij with r > . Centre of the Yangian
Let a be any Lie algebra over the field C . Consider the corresponding polynomialcurrent Lie algebra a [ z ] . In the proof of Theorem 9.3 we will use a general propertyof the universal enveloping algebra U( a [ z ]) . It is stated as the lemma below. Lemma 9.1.
Suppose that the Lie algebra a is finite-dimensional and has the trivialcentre. Then the centre of the algebra U( a [ z ]) is also trivial, that is equal to C .Proof. Consider adjoint action of the Lie algebra a [ z ] on its symmetric algebra. Itsuffices to prove that the space of invariants of this action is trivial.Let A be any element of the symmetric algebra of a [ z ] invariant under the adjointaction. Let M = dim a . Choose any basis X , . . . , X M of a and let[ X p , X q ] = M X r =1 c rpq X r where c rpq ∈ C . Let L be the minimal non-negative integer such that A = X d ,...,d M A d ...d M ( X z L ) d . . . ( X M z L ) d M where d , . . . , d M range over non-negative integers and A d ...d M is a polynomial inthe basis elements X p z s of a [ z ] with 1 p M and 0 s < L only. We havead( X p z )( A ) = 0for every index p = 1 , . . . , M . The component of the left hand side of this equationthat involves the basis elements of a [ z ] of the form X r z L +1 must be zero. Thus X d ,...,d M A d ...d M M X q,r =1 c rpq d q ( X z L ) d . . . ( X q z L ) d q − . . . ( X M z L ) d M X r z L +1 = 0 . Taking here the coefficient of X r z L +1 we obtain that X d ,...,d M A d ...d M M X q =1 c kpq d q ( X z L ) d . . . ( X q z L ) d q − . . . ( X M z L ) d M = 0 . If follows that for any non-negative integers d ′ , . . . , d ′ M we have(9.1) M X q =1 A d ′ ...d ′ q +1 ...d ′ M c rpq ( d ′ q + 1) = 0 where p, r = 1 , . . . , M . Let us now fix d ′ , . . . , d ′ M and observe that the elements X ′ q = ( d ′ q + 1) X q with q = 1 , . . . , M also form a basis of a . Since the centre of a is trivial, the system[ X p , M X q =1 a q X ′ q ] = 0 where p = 1 , . . . , M of linear equations on a , . . . , a M ∈ C has only trivial solution. It can be written as M X q =1 a q c rpq ( d ′ q + 1) = 0 where p, r = 1 , . . . , M . Hence by comparing (9.1) with the latter system we obtain that A d ′ ...d ′ q +1 ...d ′ M = 0for every q = 1 , . . . , M . It now follows that A ∈ C , and L = 0 in particular. (cid:3) Now consider the series Z ( u ) defined by (3.8). For any r > Z ( r ) be thecoefficient of this series at u − r . Just before stating Proposition 3.4 we noted that Z (1) = 0 . Hence Z ( u ) = 1 + Z (2) u − + Z (3) u − + . . . . Proposition 9.2.
For any r > the element Z ( r ) ∈ Y( gl N ) has the degree r − relative to the filtration (5.1) . Its image in the graded algebra gr ′ Y( gl N ) is equal to (1 − r ) N X i =1 e T ( r − ii . Proof.
Let us expand the factor T ki ( u + N ) appearing in the definition (3.8) as aformal power series in u − . The result has the form T ki ( u ) + N ˙ T ki ( u ) + X ki ( u )where ˙ T ki ( u ) = − T (1) ki u − − T (2) ki u − − . . . is the formal derivative of the series T ki ( u ) and X ki ( u ) = X (3) ki u − + X (4) ki u − + . . . is a series with coefficients X ( r ) ki ∈ Y( gl N ) such that deg ′ X ( r ) ki = r − r > i = j in (3.8) and summing over i = 1 , . . . , N we now get the equality(9.2) N + N X i,k =1 ( N ˙ T ki ( u ) + X ki ( u )) T ♯ki ( u ) = N Z ( u ) . Here we used the definition of the matrix T ♯ ( u ) as the transposed inverse of T ( u ) .The leading term of the series T ♯ki ( u ) is δ ik while for any r > u − r has the degree r − r > u − r of the series at the lefthand side of (9.2) equals N N X i,k =1 (1 − r ) T ( r − ki δ ik = N N X i =1 (1 − r ) T ( r − ii . Hence Proposition 9.2 follows from (9.2). Also we see once again that Z (1) = 0 . (cid:3) Theorem 9.3.
The coefficients Z (2) , Z (3) , . . . of the series Z ( u ) are free generatorsof the centre of the associative algebra Y( gl N ) . OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 19 Proof.
Let us apply Lemma 9.1 to the special linear Lie algebra a = sl N . Since thecentre of the universal enveloping algebra U( sl N [ z ]) is trivial, the decomposition gl N [ z ] = sl N [ z ] ⊕ C [ z ] N X i =1 E ii of Lie algebras implies that the centre of U( gl N [ z ]) is generated by the elements(9.3) N X i =1 E ii z r − where r > gl N [ z ] .Under the isomorphism (5.7), the elements (9.3) go respectively to the elements N X i =1 e T ( r ) ii where again r > ′ Y( gl N ) , see Proposition 5.1. On the other hand, we have alreadyproved that the elements Z (2) , Z (3) , . . . of the algebra Y( gl N ) belong to its centre,see Lemma 3.3. Hence Theorem 9.3 follows from Proposition 9.2. (cid:3) Dual Yangian
The dual Yangian for the Lie algebra gl N , denoted by Y ∗ ( gl N ), is an associativeunital algebra over the field C with a countable set of generators T ( − ij , T ( − ij , . . . where i, j = 1 , . . . , N . To write down the defining relations for these generators, introduce the series(10.1) T ∗ ij ( v ) = δ ij + T ( − ij + T ( − ij v + T ( − ij v + . . . ∈ Y ∗ ( gl N )[[ v ]] . The reason for separating the term δ ij in (10.1) will become apparent in the nextsection. Now combine all the series (10.1) into the single element(10.2) T ∗ ( v ) = N X i,j =1 T ∗ ij ( v ) ⊗ e ij ∈ Y ∗ ( gl N )[[ v ]] ⊗ End C N . We will write the defining relations of the algebra Y ∗ ( gl N ) first in their matrixform, to be compared with (2.8). For any positive integer n , consider the algebra(10.3) Y ∗ ( gl N ) ⊗ (End C N ) ⊗ n . For any index b ∈ { , . . . , n } introduce the formal power series in the variable v with the coefficients from the algebra (10.3),(10.4) T ∗ b ( v ) = N X i,j =1 T ∗ ij ( v ) ⊗ ⊗ ( b − ⊗ e ij ⊗ ( n − b ) . Here the belongs to the b -th copy of End C N . Setting n = 2 and identifying R ( u − v )with 1 ⊗ R ( u − v ) , the defining relations of Y ∗ ( gl N ) can be written as(10.5) T ∗ ( u ) T ∗ ( v ) R ( u − v ) = R ( u − v ) T ∗ ( v ) T ∗ ( u ) . The relation (10.5) is equivalent to the collection of relations( u − v ) [ T ∗ ij ( u ) , T ∗ kl ( v )] = T ∗ il ( u ) T ∗ kj ( v ) − T ∗ il ( v ) T ∗ kj ( u )for all i, j, k, l = 1 , . . . , N . We omit the proof of the equivalence, because it is similarto the proof of Proposition 2.2. The last displayed relation can be rewritten as[ T ∗ ij ( u ) , T ∗ kl ( v )] = ∞ X p =0 u − p − v p (cid:16) T ∗ il ( u ) T ∗ kj ( v ) − T ∗ il ( v ) T ∗ kj ( u ) (cid:17) . Expanding here the series in u, v and equating the coefficients at u r − v s − we get[ T ( − r ) ij , T ( − s ) kl ] = δ kj T ( − r − s ) il − δ il T ( − r − s ) kj +(10.6) s X b =1 (cid:16) T ( b − r − s − il T ( − b ) kj − T ( − b ) il T ( b − r − s − kj (cid:17) . The proof of next proposition is similar to that of Proposition 4.1 and is omitted.
Proposition 10.1.
The dual Yangian Y ∗ ( gl N ) is a bialgebra over the field C withthe counit defined ε : T ∗ ( v ) and the comultiplication defined by (10.7) ∆ : T ∗ ij ( v ) N X k =1 T ∗ ik ( v ) ⊗ T ∗ kj ( v ) . Expanding the power series in v in (10.7) and using the axiom ∆(1) = 1 ⊗
1, weget a more explicit definition of the comultiplication on the dual Yangian Y ∗ ( gl N ),(10.8) ∆ (cid:0) T ( − r ) ij (cid:1) = T ( − r ) ij ⊗ ⊗ T ( − r ) ij + N X k =1 r X s =1 T ( − s ) ik ⊗ T ( s − r − kj for r >
1; cf. (4.4). Since ε (1) = 1, for every r > ε (cid:0) T ( − r ) ij (cid:1) = 0 .The dual Yangian Y ∗ ( gl N ) is a bialgebra but not a Hopf algebra. The antipodalmap S is defined only for a completion Y ◦ ( gl N ) of Y ∗ ( gl N ) such that the element T ∗ (0) ∈ Y ◦ ( gl N ) ⊗ End C N is invertible. Then T ∗ ( v ) ∈ Y ◦ ( gl N )[[ v ]] ⊗ End C N is also invertible, and the antipode S is defined by mapping T ∗ ( v ) to its inverse.This inverse will be denoted by T ♮ ( v ). It will be used again in the end of Section 15.In order to construct such a completion, let us equip the algebra Y ∗ ( gl N ) witha descending filtration, defined by assigning to the generator T ( − r ) ij the degree r forany r >
1. Then Y ◦ ( gl N ) is defined as the formal completion of Y ∗ ( gl N ) relative tothis descending filtration. The algebra Y ◦ ( gl N ) ⊗ End C N contains the inverse of T ∗ (0) = 1 ⊗ N X i,j =1 T ( − ij ⊗ e ij . We extend the comultiplication ∆ on Y ∗ ( gl N ) to Y ◦ ( gl N ) , and also denote thisextension by ∆ . The image ∆(Y ◦ ( gl N )) lies in the formal completion of the algebraY ∗ ( gl N ) ⊗ Y ∗ ( gl N ) with respect to the descending filtration, defined by assigningto the element T ( − r ) ij ⊗ T ( − s ) kl the degree r + s . Indeed, the image ∆ (cid:0) T ( − r ) ij (cid:1) inY ∗ ( gl N ) ⊗ Y ∗ ( gl N ) is a sum of elements of degrees r and r + 1 by (10.8). OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 21 The kernel of the counit homomorphism ε : Y ∗ ( gl N ) → C consists of all theelements which of positive degree relative to the filtration, see Proposition 10.1.Therefore ε extends to the algebra Y ◦ ( gl N ). This extension is the counit map forthe Hopf algebra Y ◦ ( gl N ), it will be also denoted by ε .For any c ∈ C the assignment T ∗ ( v ) T ∗ ( v + c ) determines an automorphismof the algebra Y ◦ ( gl N ). This follows from the relations (10.5), cf. Proposition 3.1.But for c = 0 this automorphism does not preserve the subset Y ∗ ( gl N ) ⊂ Y ◦ ( gl N ),and therefore does not determine an automorphism of Y ∗ ( gl N ) .To find the square of the antipodal map S of the Hopf algebra Y ◦ ( gl N ) let T ♭ ( v )be the result of applying to the inverse of (10.2) the transposition in End C N . Write T ♭ ( v ) = N X i,j =1 T ♭ij ( v ) ⊗ e ij ∈ Y ◦ ( gl N )[[ v ]] ⊗ End C N so that S : T ∗ ij ( v ) T ♭ji ( v ) . The proof of the next lemma is similar to that of Lemma 3.3 and is omitted here.
Lemma 10.2.
There is a formal power series Z ◦ ( v ) in v with coefficients from thecentre of the algebra Y ◦ ( gl N ) such that for all indices i and j N X k =1 T ∗ ki ( v − N ) T ♭kj ( v ) = δ ij Z ◦ ( v ) . In general, the coefficients of the series Z ◦ ( v ) do not belong to the dual YangianY ∗ ( gl N ) . However, the proposition below can be derived from Lemma 10.2 just asProposition 3.4 was derived from Lemma 3.3. Hence we again omit the proof. Proposition 10.3.
The square of the map S is the automorphism of Y ◦ ( gl N )S : T ∗ ( v ) Z ◦ ( v ) − T ∗ ( v − N ) . The latter result follows just as Proposition 4.2 followed from Proposition 3.4.
Proposition 10.4.
For the series Z ◦ ( v ) defined above we have ∆ : Z ◦ ( v ) Z ◦ ( v ) ⊗ Z ◦ ( v ) . The completion Y ◦ ( gl N ) of the filtered algebra Y ∗ ( gl N ) can be described moreexplicitly. At the end of Section 12 we will show that the vector space Y ∗ ( gl N ) hasa basis parameterized by all multisets of triples ( r , i , j ) , . . . , ( r m , i m , j m ) where r , . . . , r m ∈ { , , . . . } and i , j , . . . , i m , j m ∈ { , . . . , N } while m = 0 , , , . . . . The corresponding basis vector in Y ∗ ( gl N ) is the monomial(10.9) T ( − r ) i j . . . T ( − r m ) i m j m . The ordering of the factors in this monomial can be chosen arbitrarily. Chooseany linear ordering of the basis monomials. For any positive integer r , there isonly a finite number of the basis monomials (10.9) such that r + · · · + r m r .This means that when the index of the basis monomial (10.9) in any chosen linearordering increases, then the filtration degree (10.9) r + · · · + r m → ∞ . Therefore the vector space Y ◦ ( gl N ) consists of all infinite linear combinations ofthe basis monomials (10.9), with the coefficients from the field C . Canonical pairing
There is a canonical bilinear pairing(11.1) h , i : Y( gl N ) × Y ∗ ( gl N ) → C . We shall describe the corresponding linear map β : Y( gl N ) ⊗ Y ∗ ( gl N ) → C . It willbe defined so that for all integers m, n > C N ) ⊗ m ⊗ Y( gl N ) ⊗ Y ∗ ( gl N ) ⊗ (End C N ) ⊗ n → (End C N ) ⊗ ( m + n ) given by id ⊗ β ⊗ id, will send(11.2) T ( u ) . . . T m ( u m ) ⊗ T ∗ ( v ) . . . T ∗ n ( v n ) → Y a m → Y b n R a,b + m ( u a − v b ) . Here u , . . . , u m , v , . . . , v n are independent variables. The coefficients of the series(11.3) T ( u ) , . . . , T m ( u m ) and T ∗ ( v ) , . . . , T ∗ n ( v n )belong to the algebras (2.2) and (10.3), respectively.Note that the series in u , . . . , u m and v , . . . , v n at the left hand side of (11.2)satisfy certain relations, implied by the defining relations of the algebras Y( gl N )and Y ∗ ( gl N ). The following proposition guarantees that the pairing is well-defined. Proposition 11.1.
The assignment (11.2) agrees with relations (2.8) and (10.5) .Proof.
This follows from the Yang-Baxter equation (2.6). For instance, let us con-sider the case when m = 2 and n = 1. Here we have to check that the series (cid:0) R ( u − u ) T ( u ) T ( u ) (cid:1) ⊗ T ∗ ( v )and (cid:0) T ( u ) T ( u ) R ( u − u ) (cid:1) ⊗ T ∗ ( v )with the coefficients in the algebra(End C N ) ⊗ ⊗ Y( gl N ) ⊗ Y ∗ ( gl N ) ⊗ End C N , have the same images in under the map id ⊗ β ⊗ id. These images are series withthe coefficients in (End C N ) ⊗ . Note that the second element can be rewritten as (cid:0) P T ( u ) T ( u ) P R ( u − u ) (cid:1) ⊗ T ∗ ( v )By the definition (11.2), the images of the two elements are respectively R ( u − u ) R ( u − v ) R ( u − v )and P R ( u − v ) R ( u − v ) P R ( u − u )= R ( u − v ) R ( u − v ) R ( u − u ) . The equality of two images is now evident due to (2.6). Using (2.6) repeatedly, onecan prove Proposition 11.1 for any m, n > (cid:3) Let us show that the assignments (11.2) for all m, n = 0 , , , . . . determine thevalues of the bilinear pairing (11.1) uniquely. When m = n = 0, we get from (11.2)the equality h , i = 1 . By choosing m = 1 and n = 0, we obtain from (11.2) that h T ( r ) ij , i = 0 for any r > m = 0 and n = 1, we obtain that h , T ( − s ) ij i = 0for any s > h , i = 1 obtained above. OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 23 Now suppose that m, n >
1. To determine the pairing values(11.4) (cid:10) T ( r ) i j . . . T ( r m ) i m j m , T ( − s ) k l . . . T ( − s n ) k n l n (cid:11) for any indices r , . . . , r m , s , . . . , s n ∈ { , , . . . } and i , j , . . . , i m , j m , k , l , . . . , k n , l n ∈ { , . . . , N } , the product of the rational functions R a,b + m ( u a − v b ) on the right hand side of(11.2) should be expanded as power series in the variables u − , . . . , u − m , v , . . . , v n .The series (11.3) should be then also expanded.Note that although the coefficient of v in the series (10.1) is a sum of two terms, δ ij and T ( − ij , the pairing value (11.4) can be still determined by (11.2) for anyindices s , . . . , s n > n . Namely, if some of the indices s , . . . , s n are equal to 1, the value (11.4) can be determined by (11.2), using thevalues (11.4) with n replaced by 0 , . . . , n − m = n = 1 in more detail. Then the map id ⊗ β ⊗ id maps T ( u ) ⊗ T ∗ ( v ) = N X i,j,k,l =1 e ij ⊗ (cid:18) ∞ X r =0 T ( r ) ij u − r (cid:19) ⊗ (cid:18) δ kl + ∞ X s =1 T ( − s ) kl v s − (cid:19) ⊗ e kl to the series(11.5) R ( u − v ) = 1 ⊗ − N X i,j =1 ∞ X r =1 u − r v r − e ij ⊗ e ji ;see (2.4) and (10.1). Using the equality h T ( r ) ij , i = δ r for r >
0, we get (cid:10) T ( r ) ij , T ( − s ) kl (cid:11) = − δ rs δ il δ jk for r, s > . More explicitly the value (11.4) will be determined in the course of the proofof the next lemma. This lemma describes a basic property of the bilinear pairing(11.1). It is valid for any integers m, n > Lemma 11.2. If r + · · · + r m < s + · · · + s n then the value (11.4) is zero.Proof. First suppose that s , . . . , s n >
2. Then by the definition of the pairing(11.2), the value (11.4) is the coefficient of(11.6) e i j ⊗ . . . ⊗ e i m j m ⊗ e k l ⊗ . . . ⊗ e k n l n · u − r . . . u − r m m v s − . . . v s n − n in the expansion of the product in (End C N ) ⊗ ( m + n ) [[ u − , . . . , u − m , v , . . . , v n ]] → Y a m → Y b n R a,b + m ( u a − v b ) = → Y a m → Y b n (cid:0) − ∞ X r =1 u − ra v r − b P a,b + m (cid:1) . If the coefficient of (11.6) is non-zero in this expansion then clearly we have theinequality r + · · · + r m > s + · · · + s n .Now suppose that some of the numbers s , . . . , s n are equal to 1. Without loss ofgenerality we will assume that s , . . . , s d > s d +1 , . . . , s n = 1 for some d < n .Rewrite the product at the right hand side of the definition (11.2) as → Y b d → Y a m R a,b + m ( u a − v b ) · → Y d s + · · · + s d + n − d . (cid:3) Non-degeneracy of the pairing
In Section 10 we equipped the algebra Y ∗ ( gl N ) with a descending filtration. Nowconsider the corresponding graded algebra gr Y ∗ ( gl N ). Its component of degree s will be denoted by gr s Y ∗ ( gl N ). For any s > e T ( − s ) ij the image of T ( − s ) ij in gr s Y ∗ ( gl N ). By (10.6) we immediately get Lemma 12.1.
In the graded algebra gr Y ∗ ( gl N ) , for any r, s > we have [ e T ( − r ) ij , e T ( − s ) kl ] = δ kj e T ( − r − s ) il − δ il e T ( − r − s ) kj . In Section 5 we equipped the algebra Y( gl N ) with an ascending filtration, suchthat the corresponding graded algebra gr Y( gl N ) is commutative. Its subspace ofall elements of degree s will be denoted by gr s Y( gl N ). Keeping to the notation ofSection 5, for any s > b T ( s ) ij be the image of the generator T ( s ) ij in gr s Y( gl N ).We can define a bilinear pairing(12.1) h , i : gr Y( gl N ) × gr Y ∗ ( gl N ) → C by making its value(12.2) (cid:10) b T ( r ) i j . . . b T ( r m ) i m j m , e T ( − s ) k l . . . e T ( − s n ) k n l n (cid:11) equal to (11.4) if r + . . . + r m = s + . . . + s n and by making it equal to zero otherwise.Here r , . . . , r m , s , . . . , s n > m, n > i , j , . . . , i m , j m and k , l , . . . , k n , l n may be arbitrary. This definition is self-consistent. Namely, if(12.3) r + · · · + r m = s + . . . + s n = s for some s >
1, then by Lemma 11.2 we have (cid:10) T ( r ) i j . . . T ( r m ) i m j m + X, T ( − s ) k l . . . T ( − s n ) k n l n + Y (cid:11) = (cid:10) T ( r ) i j . . . T ( r m ) i m j m , T ( − s ) k l . . . T ( − s n ) k n l n (cid:11) for any X ∈ Y( gl N ) and Y ∈ Y ∗ ( gl N ) of degrees respectively less and more than s . Proposition 12.2.
For any index s > , the restriction of the pairing (12.1) to gr s Y( gl N ) × gr s Y ∗ ( gl N ) is non-degenerate.Proof. Fix an integer s >
0. In each of two vector spaces gr s Y( gl N ) and gr s Y ∗ ( gl N )we will choose a basis so that the matrix of the bilinear pairing (12.1) relative tothese bases is lower triangular, with non-zero diagonal entries. In particular, wewill prove that these two vector spaces are of the same dimension.Let r , . . . , r m and s , . . . , s n be non-increasing sequences of positive integerssatisfying (12.3). In other words, these two sequences are partitions of s . We will OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 25 equip the set of all partitions of s with the inverse lexicographical ordering. In thisordering, the sequence r , . . . , r m precedes the sequence s , . . . , s n if for some c > r m = s n , r m − = s n − , . . . , r m − c +1 = s n − c +1 while r m − c < s n − c . Suppose that s , . . . , s d > s d +1 , . . . , s n = 1 for d >
0. Unlike in the proofof Lemma 11.2, now we do not exclude the case d = n . Take the coefficient at(12.4) u − r . . . u − r m m v s − . . . v s n − n in the expansion of the product (11.7) as a series in u − , . . . , u − m , v . . . , v n . Thiscoefficient is an element of the algebra (End C N ) ⊗ ( m + n ) . If this coefficient is non-zero, then equality r + · · · + r m = s + . . . + s n implies that each of the indices r , . . . , r m in (12.4) is a sum of some of the indices s , . . . , s n . Moreover, then each of the indices s , . . . , s n appears in these sums onlyonce. If a sequence r , . . . , r m obtained by this summation precedes the sequence s , . . . , s n in the inverse lexicographical ordering, then the two sequences mustcoincide. That is, m = n and r a = s a for every index a = 1 , . . . , m .For r = 1 , , . . . denote by S r the segment of the sequence 1 , . . . , m consisting ofall indices a such that s a = r . If the sequences r , . . . , r m and s , . . . , s n coincide,then the coefficient at (12.4) in the expansion of the product (11.7) equals( − m Y r > (cid:16) X p Y a ∈S r P a,p ( a )+ m (cid:17) where p runs through the set of all permutations of the sequence S r . Note that inthe products over r and a above, all the factors pairwise commute.The graded algebra gr Y( gl N ) is free commutative with the generators b T ( r ) ij where r > s Y( gl N ) consistingof the monomials(12.5) b T ( r ) i j . . . b T ( r m ) i m j m . The ordering of factors in (12.5) is irrelevant, let us order them in any way suchthat r > · · · > r m . Choose any linear ordering of these basis vectors, subordinateto the inverse lexicographical ordering of the corresponding sequences r , . . . , r m .The above arguments imply, that for any two basis elements, b T ( r ) i j . . . b T ( r m ) i m j m and b T ( s ) k l . . . b T ( s m ) k m l m such that the the sequence r , . . . , r m precedes the sequence s , . . . , s n , the pairingvalue (12.2) is non-zero only if m = n and for every index a = 1 , . . . , m we have r a = s a and i a = l a , j a = k a . Then the value (12.2) equals ( − m g ! h ! . . . where g, h, . . . are the multiplicities inthe sequence of the triples ( r , i , j ) , . . . , ( r m , i m , j m ) . Therefore the monomials(12.6) e T ( − r ) j i . . . e T ( − r m ) j m i m in gr s Y ∗ ( gl N ) corresponding to the basis elements (12.5) of vector space gr s Y( gl N ) ,are linearly independent. These monomials also span the vector space gr s Y ∗ ( gl N ) .The latter result follows from Lemma 12.1 by using induction on m . Hence thesemonomials form a basis in gr s Y ∗ ( gl N ) . The matrix of the pairing (12.1) relativeto the two bases is then lower triangular, with non-zero diagonal entries. (cid:3) The graded algebra gr Y ∗ ( gl N ) inherits from Y ∗ ( gl N ) the bialgebra structure.Namely, using (10.8), for any r > (cid:0) e T ( − r ) ij (cid:1) = e T ( − r ) ij ⊗ ⊗ e T ( − r ) ij and ε (cid:0) e T ( − r ) ij (cid:1) = 0 . Although the antipode S is defined only on the completion Y ◦ ( gl N ) of Y ∗ ( gl N ), itstill induces a well-defined anipodal map on the graded algebra gr Y ∗ ( gl N ),(12.8) S : e T ( − r ) ij
7→ − e T ( − r ) ij . Hence gr Y ∗ ( gl N ) becomes a Hopf algebra.Now consider the subalgebra z gl N [ z ] ∼ = gl N ⊗ ( z C [ z ]) in the polynomial currentLie algebra gl N [ z ]. The next proposition indicates the difference between the gradedalgebras gr Y( gl N ) and gr Y ∗ ( gl N ), cf. Proposition 5.1. Proposition 12.3.
The Hopf algebra gr Y ∗ ( gl N ) is isomorphic to the universalenveloping algebra U( z gl N [ z ]) .Proof. Lemma 12.1 implies that the assignment E ij z r e T ( − r ) ij for r > z gl N [ z ]) → gr Y ∗ ( gl N ) . The kernel of this homomorphism is trivial, because the monomials (12.6) in e T ( − r ) ij corresponding to basis elements (12.5) of the free commutative algebra gr Y( gl N )form a basis in gr Y ∗ ( gl N ) . This was shown in the proof of Proposition 12.2. Bycomparing the definitions (12.7),(12.8) with (5.4),(5.5) we complete the proof. (cid:3) We state the main property of the pairing h , i as the following theorem. Theorem 12.4.
The map (11.1) is a non-degenerate bialgebra pairing.Proof.
By Lemma 11.2 and Proposition 12.2 the pairing h , i is non-degenerate.Let us show that under the pairing (11.1), the multiplication and comultiplicationon Y( gl N ) become dual respectively to the comultiplication and multiplication onY ∗ ( gl N ). We have to prove that(12.10) h X, Z W i = h ∆( X ) , Z ⊗ W i and h X Y, Z i = h X ⊗ Y, ∆( Z ) i for any elements X, Y ∈ Y( gl N ) and Z, W ∈ Y ∗ ( gl N ). Here we use the convention h X ⊗ Y, Z ⊗ W i = h X, Z i h
Y, W i . For instance, let us prove the first equality in (12.10). To this end it suffices tosubstitute the series T i j ( u ) . . . T i m j m ( u m ) and T ∗ k l ( v ) . . . T ∗ k d l d ( v d ) , T ∗ k d +1 l d +1 ( v d +1 ) . . . T ∗ k n l n ( v n )for X and Z, W respectively. Here 0 d n . If d = 0 or d = n , then we substitute1 respectively for Z or for W . After these substitutions, we will have to prove that(12.11) (cid:10) T i j ( u ) . . . T i m j m ( u m ) , T ∗ k l ( v ) . . . T ∗ k n l n ( v n ) (cid:11) equals the sum N X h ,...,h m =1 (cid:10) T i h ( u ) . . . T i m h m ( u m ) , T ∗ k l ( v ) . . . T ∗ k d l d ( v n ) (cid:11) × (12.12) (cid:10) T h j ( u ) . . . T h m j m ( u m ) , T ∗ k d +1 l d +1 ( v ) . . . T ∗ k n l n ( v n ) (cid:11) . OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 27 To prove the latter equality, let us multiply (12.11) and (12.12) by the element e i j ⊗ . . . ⊗ e i m j m ⊗ e k l ⊗ . . . ⊗ e k n l n ∈ (End C N ) ⊗ ( m + n ) , taking the sum over the indices i , j , . . . , i m , j m and k , l , . . . , k n , l n . In this way,from (12.11) we obtain the product → Y a m → Y b n R a,b + m ( u a − v b )due to the definition (11.2). From (12.12) we obtain the product → Y b d → Y a m R a,b + m ( u a − v b ) · → Y d h T ( r ) i j . . . T ( r m ) i m j m , i = 0 if m > . Thus h X , i = ε ( X ) for any element X ∈ Y( gl N ). By setting m = 0 in (11.2) andusing the induction on n or, alternatively, by using Lemma 11.2, we obtain for any s , . . . , s n > h , T ( − s ) k l . . . T ( − s n ) k n l n i = 0 if n > . Thus h , Z i = ε ( Z ) for any element Z ∈ Y ∗ ( gl N ) . Therefore the counit and theunit maps for the bialgebra Y( gl N ) are dual respectively to the unit and the counitmaps for the bialgebra Y ∗ ( gl N ). (cid:3) Due to Theorem 8.3, the vector space Y( gl N ) has a basis parameterized by allmultisets of triples ( r , i , j ) , . . . , ( r m , i m , j m ) where r , . . . , r m ∈ { , , . . . } and i , j , . . . , i m , j m ∈ { , . . . , N } while m = 0 , , , . . . . The corresponding basis vector in Y( gl N ) is the monomial(12.13) T ( r ) i j . . . T ( r m ) i m j m . The ordering of the factors in this monomial can be chosen arbitrarily. Supposethat here r > · · · > r m . Then the sequence r , . . . , r m can be regarded as apartition of r + · · · + r m . Equip the set of all partitions of 0 , , , . . . with thefollowing ordering. If r < s , the partitions of r precede those of s . For any given r , the set of partitions of r is equipped with the inverse lexicographical ordering ;see the proof of Proposition 12.2. Choose any linear ordering of the basis elements(12.13), subordinate to the above described ordering of their sequences r , . . . , r m .The proof of Proposition 12.2 implies that the monomials T ( − r ) j i . . . T ( − r m ) j m i m corresponding to the basis elements (12.13) form a basis of the vector space Y ∗ ( gl N ).The matrix of the pairing (11.1) relative to these two bases is lower triangular withnon-zero diagonal entries; see also Lemma 11.2. Here the basis elements of Y ∗ ( gl N )are linearly ordered as the corresponding basis elements (12.13) of Y( gl N ) . Universal R -matrix Consider the formal completion Y ◦ ( gl N ) of the filtered algebra Y ∗ ( gl N ) definedin Section 10. By Proposition 11.2 the canonical pairing (11.1) extends to a pairing h , i : Y( gl N ) × Y ◦ ( gl N ) → C . Choose any basis X , X , . . . in the vector space Y( gl N ). Proposition 13.1.
The completion Y ◦ ( gl N ) does contain the system of elements X ′ , X ′ , . . . dual to X , X , . . . so that h X r , X ′ s i = δ rs for any r and s .Proof. As we explained at the end of Section 10, one can choose a basis Y , Y , . . . in Y( gl N ) and a basis Y ∗ , Y ∗ , . . . in Y ∗ ( gl N ) so that the filtration degree(13.1) deg Y ∗ s → ∞ when s → ∞ , and so that the matrix of the pairing (11.1) relative to these bases is lower triangularwith non-zero diagonal entries. Let [ g rs ] be its inverse matrix. The formal sums(13.2) Y ′ s = ∞ X r =1 g rs Y ∗ r satisfy the equations h Y r , Y ′ s i = δ rs for all indices r and s . Each of these sums iscontained in Y ◦ ( gl N ) due to (13.1). Moreover, because the the matrix [ g rs ] is alsolower triangular, the property (13.1) implies that(13.3) deg Y ′ s → ∞ when s → ∞ . Now let X , X , . . . be any basis in Y( gl N ). Let [ h rs ] be the coordinate changematrix from the basis Y , Y , . . . so that for any index r we have Y s = ∞ X r =1 h rs X r . This sum must be finite, so that for any fixed index s there are only finitely manynon-zero coefficients h rs . The sums(13.4) X ′ r = ∞ X s =1 h rs Y ′ s satisfy the equations h X r , X ′ s i = δ rs as required. Each of these sums is containedin the completion Y ◦ ( gl N ) due to the property (13.3). (cid:3) Consider an infinite sum of elements of the tensor product Y ◦ ( gl N ) ⊗ Y( gl N )(13.5) R = ∞ X r =1 X ′ r ⊗ X r . This sum does not depend on the choice of the basis X , X , . . . in the vector spaceY( gl N ) in the following sense. Let Y , Y , . . . be the basis in Y( gl N ) used in theproof of Proposition 13.1. Using the formula (13.4) for every r = 1 , , . . . expandthe vectors X ′ , X ′ , . . . in (13.5). Then fix an index s and consider the sum of terms ∞ X r =1 ( h rs Y ′ s ) ⊗ X r OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 29 corresponding to the vector Y ′ s in (13.4). Only finite number of these terms arenon-zero, and their sum is equal to Y ′ s ⊗ Y s . In this sense, the sum in (13.5) equals ∞ X s =1 Y ′ s ⊗ Y s . The infinite sum R is called the universal R-matrix for the Yangian Y( gl N ) .Any element of the vector space Y ◦ ( gl N ) ⊗ Y( gl N ) determines a linear operatoron the vector space Y( gl N ) . If A is the operator corresponding to an element Z ⊗ Y ∈ Y ◦ ( gl N ) ⊗ Y( gl N ), then(13.6) A ( X ) = h X , Z i Y for any X ∈ Y( gl N ) . By the above argument, the series of operators corresponding to (13.5) pointwiseconverges to the identity operator id : X X on the vector space Y( gl N ) . Proposition 13.2.
For the comultiplication on Y( gl N ) and Y ◦ ( gl N ) we have (13.7) (id ⊗ ∆) ( R ) = R R and (∆ ⊗ id) ( R ) = R R where R = ∞ X r =1 X ′ r ⊗ X r ⊗ , R = ∞ X r =1 X ′ r ⊗ ⊗ X r , R = ∞ X r =1 ⊗ X ′ r ⊗ X r . Proof.
Let us prove the first of the two equalities (13.7). This is an equality ofinfinite sums of elements from the tensor product Y ◦ ( gl N ) ⊗ Y( gl N ) ⊗ Y( gl N ). Itmeans the equality of the corresponding operators Y( gl N ) → Y( gl N ) ⊗ Y( gl N ).By applying the linear operator corresponding to the infinite sum (id ⊗ ∆) ( R ) toany fixed element X ∈ Y( gl N ) we get the element ∆( X ). By applying to X theoperator corresponding to R R we obtain the sum ∞ X r,s =1 h X , Y ′ r Y ′ s i Y r ⊗ Y s = ∞ X r,s =1 h ∆( X ) , Y ′ r ⊗ Y ′ s i Y r ⊗ Y s = ∆( X ) . Here we used the first equality in (12.10), and non-degeneracy of the pairing (11.1).The property (13.3) guarantees that in both sums over r and s displayed above,only finite number of summands are non-zero when X is fixed; see Lemma 11.2.We have thus proved the first equality in (13.7). The second equality is deducedfrom the second equality in (12.10) in a similar way. (cid:3) Proposition 13.3.
For the counit maps on Y( gl N ) and Y ◦ ( gl N ) , (id ⊗ ε ) ( R ) = 1 and ( ε ⊗ id) ( R ) = 1 . Proof.
Because ε ( X ) = h X , i for any element X ∈ Y( gl N ) by Theorem 12.4,(id ⊗ ε ) ( R ) = ∞ X s =1 h Y s , i Y ′ s = 1 . Similarly, because ε ( Z ) = h , Z i for any element Z ∈ Y ◦ ( gl N ), we also have( ε ⊗ id) ( R ) = ∞ X s =1 h , Y ′ s i Y s = 1where only finitely many summands are non-zero due to (13.3), see Lemma 11.2. (cid:3) The infinite sum in (13.5) can be also regarded as an element of a completionof the tensor product Y ∗ ( gl N ) ⊗ Y( gl N ). Namely, let us extend the descendingfiltration from the algebra Y ∗ ( gl N ) to the tensor product Y ∗ ( gl N ) ⊗ Y( gl N ) bygiving the degree r to each element of the form T ( − r ) ij ⊗ X where X ∈ Y( gl N ) and r >
1. The element 1 ⊗ X of Y ∗ ( gl N ) ⊗ Y( gl N ) is given the zero degree. Take theformal completion of the algebra Y ∗ ( gl N ) ⊗ Y( gl N ) relative to this filtration. Thiscompletion contains the tensor product Y ◦ ( gl N ) ⊗ Y( gl N ), but does not coincidewith it because the algebra Y( gl N ) is infinite-dimensional.The next corollary shows in particular, that the sum in (13.5) is invertible as anelement of the completion of the algebra Y ∗ ( gl N ) ⊗ Y( gl N ). Corollary 13.4.
For the antipodal maps on Y( gl N ) and Y ◦ ( gl N ) we have (id ⊗ S) ( R ) = R − and (S ⊗ id) ( R ) = R − . Proof.
Regard the first equality in (13.7) as that of the elements of the completionof the algebra Y ∗ ( gl N ) ⊗ Y( gl N ) ⊗ Y( gl N ). On this algebra, the descending filtrationis defined by giving the degree r to each element of the form T ( − r ) ij ⊗ X ⊗ Y where X, Y ∈ Y( gl N ) and r >
1. The element 1 ⊗ X ⊗ Y is then given the degree zero.Let µ : Y( gl N ) ⊗ Y( gl N ) → Y( gl N ) be the map of algebra multiplication, and δ : C → Y( gl N ) be the unit map: δ (1) = 1 . Let us apply the map id ⊗ S ⊗ id, andthen the map id ⊗ µ to to both sides of the first equality in (13.7). At the righthand side we get the element ((id ⊗ S)( R )) · R . At the left hand side we get theelement of the tensor product Y ◦ ( gl N ) ⊗ Y( gl N ),((id ⊗ µ ) (id ⊗ S ⊗ id) (id ⊗ ∆)) ( R ) = ((id ⊗ δ ) (id ⊗ ε )) ( R ) = 1 ⊗ . Here we used the first axiom of antipode from Section 4 in the case A = Y( gl N ),and the first equality of Proposition 13.3. Hence the first equality of Corollary 13.4follows from the first equality in (13.7).Similarly, using the first axiom of antipode in the case A = Y ◦ ( gl N ) and thesecond equality of Proposition 13.3, the second equality of Corollary 13.4 followsfrom the second equality in (13.7). The last equality should be regarded here asthat of the elements of the completion of the algebra Y ∗ ( gl N ) ⊗ Y ∗ ( gl N ) ⊗ Y( gl N ) . On this algebra a descending filtration is defined by giving the degree r + s to anyelement of the form T ( − r ) ij ⊗ T ( − s ) kl ⊗ X .
Then the elements T ( − r ) ij ⊗ ⊗ X and 1 ⊗ T ( − r ) ij ⊗ X are given the degree r , whilethe element 1 ⊗ ⊗ X is given the degree zero. Here X ∈ Y( gl N ) and r, s > (cid:3) Let us now replace the complex parameter c in the definition (6.2) of a covectorrepresentation ρ c of Y( gl N ) by the formal variable v . Then we get a homomorphism(13.8) ρ v : Y( gl N ) → End C N [ v ];it is defined by the assignment T ( u ) R ( u − v ) of formal power series in u − .Similarly, the assignment T ∗ ( v ) R ( u − v ) of formal power series in v definesa homomorphism(13.9) ρ ∗ u : Y ∗ ( gl N ) → End C N [ u − ] . OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 31 To prove the homomorphism property using the matrix form (10.5) of the definingrelations of the algebra Y ∗ ( gl N ), we have to check the equality of rational functionsin the variables u, v and w with the values in the algebra (End C N ) ⊗ , R ( u − v ) R ( u − w ) R ( v − w ) = R ( v − w ) R ( u − w ) R ( u − v ) . This equality follows from (2.6). Here we use the indices 0 , , , , C N ) ⊗ . By comparing the expansions (10.2) and(11.5), we see that(13.10) ρ ∗ u : T ( − r ) ij
7→ − u − r e ji for any r > . Obviously, the homomorphism ρ ∗ u extends to a homomorphismY ◦ ( gl N ) → End C N [[ u − ]] . We shall keep the notation ρ ∗ u for the extended homomorphism. Proposition 13.5.
We have equalities of formal power series in u − and v , ( ρ ∗ u ⊗ id) ( R ) = T ( u ) and (id ⊗ ρ v ) ( R ) = T ∗ ( v ) . Proof.
By the definition (11.2) of the pairing Y( gl N ) ⊗ Y ∗ ( gl N ) → C for any n > T ( u ) ∈ End ( C N ) ⊗ Y( gl N )[[ u − ]] has the property that T ( u ) ⊗ T ∗ ( v ) . . . T ∗ n ( v n ) R ( u − v ) . . . R ,n +1 ( u − v n )under the linear mapid ⊗ β ⊗ id : End C N ⊗ Y( gl N ) ⊗ Y ∗ ( gl N ) ⊗ (End C N ) ⊗ n → (End C N ) ⊗ ( n +1) . Because our pairing is non-degenerate, the same property for the element( ρ ∗ u ⊗ id ) ( R ) = ∞ X s =1 ρ ∗ u ( Y ′ s ) ⊗ Y s will imply the first equality of Proposition 13.5. Note that when s → ∞ , then thedegree in u − of the image ρ ∗ u ( Y ′ s ) tends to infinity due to (13.3) and (13.10). Hencethe above displayed sum over s = 1 , , . . . is contained in End ( C N ) ⊗ Y( gl N )[[ u − ]] .Thus to prove the first equality of Proposition 13.5, we have to show that underthe linear map id ⊗ β ⊗ id , ∞ X s =1 ρ ∗ u ( Y ′ s ) ⊗ Y s ⊗ T ∗ ( v ) . . . T ∗ n ( v n ) R ( u − v ) . . . R ,n +1 ( u − v n ) . Since the system of vectors Y ′ , Y ′ , . . . is dual to the basis Y , Y , . . . of Y( gl N ), thisis equivalent to showing that T ∗ ( v ) . . . T ∗ n ( v n ) R ( u − v ) . . . R ,n +1 ( u − v n )under the linear map ρ ∗ u ⊗ id : Y ∗ ( gl N ) ⊗ (End C N ) ⊗ n → (End C N ) ⊗ ( n +1) [ u − ] . The latter property follows directly from the definition of the homomorphism ρ ∗ u .The proof of the second equality of Proposition 13.5 is similar and is omitted. (cid:3) Corollary 13.6.
We have the equality of formal power series in u − and v , ( ρ ∗ u ⊗ ρ v ) ( R ) = R ( u − v ) . Double Yangian
Let ∆ ′ be the comultiplication on Y ∗ ( gl N ) opposite to the comultiplication ∆defined by (10.7). By definition, the map∆ ′ : Y ∗ ( gl N ) → Y ∗ ( gl N ) ⊗ Y ∗ ( gl N )is the composition of the comultiplication ∆ with the linear operator on the tensorproduct Y ∗ ( gl N ) ⊗ Y ∗ ( gl N ) exchanging the tensor factors.The double Yangian of gl N is defined as an associative unital algebra DY( gl N )over C generated by the elements of Y( gl N ) and Y ∗ ( gl N ) subject to the relations(14.1) R ∆( W ) = ∆ ′ ( W ) R for every W ∈ Y ∗ ( gl N ) . In the rest of this section we will provide a more explicit description of the algebraDY( gl N ) , see Theorem 14.4 below. In Section 15 we will show that the defininghomomorphisms of Y( gl N ) and Y ∗ ( gl N ) to DY( gl N ) are in fact embeddings. At theend of that section we will also provide an equivalent definition of the DY( gl N ) .In (14.1) we have an equality of infinite sums of elements of the tensor productY ◦ ( gl N ) ⊗ DY( gl N ). It means the equality of the corresponding linear operatorsY( gl N ) → DY( gl N ), cf. (13.6). For instance, let us consider the infinite sum R ∆( W ) = ∞ X s =1 ( Y ′ s ⊗ Y s ) ∆( W )at the right hand side of the equality postulated in (14.1). Note that for any fixed X ∈ Y( gl N ) and Z ∈ Y ∗ ( gl N ), only finitely many summands in the infinite sum ∞ X s =1 h X , Y ′ s Z i Y s are non-zero; see Lemma 11.2 and the property (13.3). This observation shows thatthe linear operator Y( gl N ) → DY( gl N ) corresponding to the infinite sum R ∆( W )is well-defined for any element W ∈ Y ∗ ( gl N ).Now take the pair of homomorphisms ρ u and ρ ∗ u where we use the same formalvariable u , see (13.8) and (13.9). Proposition 14.1.
The associative algebra homomorphisms ρ u , ρ ∗ u extend to ahomomorphism DY( gl N ) → End C N [ u, u − ] . Proof.
Using (14.1), for any W ∈ Y ∗ ( gl N ) we have to check the equality(id ⊗ ρ u ) ( R ) (id ⊗ ρ ∗ u ) (∆( W )) = (id ⊗ ρ ∗ u ) (∆ ′ ( W )) (id ⊗ ρ u ) ( R )of formal series in u with coefficients in the algebra Y ◦ ( gl N ) ⊗ End C N . It sufficesto substitute here the series T ∗ ij ( v ) for the element W . Due to the definition (10.7)and to Proposition 13.5, the result of the substitution is the relation N X k =1 T ∗ ( u ) ( T ∗ ik ( v ) ⊗ ρ ∗ u ( T ∗ kj ( v ))) = N X k =1 ( T ∗ kj ( v ) ⊗ ρ ∗ u ( T ∗ ik ( v ))) T ∗ ( u ) . Let us take the tensor products of both sides of the latter relation with the element e ij ∈ End C N , and then sum over i, j = 1 . . . , N . Using the identity e ij = e ik e kj OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 33 we then get the relation N X i,j,k =1 ( T ∗ ( u ) ⊗ T ∗ ik ( v ) ⊗ ρ ∗ u ( T ∗ kj ( v )) ⊗ e ik e kj ) =(14.2) N X i,j,k =1 ( T ∗ kj ( v ) ⊗ ρ ∗ u ( T ∗ ik ( v )) ⊗ e ik e kj ) ( T ∗ ( u ) ⊗ u, v with the coefficients in Y ∗ ( gl N ) ⊗ End C N ⊗ End C N .Note that by the definition of the homomorphism (13.9), N X i,j =1 ρ ∗ u ( T ∗ ij ( v )) ⊗ e ij = R ( u − v ) . Therefore the relation (14.2) can be rewritten as T ∗ ( u ) T ∗ ( v ) (1 ⊗ R ( u − v )) = (1 ⊗ R ( u − v )) T ∗ ( v ) T ∗ ( u ) . But this is just the defining relation for the algebra Y ∗ ( gl N ), see (10.5). (cid:3) Let c be any non-zero complex number. In Proposition 14.1, we can specializethe formal variable u to c . Then we obtain a representation DY( gl N ) → End C N .We call it a covector representation of the algebra DY( gl N ), it extends the covectorrepresentation (6.2) of the algebra Y( gl N ).The vector representation (6.3) of Y( gl N ) can be extended to a representationof DY( gl N ), by mapping T ∗ ( v ) R t ( v − u ). We call it a vector representation ofthe algebra DY( gl N ) and denote it by σ c . Note that then(14.3) σ c : T ( − r ) ij c − r e ij for any r > . The proof that these assignments together with (6.3) define a representation of thealgebra DY( gl N ) is similar to that of Proposition 14.1, and is omitted here.To write down commutation relations in the algebra DY( gl N ), we will use thetensor product End C N ⊗ DY( gl N ) ⊗ End C N . There is a natural embedding of thealgebra End C N ⊗ End C N into this tensor product, such that x ⊗ y x ⊗ ⊗ y for any elements x, y ∈ End C N . In the next proposition, the Yang R -matrix (2.4)is identified with its image relative to this embedding. Proposition 14.2.
In the algebra
End C N ⊗ DY( gl N ) ⊗ End C N [[ u − , v ]] we have (14.4) ( T ( u ) ⊗ R ( u − v ) (1 ⊗ T ∗ ( v )) = (1 ⊗ T ∗ ( v )) R ( u − v ) ( T ( u ) ⊗ . Proof.
Let us substitute T ∗ ij ( v ) for W in the equality in (14.1), and then apply thehomomorphism ρ ∗ u ⊗ id to the resulting equality. Due to the definition (10.7) andto Proposition 13.5, we get an equality of formal power series in u − and v withthe coefficients from End C N ⊗ DY( gl N ), N X k =1 T ( u ) ( ρ ∗ u ( T ∗ ik ( v )) ⊗ T ∗ kj ( v )) = N X k =1 ( ρ ∗ u ( T ∗ kj ( v )) ⊗ T ∗ ik ( v )) T ( u ) . Let us now take the tensor products of both sides of this equality with the element e ij ∈ End C N , and then sum over i, j = 1 . . . , N . Using the identity e ij = e ik e kj we obtain an equality of series with coefficients from End C N ⊗ DY( gl N ) ⊗ End C NN X i,j,k =1 ( T ( u ) ⊗ ρ ∗ u ( T ∗ ik ( v )) ⊗ T ∗ kj ( v ) ⊗ e ik e kj ) =(14.5) N X i,j,k =1 ( ρ ∗ u ( T ∗ kj ( v )) ⊗ T ∗ ik ( v ) ⊗ e ik e kj ) ( T ( u ) ⊗ . By using the definition of ρ ∗ u the equality (14.5) can be rewritten as (14.4). (cid:3) Proposition 14.3.
Relation (14.4) is equivalent to the collection of relations (14.1) .Proof.
By Proposition 14.2 the relation (14.4) follows from (14.1). Let u , . . . , u m be independent variables. Define the homomorphism(14.6) ρ ∗ u ...u m : Y ∗ ( gl N ) → (End C N ) ⊗ m [ u − , . . . , u − m ]as the composition of the m -fold comultiplication Y ∗ ( gl N ) → Y ∗ ( gl N ) ⊗ m and of thetensor product of the homomorphisms (13.9) where u = u , . . . , u m . By using thedescending filtration on Y ∗ ( gl N ) and the surjective homomorphism (12.9) we canprove that when the number m vary, the kernels of all homomorphisms ρ ∗ u ...u m have only zero intersection. The proof is similar that of Proposition 8.1 and isomitted here. It now suffices to derive from (14.4) that for any W ∈ Y ∗ ( gl N )(14.7) ( ρ ∗ u ...u m ⊗ id ) ( R ∆( W )) = ( ρ ∗ u ...u m ⊗ id ) (∆ ′ ( W ) R ) . Here the homomorphism (14.6) is extended to a homomorphismY ◦ ( gl N ) → (End C N ) ⊗ m [[ u − , . . . , u − m ]]and the extension is still denoted by ρ ∗ u ...u m . Using Propositions 13.2 and 13.5, therelation (14.7) can be rewritten as T ( u ) . . . T m ( u m ) ( ρ ∗ u ...u m ⊗ id ) (∆( W ))= ( ρ ∗ u ...u m ⊗ id) (∆ ′ ( W )) T ( u ) . . . T m ( u m ) . It suffices to verify the latter relation for each of the series T ∗ ij ( v ) being substitutedfor the element W . By the definition (10.7), the substitution yields the relation ofthe formal power series in u − , . . . , u − m and v with the coefficients in the algebra(End C N ) ⊗ m ⊗ DY( gl N ), T ( u ) . . . T m ( u m ) × N X k ,...,k m =1 ρ ∗ u ( T ∗ ik ( v )) ⊗ ρ ∗ u ( T ∗ k k ( v )) ⊗ . . . ⊗ ρ ∗ u m ( T ∗ k m − k m ( v )) ⊗ T ∗ k m j ( v ) = N X k ,...,k m =1 ρ ∗ u ( T ∗ k k ( v )) ⊗ . . . ⊗ ρ ∗ u m − ( T ∗ k m − k m ( v )) ⊗ ρ ∗ u m ( T ∗ k m j ( v )) ⊗ T ∗ ik ( v ) × T ( u ) . . . T m ( u m ) . Let us now take the tensor products of both sides of this relation with the element e ij ∈ End C N , and then sum over the indices i, j = 1 . . . , N . By using the identity e ij = e ik e k k . . . e k m − k m e k m j OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 35 in End C N , we arrive at the following relation of series with the coefficients fromthe tensor product (End C N ) ⊗ m ⊗ DY( gl N ) ⊗ End C N :( T ( u ) . . . T m ( u m ) ⊗ R ,m +1 ( u − v ) . . . R m,m +1 ( u m − v ) (1 ⊗ T ∗ ( v ))= (1 ⊗ T ∗ ( v )) R ,m +1 ( u − v ) . . . R m,m +1 ( u m − v ) ( T ( u ) . . . T m ( u m ) ⊗ . Here the subscript m + 1 labels the last tensor factor End C N , which comes afterDY( gl N ). This relation can be proved by using (14.1) repeatedly, i.e. m times. (cid:3) We have now established the following theorem explicitly decribing DY( gl N ) . Theorem 14.4.
The algebra
DY( gl N ) is generated by elements T ( r ) ij , T ( − r ) ij with i, j N and r > subject only to the relations (2.8) , (10.5) and (14.4) . Note that the relation (14.4) is equivalent to the collection of relations( u − v ) [ T ij ( u ) , T ∗ kl ( v )] = N X m =1 (cid:16) δ jk T im ( u ) T ∗ ml ( v ) − δ il T ∗ km ( v ) T ∗ mj ( u ) (cid:17) for all i, j, k, l = 1 , . . . , N . We omit the proof of the equivalence, as it is very similarto the proof of Proposition 2.2. The last displayed relation can be rewritten as[ T ij ( u ) , T ∗ kl ( v )] = ∞ X p =0 N X m =1 u − p − v p (cid:16) δ jk T im ( u ) T ∗ ml ( v ) − δ il T ∗ km ( v ) T mj ( u ) (cid:17) . Expanding here the series in u, v and equating the coefficients at u − r v s − we get[ T ( r ) ij , T ( − s ) kl ] = r X a =max(1 ,r − s +1) (cid:16) δ jk (cid:16) δ a,r − s +1 T ( r − s ) il + N X m =1 T ( a − im T ( r − s − a ) ml (cid:17) − δ il (cid:16) δ a,r − s +1 T ( r − s ) kj + N X m =1 T ( r − s − a ) km T ( a − mj (cid:17)(cid:17) for any indices r, s >
1. Here we keep to the notation T (0) ij = δ ij .We will complete this section with describing a bialgebra structure on DY( gl N ).The algebra DY( gl N ) is generated by its two subalgebras, Y( gl N ) and Y ∗ ( gl N ). Wehave already shown that the assignments (4.1) and (10.7) define comultiplicationson these two subalgebras, while the assignments ε : T ( u ) ε : T ∗ ( v ) ∗ ( gl N ) by its opposite comultiplication ∆ ′ . Proposition 14.5.
The double Yangian
DY( gl N ) is a bialgebra over C with thecomultiplication defined by extending ∆ on Y( gl N ) and ∆ ′ on Y ∗ ( gl N ) , and withthe counit defined by mapping T ( u ) , T ∗ ( v ) .Proof. Using the equivalent form (14.4) of the defining relations (14.1), the proofis similar to that of the proof of Proposition 4.1. Here we omit the details. (cid:3)
Filtration on the double Yangian
In Section 5 we explained that the associative algebra Y( gl N ) can be regarded asa flat deformation of the universal enveloping algebra U( gl N [ z ]). Our explanationwas based on Proposition 5.1. In the present section we establish an analogue ofthat result for the double Yangian DY( gl N ) . In order to do so, let us replace the descending filtration on the algebra Y ∗ ( gl N )by an ascending filtration, such that any generator T ( − r ) ij with r > − r . Relative to this ascending filration on Y ∗ ( gl N ), the subspace of elements ofdegree not more than − r coincides with the subspace of the elements of degree notless than r relative to the descending filtration. Let us now combine the ascendingfiltration on Y ∗ ( gl N ) with the ascending filtration on Y( gl N ) used in Section 8.That is, now introduce an ascending Z -filtration on the algebra DY( gl N ) by settingdeg ′ T ( r ) ij = r − ′ T ( − r ) ij = − r for each index r >
1. Denote by gr ′ DY( gl N ) the corresponding Z -graded algebra.Keeping to the notation of Section 8, for any r > e T ( r ) ij be the image of T ( r ) ij inthe degree r − ′ DY( gl N ) . Since we are now using an ascendingfiltration on Y ∗ ( gl N ) instead of the descending one, for any r > e T ( − r ) ij the image of T ( − r ) ij in the degree − r component of gr ′ DY( gl N ) . So e T ( − r ) ij now formally gets a new meaning, which should not cause any confusion however. Lemma 15.1.
In the graded algebra gr ′ DY( gl N ) for any r, s > we have [ e T ( r ) ij , e T ( − s ) kl ] = δ kj e T ( r − s ) il − δ il e T ( r − s ) kj if r − s > ,δ kj e T ( r − s − il − δ il e T ( r − s − kj if r − s . Proof.
This follows from the relation displayed in Section 14 last. Indeed, relativeto the ascending filtration on DY( gl N ) the commutator at the left hand side of thatrelation has the degree r − s − r, s >
1. For r − s > δ jk T ( r − s ) il − δ il T ( r − s ) kj plus terms of degree not more that r − s − r − s = 0 that sum equals δ jk (cid:0) δ il + T ( − il (cid:1) − δ il (cid:0) δ kj + T ( − kj (cid:1) = δ jk T ( − il − δ il T ( − kj plus terms of degree not more that −
2. Finally, for r − s < δ jk T ( r − s − il − δ il T ( r − s − kj plus terms of degree not more that r − s − (cid:3) The graded algebra gr ′ DY( gl N ) inherits from DY( gl N ) a bialgebra structure,see Proposition 14.5. Moreover gr ′ DY( gl N ) is a Hopf algebra, see the remarks wemade just before Proposition 12.3. Proposition 15.2.
The graded Hopf algebra gr ′ DY( gl N ) is isomorphic to universalenveloping algebra U( gl N [ z, z − ]) .Proof. Consider the subalgebras gr ′ Y( gl N ) and gr ′ Y ∗ ( gl N ) of the graded algebragr ′ DY( gl N ) . We have an isomorphism (5.7) of graded algebras defined by theassignments (5.6). Further, due to Lemma 12.1 a surjective homomorphismU( z − gl N [ z − ]) → gr ′ Y ∗ ( gl N )can be defined by E ij z − r e T ( − r ) ij for r > . Lemma 15.1 ensures that these two homomorphisms extend to a homomorphism(15.1) U( gl N [ z, z − ]) → gr ′ DY( gl N ) . OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 37 This homomorphism is surjective and we will prove that it is injective as well. Ourproof will be similar to the proof of injectivity of the homomorphism (5.7) given atthe end of Section 8. But now we will use Propositions 14.1 and 14.5.Take any finite linear combination C of the products( E i j z s ) . . . ( E i m j m z s m ) ∈ U( gl N [ z, z − ])with certain coefficients C s ...s m i j ...i m j m ∈ C where the indices s , . . . , s m ∈ Z and the number m > i , j , . . . , i m , j m may vary as well. Suppose C = 0 as an element of U( gl N [ z, z − ]) .The algebra U( gl N [ z, z − ]) comes with a natural Z -grading such that for any integer s the generator E ij z s has the degree s . The homomorphism (15.1) preserves thisgrading. Without loss of generality, suppose that the element C is homogeneous ofdegree d with respect to this grading. That is, C s ...s m i j ...i m j m = 0 if s + · · · + s m = d. Now define the element A ∈ DY( gl N ) as the sum X s + ··· + s m = d C s ...s m i j ...i m j m T ( r ) i j . . . T ( r m ) i m j m where for every k = 1 , . . . , m we set r k = s k if s k < r k = s k + 1 if s k > B be the image of A in the d -th component of the graded algebra gr ′ DY( gl N ) .The element B coincides with the image of C under the homomorhism (15.1).For any non-zero complex number c the evaluation representation (6.4) of thealgebra U( gl N [ z ]) can be extended to a representation e σ c of U( gl N [ z, z − ]) so that e σ c : E ij z s c s e ij for any s ∈ Z . Then by (6.3) and (14.3) we have e σ c ( E ij z s ) = ( σ c ( T ( s ) ij ) if s < ,σ c ( T ( s +1) ij ) if s > . Now let c , . . . , c n be any non-zero complex numbers. Let D ∈ (End C N ) ⊗ n bethe image of C under the tensor product of the representations e σ c , . . . , e σ c n of thealgebra U( gl N [ z, z − ]). Denote by σ c ...c n the tensor product of the representations σ c , . . . , σ c n of the algebra DY( gl n ); here we use Proposition 14.5. The image of A ∈ DY( gl N ) under the representation σ c ...c n is a Laurent polynomial in c , . . . , c n .The degree of this polynomial does not exceed d , see (4.4) and (10.8). The sum ofthe terms of degree d of this polynomial equals D , see the proof of Proposition 8.1.For any finite-dimensional Lie algebra a there is an analogue of Lemma 7.1 for a [ z, z − ] instead of a [ z ]. The proof of that analogue is similar to that of Lemma 7.1itself and is omitted here. Using that analogue, we can choose n and c , . . . , c n = 0so that D = 0 . Then deg ′ A = d . Indeed, if we had deg ′ A < d then the degree ofthe Laurent polynomial σ c ...c n ( A ) would be also less then d . This would contradictto the non-vanishing of D . By the definition of the element B ∈ gr ′ Y( gl N ), theequality deg ′ A = d means that B = 0. So the homomorphism (15.1) is injective.Comparing the definitions (5.2),(5.3) and (12.7),(12.8) with general definitions(5.4),(5.5) now completes the proof of the proposition. (cid:3) By applying the Poincar´e–Birkhoff–Witt theorem [2, Section 2.1] to the currentLie algebra gl N [ z, z − ] we now obtain its analogue for the double Yangian DY( gl N ) . Theorem 15.3.
Given any linear ordering of the set of generators T ( r ) ij and T ( − r ) ij with r > , any element of the algebra DY( gl N ) can be uniquely written as a linearcombination of ordered monomials in these generators. Corollary 15.4.
The defining homomorphisms of the algebras Y( gl N ) and Y ∗ ( gl N ) to DY( gl N ) are embeddings. We will now use our ascending filtration on DY( gl N ) to show that in the initialdefinition of this algebra, the relations (14.1) can be replaced by the relations(15.2) ∆( X ) R = R ∆ ′ ( X ) for every X ∈ Y( gl N ) . Here ∆ ′ is the comultiplication on Y( gl N ) opposite to (4.1). The infinite sums atboth sides of the relations (15.2) can be regarded as elements of the tensor productof Y( gl N ) and of the completion of DY( gl N ) relative to our ascending filtration. Thecompletion of Y ∗ ( gl N ) as a subalgebra of DY( gl N ) then coincides with Y ◦ ( gl N ) . Proposition 15.5.
Relations (15.2) in the algebra
DY( gl N ) are equivalent to (14.1) .Proof. Let Y , Y , . . . be the basis of Y( gl N ) from the proof of Proposition 13.1. Let Y p Y q = ∞ X r =1 a rpq Y r and ∆( Y r ) = ∞ X p,q =1 b rpq Y p ⊗ Y q so that a rpq , b rpq ∈ C are the structure constants of the bialgebra Y( gl N ) relative tothis basis. Since the system of vectors Y ′ , Y ′ , . . . of Y ◦ ( gl N ) is dual to the system Y , Y , . . . relative to the bialgebra pairing (11.1), we also have the equalities Y ′ p Y ′ q = ∞ X r =1 b rpq Y ′ r and ∆( Y ′ r ) = ∞ X p,q =1 a rpq Y ′ p ⊗ Y ′ q . Here we extend the comultiplication ∆ on Y ∗ ( gl N ) to Y ◦ ( gl N ) as we did just afterstating Proposition 10.1.It suffices to take X = Y r with r = 1 , , . . . in the relations (15.2). Hence we get ∞ X p,q,s =1 b rpq ( Y p Y ′ s ) ⊗ ( Y q Y s ) = ∞ X p,q,s =1 b rpq ( Y ′ s Y q ) ⊗ ( Y s Y p )or ∞ X p,q,s,t =1 a tqs b rpq ( Y p Y ′ s ) ⊗ Y t = ∞ X p,q,s,t =1 a tsp b rpq ( Y ′ s Y q ) ⊗ Y t . So the relations (15.2) are equivalent to the relations in our completion of DY( gl N )(15.3) ∞ X p,q,s =1 a tqs b rpq Y p Y ′ s = ∞ X p,q,s =1 a tsp b rpq Y ′ s Y q where r , t = 1 , , . . . . The vectors Y ′ , Y ′ , . . . have been determined by (13.2) using a basis Y ∗ , Y ∗ , . . . of Y ∗ ( gl N ) . We also have the equalities(15.4) Y ∗ s = ∞ X r =1 f rs Y ′ r OUBLE YANGIAN AND THE UNIVERSAL R -MATRIX 39 where f rs = h Y r , Y ∗ s i . The matrix [ g rs ] used in (13.2) is inverse to [ f rs ] . Due to(13.2) and (15.4) we can replace W ∈ Y ∗ ( gl N ) by Y ′ t ∈ Y ◦ ( gl N ) with t = 1 , , . . . in the relations (14.1). In this way we get ∞ X p,q,s =1 a tpq ( Y ′ s Y ′ p ) ⊗ ( Y s Y ′ q ) = ∞ X p,q,s =1 a tpq ( Y ′ q Y ′ s ) ⊗ ( Y ′ p Y s )or ∞ X p,q,r,s =1 a tpq b rsp Y ′ r ⊗ ( Y s Y ′ q ) = ∞ X p,q,r,s =1 a tpq b rqs Y ′ r ⊗ ( Y ′ p Y s ) . So the relations (14.1) are equivalent to the relations in our completion of DY( gl N ) ∞ X p,q,s =1 a tpq b rsp Y s Y ′ q = ∞ X p,q,s =1 a tpq b rqs Y ′ p Y s where r , t = 1 , , . . . . By cyclically permuting the summation indices in these relations we get (15.3). (cid:3)
Corollary 15.6.
The coefficients of the series Z ( u ) lie in the centre of DY( gl N ) .Proof. The coefficients of Z ( u ) lie in the centre of Y( gl N ) by Lemma 3.3. To provethat they commute with the elements of Y ∗ ( gl N ) as a subalgebra of DY( gl N ) let ussubstitute the series Z ( u ) for X ∈ Y( gl N ) in (15.2). Due to Proposition 4.2 we get ∞ X s =1 ( Z ( u ) Y ′ s ) ⊗ ( Z ( u ) Y s ) = ∞ X s =1 ( Y ′ s Z ( u )) ⊗ ( Y s Z ( u )) . As the coefficients of Z ( u ) are central in Y( gl N ) , dividing this by 1 ⊗ Z ( u ) yields ∞ X s =1 ( Z ( u ) Y ′ s ) ⊗ Y s = ∞ X s =1 ( Y ′ s Z ( u )) ⊗ Y s . It follows that the coefficients of Z ( u ) commute with every Y ′ s in our completion ofthe algebra DY( gl N ) . By using the relations (15.4) we now get the corollary. (cid:3) Now consider the series Z ◦ ( v ) appearing in Lemma 10.2. Arguing as in the proofof the Corollary 15.6, but using the relations (14.1) and Proposition 10.4 instead ofthe relations (15.2) and Proposition 4.2, we can show that the coefficients of Z ◦ ( v )belong to the centre of our completion of the algebra DY( gl N ) . However, in generalthese coefficients do not belong to the algebra DY( gl N ) itself, see Section 10 again.Our completion of the algebra DY( gl N ) can also be used to rewrite the relations(10.5) and (14.4) similarly to (2.8). Take the element T ♮ ( v ) inverse to T ∗ ( v ) . In thenotation analogous to (10.4) the equality (10.5) of series in u and v with coefficientsin Y ∗ ( gl N ) ⊗ (End C N ) ⊗ can be then rewritten as the equality R ( u − v ) T ♮ ( u ) T ♮ ( v ) = T ♮ ( v ) T ♮ ( u ) R ( u − v )of series with coefficients in Y ◦ ( gl N ) ⊗ (End C N ) ⊗ . The (14.4) can be rewritten as R ( u − v ) ( T ( u ) ⊗ ⊗ T ♮ ( v )) = (1 ⊗ T ♮ ( v )) ( T ( u ) ⊗ R ( u − v ) . References [1] D. Bernard and A. LeClair,
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