On Yangian and Long Representations of the Centrally Extended su(2|2) Superalgebra

The centrally extended su(2|2) superalgebra is an asymptotic symmetry of the light-cone string sigma model on AdS5 x S5. We consider an evaluation representation of the conventional Yangian built over a particular 16-dimensional long representation of the centrally extended su(2|2). Interestingly, we find that S-matrices compatible with this evaluation representation do not exist. On the other hand, by requiring centrally extended su(2|2) invariance and explicitly solving the Yang-Baxter equation, we find a scattering matrix for long-short representations of the Lie superalgebra. We notice that this S-matrix is invariant under a different representation of non-evaluation type, induced from the tensor product of short representations. Our findings concern the conventional Yangian only, and are not applied to possible algebraic extensions of the latter.


Introduction
An important progress towards the solution of the finite-size spectral problem of the planar AdS/CFT system has been recently made. On the one hand, a generalized Lüscher approach for treating the leading wrapping effects was developed [1][2][3][4][5][6][7][8][9][10][11][12][13] and successfully confronted against direct field-theoretic computations [14,15]. On the other hand, the Thermodynamic Bethe Ansatz based on the mirror theory [16] was advanced as a tool to capture the exact string spectrum in both the 't Hooft coupling constant λ and the size L of the system [17][18][19][20][21][22][23][24][25][26][27][28][29][30]. In many respects the success of this research is based on the existence of an asymptotic symmetry, which consists of (two copies of) the su(2|2) superalgebra [31][32][33] centrally extended by two central charges. In particular, this superalgebra and the associated Yangian [34] have been used to explicitly determine the S-matrices describing scattering of fundamental and bound-state particles of the light-cone string sigma model [35,36], which is important for setting up the Thermodynamic Bethe Ansatz approach.
Albeit nice, some of the recent developments were based on certain assumptions and clever guesses, which provides us with further motivation to better understand the representation theory of the centrally extended su(2|2) superalgebra, as well as its implications. So far mainly short (atypical) representations have been the subject of interest in the context of the string sigma model. This is because these representations correspond to bound states of fundamental particles [37] and, together with the latter, they constitute the asymptotic spectrum of the sigma model. On the other hand, long (typical) representations naturally enter in the construction of the large L asymptotic solution of the TBA equations via the so-called Y-functions [21]. It is therefore interesting to look for at the scattering theory involving long representations. Another independent aspect where long representations should play a role concerns the issue of the universal R-matrix. If such a quantity exists as an abstract element in A ⊗ A , where A is a Hopf algebra, then it can be evaluated in any two representation of A . In the case of (the Yangian of) centrally-extended su(2|2), these must include the long ones. Therefore, there should exist a concrete (matrix) realization for an intertwiner of symmetry generators in the tensor product of the corresponding (long) representations (the S-matrix).
In this paper, we will make a first step towards understanding the scattering problem involving long representations of the centrally extended su(2|2). We start with building such long representations by applying an outer sl (2) automorphism to the representations of the unextended su(2|2) superalgebra [38]. These representations, in turn, can be obtained from those constructed for gl(2|2) by Gould and Zhang [39], see also [40,41]. They are parameterized by a continuous parameter q ∈ C, which is the value of the unique central charge (the Hamiltonian) in a given representation. An outer sl(2) automorphism acting on su(2|2) can be used to generate two extra central charges, which depend on additional parameters P and g. Here P is identified with the (generically complex) particle momentum, while g with the coupling constant. We will be interested in the lowest (16-dimensional) long representation, for which we will obtain an explicit realization in terms of 16 × 16 matrices depending on q, P and g. As usual, special values of q correspond to the shortening conditions. In particular, q = 1 corresponds to an indecomposable formed out of two short 8-dimensional representations.
Given an explicit realization of the long 16-dimensional representation, we construct the corresponding evaluation representation for the Yangian introduced by Beisert [34].
The defining relations are given in Appendices 5.2 and 5.3. We will refer to this Yangian, exclusively built upon the centrally extended su(2|2) superalgebra, as the conventional Yangian. Whenever the term 'Yangian' will be used throughout the paper, it will always be understood as conventional, even if we will not mention it explicitly.
We will then try to find an S-matrix which scatters the long evaluation with the long evaluation or the long evaluation with a short four-dimensional representation. Namely, we try to find an S-matrix which acts as the following intertwiner: where J is a generator of centrally extended su(2|2), J is the corresponding Yangian generator in the evaluation representation, and ∆ and ∆ op are the coproduct and its opposite (see section 2 for the precise definitions). We recall that this construction proved to work well for the fundamental or bound-state, i.e. short, representations, and it lead to the complete determination of the corresponding bound state scattering matrix [35,42]. However, where one of the representations involved is long evaluation, we find that the S-matrix satisfying the invariance conditions above does not exist.
The origin of this problem can be clearly seen in Drinfeld's second realization [43] where it can be traced back to non-co-commutativity of the higher Yangian central charges C n with n ≥ 2. Co-commutativity means that the coproduct map coincides with its opposite, namely ∆ = ∆ op . When a coproduct will be called co-commutative without further specification, it means it is such when it acts on all the generators of the algebra in question. We will often use the terminology co-commutativity of the generator X as a shortcut for co-commutativity of the coproduct when acting on generator X, namely ∆(X) = ∆ op (X). The term quasi-co-commutative referred to a coproduct means that the coproduct map is equal to its opposite up to conjugation with an invertible element S, namely ∆ op S = S ∆. This fact can happen in some representations, and not in others. If it happens in all representations, and the element S can be determined in a representationindependent way as an abstract object, we will call it the universal R-matrix. If only some representations allow the coproduct to be related to its opposite by conjugation with a invertible matrix S, we will say that those representations admit an S-matrix, but there is no universal R-matrix.
Since the coproducts of the Yangian central charges only involve central elements (see formula (5.8)), co-commutativity of the central charges in a specific representation is a necessary condition for the existence of an S-matrix in that representation. In fact, one from which invertibility of S implies ∆ op (C n ) = ∆(C n ). Finding at least one representation of the Yangian where this necessary condition is not satisfied 2 implies that the corresponding universal R-matrix does not exist.
Even if the Yangian evaluation representation does not admit an S-matrix, one can still look for centrally extended su(2|2)-invariant solutions of the Yang-Baxter equation.
Indeed, we find two such solutions. It is important to understand if there is still some extended symmetry they correspond to. To answer this question, we recall that, generically, a product of two short representations gives an irreducible long representation [38]. In particular, Here, V 4d (p) is a fundamental 4-dimensional representation which depends on the particle momentum and the coupling constant. Analogously, V 16d (P, q) is a long 16-dimensional representation described by the momentum P , the coupling constant g and the parameter q. We find an explicit relation between the pairs (p 1 , p 2 ) and (P, q) at fixed g, in particular P = p 1 +p 2 . Obviously, for a given p 1 and p 2 there is a unique long representation. However, any long representation can be written as a tensor product of two short representations in two different ways.
The observed relationship between long and short representations suggests that the Smatrix S LS , which scatters a long representation with a short one, can simply be composed as a product of two S-matrices S 13 and S 23 describing the scattering of the corresponding short representations, i.e.
In this formula, the tensor product of two short representations in the spaces 1 and 2 with momenta p 1 and p 2 gives a long representation (P, q), which scatters with a short representation in the third space with momentum p 3 . We then verify that the two S-matrices we found by solving the Yang-Baxter equations indeed coincide with the product of two "short" S-matrices. The fact that we find two matrices is explained by the double-covering relationship between (p 1 , p 2 ) and (P, q). This finding also shows that the Yangian symmetry can be induced on long representation from the one defined on the short ones, and this tensor product representation automatically admits an S-matrix. It naturally does that in 2 For instance, if one takes the coproduct of C2 in a long ⊗ long representation, and subtracts from it its opposite, one obtains an expression which expands semiclassically as , with x a classical rapidity.
both branches of the double-covering. Importantly, this (double-branched) tensor product representation of the Yangian is not isomorphic to the long evaluation representation we were discussing before, even though the two short representations composing it are the short evaluation representations of the Yangian described in [34,43]. This is also clear from the fact that the long evaluation representation discussed before does not admit an S-matrix.
The existence of two solutions for S LS , corresponding to two Yangian representations induced from short representations, is an unexpected feature which we do not have a good explanation for. Both S-matrices come with the canonical normalization and, therefore, they cannot be related to each other by any multiplicative factor (an extra dressing phase).
They are not related by a similarity transformation either. We note, however, that at the special value q = 1 where the long multiplet becomes reducible, the two matrices S LS become of the form (the block structure refers to the split into the 8-dimensional sub-and factor representations one finds at q = 1, as we will discuss in the paper) where only the scalar coefficient α is different for the two solutions. Here, D corresponds to the factor representation (symmetric), and coincides with (the inverse of) the known symmetric bound-state S-matrix S AB [44]. This is in agreement with the fact that there is a unique bound-state S-matrix.
The paper is organized as follows. In the next section, starting from a construction of long representations, we discuss the Yangian and prove the non-existence of a universal Rmatrix. This no-go theorem applies to the conventional Yangian only, and it may not hold when considering algebraic extensions of the latter 3 . In section 3 we find the "long-short" S-matrix by solving the corresponding Yang-Baxter equation. In section 4 we present an alternative construction of long representations and the associated S-matrix via tensor product of short ones. In appendices 5.1-5.3 we provide several computational details.
Finally, in appendix 5.4 we discuss some aspects of the Hirota equations related to the long representations we construct in the paper. Most of the corresponding discussion should be known to experts, and we include it only for completeness.

Long representations and Yangian
We start with discussing the representation theory of sl(2|2) and its generalization to the centrally extended case. We provide a matrix realization of the simplest 16-dimensional long multiplet. Then we discuss the evaluation representation of the corresponding Yangian algebra based on this long multiplet and show the absence of a universal R-matrix.

Constructing long representations
The paper [39] explicitly constructs all finite-dimensional irreducible representations of gl(2|2) in an oscillator basis. Generators of gl(2|2) are denoted by E ij , with commutation Indices i, j, k, l run from 1 to 4, and the fermionic grading is assigned as d Finite dimensional irreps are labelled by two half-integers j 1 , j 2 = 0, 1 2 , ..., and two complex numbers q and y. These numbers correspond to the values taken by appropriate generators on the highest weight state |ω of the representation, defined by the following conditions: The generator N never appears on the right hand side of the commutation relations, therefore it is defined up to the addition of a central element βI, with β a constant 4 . This also means that we can consistently mod out the generator N , and obtain sl(2|2) as a subalgebra of the original gl(2|2) algebra 5 . In order to construct representations of the centrally-extended su(2|2) Lie superalgebra 6 , we then first mod out N , and subsequently perform an sl(2) rotation by means of the outer automorphism of su(2|2) [38].
As usual for superalgebras, irreps are divided into typical (long), which have generic values of the labels j 1 , j 2 , q, and atypical (short), for which special relations are satisfied by these labels. Short representations occur here for ±q = j 1 − j 2 and ±q = j 1 + j 2 + 1. When these relations are satisfied, the dimension of the representation is smaller than what it would generically be for the same values of j 1 , j 2 , but q arbitrary (that is, the dimension is in these cases smaller than 16(2j 1 + 1)(2j 2 + 1)). One notices also that, when starting from 4 We decided to drop the term βI since it will not affect our discussion. 5 Further modding out of the center I produces the simple Lie superalgebra psl(2|2). Its representations can be understood as that of sl(2|2) for which q = 0. Correspondingly, (2|2) has long irreps of dimension 16(2j1 + 1)(2j2 + 1) with j1 = j2 and short irreps with j1 = j = j2 of dimension 16j(j + 1) + 2. For a discussion of the tensor product decomposition of psl(2|2), see [45]. 6 The reality condition on the algebra will be imposed later, and will not affect the present discussion.
a long irrep and reaching these special values by continuous variation of the parameter q, one generically ends up into a reducible but indecomposable representation.
Second, we consider the simplest long representation of dimension 16. In terms of the gl(2|2) labels introduced above, this is the 16-dimensional long representation characterized by j 1 = j 2 = 0, and arbitrary q. It is instructive to see how it branches under the su(2)⊕su(2) algebra. We denote as [l 1 , l 2 ] the subset of states which furnish a representation of su(2) ⊕ su(2) with angular momentum l 1 w.r.t the first su(2), and l 2 w.r.t the second su(2), respectively. The branching rule is One can straightforwardly verify that the total dimension adds up to 16, since [l 1 , l 2 ] has dimension (2l 1 + 1) × (2l 2 + 1).
For generic values of q, the corresponding long representations have no interpretation in terms of Young tableaux. However, when q is a certain integer, such an interpretation becomes possible. Consider rectangular Young tableaux, with one side made of 2 boxes, and the other side made of arbitrarily many boxes. These are long representations, denoted by (2, s) and (a, 2) according to the length (in boxes) of their sides. Together with the short irreps, denoted accordingly as (1, s) (symmetric) and (a, 1) (antisymmetric), they span all the admissible rectangular representations. In fact, every allowed representation has to have its associated Young tableaux fit into the so-called "fat hook" [49], which has branches of width equal to two boxes. All representations (2, s) (respectively, (a, 2)) with s ≥ 2 (respectively, a ≥ 2) have dimension 7  can be found in [50].
As a first step of our study, we have explicitly constructed the oscillator representation by using the formulas of [39], and derived from it the 16 × 16 matrix realization of the algebra generators. We have done this before acting with the outer automorphism, in such a way that the subsequent sl(2) rotation provides an explicit matrix representation of centrally-extended su(2|2). This explicit realization is reported in appendix 5.1. Below we discuss some of the salient features of this realization.
The way the outer automorphism is implemented is by mapping the gl(2|2) nondiagonal generators into new generators as follows: subject to the constraint Diagonal generators are automatically obtained by commuting positive and negative roots.
In particular, from the explicit matrix realization one obtains the following values of the central charges: (½ is the 16-dimensional identity matrix), satisfying the condition (2.7) When q 2 = 1, this becomes a shortening condition. In fact, for q = 1 the 16-dimensional representation becomes reducible but indecomposable. Its subrepresentation [45] is a short anti-symmetric 8-dimensional representation. Formula (2.7) above, however, tells us that we can conveniently think of q as a generalized bound state number, since for short representations 2q would be replaced by the bound state number M in the analogous formula for the central charges. This is particularly useful, since it allows us to parameterize the labels a, b, c, d in terms of the familiar bound state variables 8 x ± , just replacing the bound state number M by 2q. The explicit parameterization is given by and As in the case of short representations, there exist a uniformizing torus with variable z and periods depending on q [51] . The choice (2.9) for η is historically preferred in the string theory analysis [16,35,44,52], and will actually ensure our S-matrix to be symmetric.
Finally, we point out that positive and negative values of q correspond to positive and negative energy representations, respectively.

Hopf algebra and Yangian
Having in mind the derivation of an S-matrix in the above described long representation, we equip the symmetry algebra with the deformed Hopf-algebra coproduct 8 [53,54] where J is any generator of centrally-extended su(2|2), [[J]] = 0 for the bosonic su(2)⊕su (2) generators and for the energy generator H, [[J]] = 1 (resp., −1) for the Q (resp., G) supercharges, and [[J]] = 2 (resp., −2) for the central charge C (resp., C † ). The value of U is determined by the consistency requirement that the coproduct is co-commutative on the center, which is a necessary condition for the existence of an S-matrix S satisfying 9 This produces the algebraic condition for some representation-independent constant κ. With our choice of parametrization (2.8), κ gets re-expressed via the coupling constant g as κ = 2 ig , and we obtain the familiar relation (2.14) The advantage of the choice (2.8) is that the above relations are valid as they stand both for long and short representations. This will be particularly useful, since we plan to project 9 S acts as S : V1 ⊗ V2 → V1 ⊗ V2 on representation modules.
the coproduct (2.11) into a long representation in the first space, and a short one in the second space. This is so because we will be primarily interested in the S-matrix scattering long representations against short ones.
In order to have a complete realization of the Hopf algebra, one needs to remember the antipode map S [44,51,54], and specify the charge conjugation matrix C that implements the antipode in the long representations 10 . One has in particular where J is the antiparticle representation associated to the representation we choose on the l.h.s. of (2.15). One finds that the charge conjugation matrix in the 16-dimensional long representation is given by The blocks in (2.16) refer to the branching rule (2.3), with the ordering of states given in section 3.1. The antiparticle representation J is still defined by sending p → −p (together with changing sign to the eigenvalue of the energy generator H), which means exactly as in the case of short representations. Once again, on the uniformizing torus (cf. comment to (2.8)), applying the particle to anti-particle transformation four times gives the identity, which corresponds to the Z 4 graded Lie-algebra structure of centrally extended su(2|2).
The next step is to study the Yangian in this representation. One can prove that the defining commutation relations of Drinfeld's first realization of the Yangian given in [34] (see appendix 5.2) are satisfied (by the generators and their coproducts) if we assume the evaluation representation 11 where the spectral parameter u assumes the familiar form The Yangian coproducts are given by the same formulas used in [35], and the above value of u is determined by requiring co-commutativity of the Yangian central charges C, C † .
We report the details in appendix 5.2 for convenience of the reader.
Drinfeld's second realization is also obtained by applying a similar (Drinfeld's) map as in [43] 12 . This ensures the fulfilment of the Serre relations (see also [55]). All defining relations in [43] are satisfied (see appendix 5.3), although the representation one obtains after Drinfeld map is not any longer of a simple evaluation-type, but more complicated. In fact, level-n simple roots J n are not obtained from level-zero ones via multiplication by a (possibly shifted) spectral parameter to the power n. Nevertheless, the representation we obtain for Drinfeld's second realization of the Yangian is consistent, and the coproducts obtained after Drinfeld's map respect all commutation and Serre relations 13 . We give details of this realization in appendix 5.3.
However, surprisingly, it turns out that the Yangian in this representation, both for coproducts projected into long ⊗ short and for long ⊗ long representations, does not admit an S-matrix. This is easily seen by considering the Yangian central charges C n , C † n . While for n = 0, 1, their coproducts are co-commutative, this is not so for n ≥ 2. Only for the special case q 2 = 1 the Yangian central charges appear to be co-commutative also for n = 2 and higher 14 . Nevertheless, even for the special case q 2 = 1, the Yangian still does not seem to admit an S-matrix in this representation. One way to see it is by noticing that the when applied to certain combinations of generators and on particular states (for instance, of highest weight w.r.t. to the su(2) ⊕ su(2) splitting (2.3)), leads to a contradiction when 11 We use the conventions of [35]. 12 We have checked that the map we use in this paper (see appendix 5.3) also works for the fundamental representation equally well, and is, in this sense, universal. This map might be related to the one used in [43] by redefinitions of the generators in the various realizations. 13 Antipode and charge conjugation are also perfectly consistent with Drinfeld's second realization. 14 For these special values of q we actually checked co-commutativity only up to n = 4.
the explicit matrix realization is used. This means that such an S-matrix does not exist for this representation of the Yangian, which also implies that a universal R-matrix for the Yangian [34] does not exist.
However, as we will discuss in section 4, a different Yangian representation, for which an S-matrix does indeed exists, can be induced on the space of long representations. This Yangian representation is obtained via the decomposition of long representations into short ones, and is therefore built upon the Yangian representations that have been already built on short representations. This induced representation is quite different from the one described above (cf. (2.18)), and, in particular, it is not related to (2.18) via any similarity transformation combined with redefinition of the spectral parameters.
A remark is in order. In principle, it should be possible to deduce non-co-commutativity of the higher central charges directly from the corresponding formulas for the coproducts written in terms of algebra generators, without referring to a specific representation. These formulas should also imply that the non-co-commutative part must disappear for representations which satisfy the shortening conditions. However, the abstract formulation of the coproducts is quite cumbersome, and we find it more illuminating to exhibit a concrete representation for which the higher central charges show non-co-commutativity, cf. footnote 2.

The Long-Short S-matrix
Let us start by deriving all S-matrices satisfying the relation (2.12) exclusively for the (level-zero) algebra generators. Such an S-matrix describes the scattering of an excitation in the long representation with momentum P and parameter q, against a fundamental particle with momentum p: This S-matrix should relate the Hopf algebra structure to the opposite Hopf algebra one.
This means that S should satisfy (2.12).

Kinematic Structure
It is useful to apply a procedure similar to the one performed in [35], this time without the explicit help of Yangian symmetry. We again begin by using the su(2) ⊕ su (2) invariance (with trivial coproduct) to divide the space V 16d (P, q) ⊗ V 4d (p) into blocks with definite su(2) L ⊕ su(2) R weights (a; b). Let us denote the basis-vectors of the long representation by f i and the basis for the short representation by e i , respectively. They correspond to 16-dimensional (resp., 4-dimensional) vectors with all zeroes, except in position i, where there is a 1. The su(2) L ⊕ su(2) R weights are explicitly given in the following tables: From these tables it is now easy to read off the blocks corresponding to weights (a; b).
Explicitly, we find four one-dimensional blocks corresponding to vectors that have maximum/minumum weight. These subspaces are related in the following way: and Both sets of subspaces are again related via the su(2) ⊕ su(2) generators as is indicated in Finally, there are four nine-dimensional blocks Because of the relations between the different subspaces one only has to find the action of the S-matrix on the following subspaces Let us conveniently denote the vectors in these spaces by The final step consists in introducing a (opposite) coproduct basis that allows for a quick derivation of the S-matrix. It turns out that we must use as building blocks both | 3 2 ; 0 and |0; 3 2 . We find for the aforementioned three-dimensional subspaces For the relevant nine-dimensional spaces we find where i = 1, 2, 3.

S-Matrix
From the coproduct basis it is easily seen that the S-matrix will be fixed upon specifying its action on | 3 2 ; 0 and |0; 3 2 . Since these vectors both form a one-dimensional block, they are mapped onto themselves by the S-matrix. We will normalize the S-matrix as follows: Let us start considering the action of the S-matrix on the three-dimensional block V (1; 1 2 ) . We first define q ± ≡ q ± 1. (3.16) The basis transformation that relates the standard basis to the coproduct basis and the opposite coproduct basis can be written in terms of the following matrix: More precisely one finds that the basis transformations Λ 3d and Λ op 3d are given by Λ 3d = Λ 3d,tot (1, 2, 1), Λ op 3d = Λ 3d,tot (3, 4, X ). (3.18) We use the coefficients (5.9,5.10) that explicitly include the braiding factors. By construction, the action of the S-matrix is now given by The other three-dimensional space V ( 1 2 ;1) has transformation matrix One again findsΛ 3d =Λ 3d,tot (1, 2, 1),Λ op 3d =Λ 3d,tot (3, 4, X ). (3.22) This in turn leads to The S-matrices in the other three-dimensional blocks are also described by the above expressions. From Figure 1 we see that they are isomorphic via the su(2) operators. They are related in a straightforward way; the specified maps map basis vectors to basis vectors, e.g.
This leads to For the nine-dimensional blocks we again have two distinct cases. We start with V ( 1 2 ;0) and write Since we have expressed the coproduct basis in terms of the three-dimensional subspaces, we find, in analogy with the previous discussion, The matrices Λ 9d and Λ op 9d are given by On the other hand, we have Here we find The matricesΛ 9d andΛ op 9d are given bȳ Again one can use the su(2) ⊕ su(2) generators to relate these two nine-dimensional Smatrices to the two remaining ones. However, the relation is slightly less straightforward.
In particular we find It is easily checked that this S-matrix is symmetric and is indeed invariant under the full centrally extended su(2|2) algebra. By construction, one can also see that the S-matrix is automatically obtained in the factorized form S = F 21 F −1 12 (Drinfeld twist) [56].

Yang-Baxter Equation
From the previous section we saw that invariance under the symmetry algebra is not enough to fix the S-matrix completely. We still have a free parameter X . This parameter can be fixed by imposing that the S-matrix solves the Yang-Baxter equation: we obtain two quadratic equations for X of the form A + BX (P, p 2 ) + CX (P, p 3 ) + DX (P, p 2 )X (P, p 3 ) = 0, (3.38) where A, B, C, D are functions of P, p 2 , p 3 . It is easily seen that there are two different solutions to these equations. This means that we find two S-matrices, and they are not related by a similarity transformation. The solutions for X appear however rather complicated and we refrain from giving their explicit expressions. It can be checked that both solutions for satisfy the following relations Unitarity: Hermiticity: CPT Invariance: S 12 = S t 12 . Yang-Baxter: This completes our derivation of the S-matrices based on the su(2|2) symmetry.

Long representations via tensor product of short ones
The scope of this section is to prove that the two S-matrices we have just derived, relying only on the Lie superalgebra symmetry and the Yang-Baxter equation, can both be obtained from the tensor product of two short evaluation representations of the Yangian. In principle from the previous analysis we could have found more solutions than those related to short evaluation representations, but we will show that this is not the case.
Consider the tensor product of two short representations labelled by momentum (p 1 , p 2 ), This vector space naturally carries a representation of centrally extended su(2|2) via the (opposite) coproduct, i.e. for any generator J we have It is easily seen by considering the central charges on this space that we are dealing with a long representation. To be precise, we find where the energy E(p) is E(p) 2 = 1 + 4g 2 sin 2 p 2 . (4.4) One has therefore The dispersion relation (4.4) has two branches, corresponding to particles and anti-particles.
Fixing momentum p and a choosing a branch specifies the fundamental representation completely. Then, the tensor product of two such representations is identified with a unique 16-dimensional long representation with momentum P and the central charge q specified above.
Consider now the inverse problem, i.e. suppose we are given a long representation (P, q) and we want to factorize it into the tensor product of two fundamental representations. It is convenient to label the representation space corresponding to particles as V + and the one corresponding to anti-particles as V − . Thus, the carrier space of the long representation can be identified with one of the following four spaces: In the emerging solutions of the factorization problem the momenta p 1 and p 2 can be ordered, and we always assume that the ordering is such that p 1 ≺ p 2 15 . Assuming for simplicity that q is real, we find that to a long representation V (P, q) one can associate two solutions of the factorization problem. For instance, for q positive the two solutions are both associated with the case I, or one of the solutions is from I and the second is from II. Analogous situation takes place for q negative. Thus, any long representation can be written as a tensor product of two different short representations. Actually, this observation 15 The details of the ordering are irrelevant, since its only function is to choose a unique representative between the couple (p1, p2) and its permuted couple (p2, p1).
was reflected earlier in the fact that we found two independent long-short S-matrices that solve the Yang-Baxter equation.
Instead of particle momenta p i and P one can use the corresponding rapidity variables u i and u. The equation for q and the momentum conservation take the form and maps the u-plane on the kinematic region of the string theory [16]. The energy of the long representation is given by In general, given u 1 and u 2 , the variable q will appear as a complicated function q ≡ q(u 1 , u 2 , g). However, there are two special cases, where q is a constant independent of u i and g. Indeed, for u 1 = u ± i g and u 2 = u ∓ i g one gets q = 1. Analogously, for u 1 = −u ± i g and u 2 = −u ∓ i g one finds q = −1. These values of q correspond to the shortening conditions, for which the long multiplet becomes reducible but indecomposable.
Imposing the ordering u 1 ≺ u 2 we get e.g. for q = 1 only one solution. This is an artifact of our parametrization x(u) in eqs.(4.8) and (4.9). As is known, the u-plane covers through the map x(u) only the string region on the z-torus [16]. To find the other solution, one has to change the map x(u) for the one which covers the mirror regions on the z-torus. For q = 1 both solutions are from V + ⊗ V + , which is also the case for q close to one. However, when q deviates from q = 1 sufficiently enough, two solutions can occur in V + ⊗ V + and We further note that one can explicitly find the similarity transformation that relates the long algebra generators to the ones that arise from the coproduct. It is convenient to first express the coefficients a L , b L , c L , d L parameterizing long representations via a 1 , etc. that describe the short representations (again we use the coefficients that already include braiding factors, (5.9,5.10)) In terms of these coefficients we find that the algebra generators are related via a similarity transformation where the coefficients are given by , (4.14) .
The coproduct on three short representation is given by (∆ ⊗ ½)∆. It is easily seen that Thus we see that S 13 S 23 intertwines the coproduct on the tensor product of a long and a short representation. By the above similarity transformation, this means that we can interpret S as being built up out of fundamental S-matrices. The two different choices of short representations that give rise to the long representation then indeed gives two different solutions for S. They exactly coincide with the ones that are found from the Yang-Baxter equation.
As we discussed in section 2.2, the fact that the S-matrix in short representation possesses Yangian symmetries (in evaluation representations) automatically induces, via the above mentioned tensor product procedure, a Yangian representation associated to the long representation. The generators are simply given by ∆ is projected into short ⊗ short Yangian representations, the latter being characterized by the known ('short') spectral parameters u 1 and u 2 (on the first and second factor of the tensor product, respectively). These short spectral parameters are linked to the parameters of the two corresponding short representations as in (2.19). When using the formulas in appendices 5.  Figure 2: The various representations, their q → 1 limits, and the block structure of the corresponding S-matrix in this limit. We denoted with V ZG the representation obtained from [39]. As a consequence of the upper-triangular structure, the bottom-right block of the limiting S-matrices satisfies the YBE by itself.
bound-state Yangian invariance in the evaluation representation. As we said, this unitary representation at q = 1 is not isomorphic to a tensor (or more precisely, to the co-) product of short fundamental representations, and it turns out to furnish a way of extracting the subrepresentation and factor representation from the indecomposable one we are dealing with. We have summarized the situation for convenience in Figure 2.

Explicit Parameterization
We list in this appendix the generators of centrally extended su(2|2) in the long representation. We only report explicitly the simple roots for a distinguished Dynkin diagram, the remainder of the algebra being generated via commutation relations. We present the roots in a unitary representation. To achieve this, we perform a similarity transformation on the generator constructed directly from the oscillator basis of [39], in oder to obtain hermitean matrices. First, the bosonic su(2) ⊕ su(2) roots are given by Next, we show two fermionic roots as an example: Finally, the similarity transformation that relates the unitary representation to the one from [39] is given by where q ± have been defined in (3.16). We notice that this transformation is singular for q 2 = 1, where the representation becomes reducible but indecomposable.

Yangians and Coproducts: Drinfeld's first realization
The double Yangian [57] DY (g) of a (simple) Lie algebra g is a deformation of the universal enveloping algebra U (g[u, u −1 ]) of the loop algebra g[u, u −1 ]. The Yangian is obtained by adding to the Lie algebra a set of partner generators J A n , n ∈ Z satisfying the commutation relations where F AB C are the structure constants of g. The centrally-extended su(2|2) Yangian has the following coproduct 16 [34,53,54]: where U, [[A]] are given in section 2.2.
The evaluation representation we have been discussing in section 2.2 is obtained as [34]. In this representation the coproduct structure is fixed in terms of the 16 Here, = 1 g , and it can be reabsorbed in the definition of the algebra generators, which is the convention we use in the paper. We display in this particular formula just to show how the 'tail' of the Yangian coproduct organizes itself. The terms we omit from (5.7) are also completely determined by the knowledge of the level zero and level one coproducts, in a recursive fashion. coproducts of J, J. As in the case of short representations, it is possible to absorb the factors arising due to the presence of U into a non-local redefinition of the representation labels a i , b i , c i , d i . We will give here the formulas for the Yangian coproducts and for these redefined labels.
We have used in the above formulas the shorthand notation J 1 Y 2 = J ⊗ Y. In case of long representation in space 1 and short representation in space 2 of the tensor product, the labels used in ∆ are given by: Accordingly, the labels used in ∆ op are given by: (5.10) The non-trivial braiding factors are all hidden in the parameters of the four representations involved.

Yangians and Coproducts: Drinfeld's second realization
The second realization of the Yangian [58] is given in terms of Chevalley-Serre type generators and relations. The formulas for the centrally-extended su(2|2) case have been given in [43]. They are expressed in terms of Cartan generators κ i,m and fermionic simple roots ξ ± i,m , i = 1, 2, 3, m = 0, 1, 2, . . . , subject to the following relations: 14) The isomorphism (Drinfeld's map) between the first and the second realization is given as follows: where H i , E i , F i are the Yangian partners of H i , E i , F i in the first realization, and the special elements are given by By knowing level-zero and level-one generators, one can recursively construct all higherlevel generators by repeated use of the relations (5.11). We have performed extensive checks of the consistency of the (long) representation we find after Drinfeld's map with all relations (5.11). The explicit form of these generators is not particularly illuminating and we omit to report it here. The only interesting point is that it is not of a simple evaluation type, but rather more complicated. The Cartan generators at level one, for instance, are not represented by diagonal matrices, still with all relations (5.11) being satisfied.
in Drinfeld's second realization by knowing the coproducts in Drinfeld's first realization (see previous section) and using the homomorphism property ∆(ab) = ∆(a)∆(b). Same consistency we have found for coproducts and other Hopf algebra structures.

A remark on long representations and Hirota equations
The large L asymptotic solution for the Y-system (see the Introduction) is most conveniently written in terms of certain transfer-matrices associated with the underlying symmetry group of the model [59]. In the context of the string sigma model the corresponding asymptotic solution was presented in [21]. In this solution the corresponding Y-functions are re-expressed in terms of suitable T-functions T a,s . The latter must obey the so-called where f ± (u) = f (u± i g ). These equations are formally solved by the Bazhanov-Reshetikhin (BR) determinant formula [60] T a,s (u) = det 1≤i,j≤s T a+i−j,1 (u + i g (s + 1 − i − j)) = = det 1≤i,j≤a T 1,s+i−j (u + i g (a + 1 − i − j)) , (5.18) which expresses all T a,s either in terms of T 1,s or T a,1 . In the large L-limit the T-function T a,s is supposed to coincide with (the eigenvalues of) the transfer matrix evaluated in the rectangular representation (a, s) of the centrally extended sl(2|2). For the case without central extension, this fact has been proved in [61]. Here we will be concerned with centrally-extended sl(2|2). Rather than developing a general theory, we will construct explicitly for one simple example the corresponding transfer matrices, i.e. without appealing to the BR formula, and show that the Hirota equations are indeed satisfied.
Our construction also allows one to better understand the role of long representations giving rise to a generic transfer matrix T L . Namely, long representations for which the central charge q satisfies the shortening condition become reducible but indecomposable.
Using the relationship between long and short representations, we show that the transfer matrix T L admit a factorization into a tensor product of the transfer-matrices corresponding to short representations. We carry out this construction for our simplest 16 All transfer matrix eigenvalues T 1,s has been obtained with the help of the Algebraic Bethe Ansatz technique in [36]. Alternatively, they can be found with the help of the quantum characteristic function [38]. Here, we first need the eigenvalues which correspond to the su(2) sector. They are given by This transfer matrix is associated with the canonically normalized S-matrix which is equal to unity on the su(2) vacuum. We recall that the fundamental representation can be realized on the space of two bosonic variables w 1 and w 2 , and two fermionic variables θ 3 and θ 4 [44]. The su(2) vacuum state is composed of a chain of w 1 's, i.e. (w 1 ) ⊗K I , where K I is the number of excited particles with rapidities v i and kinematic variables x ± i = x v i ± i g . In the formula (5.21) the quantities x ± are the kinematic variables corresponding to the auxiliary bound-state particle with rapidity u: x ± = x u ± i g s . Finally, the quantities λ ± are given by 17 Notice that this can also be interpreted in the light of the discussion at the end of section 4. The transfer matrix is insensitive to a similarity transformation in the auxiliary space, therefore the unitary representation should really give the same result as the tensor (or co-) product of short ones. In fact, in the unitary representation the above splitting in the limit of q → 1 is an obvious consequence of decomposability into the bound-state irreducible components.
On the other hand, the eigenvalue of the transfer matrix on the sl(2) vacuum, i.e. on a fermionic state (θ 3 ) ⊗K I , takes the form [36] T sl (2) (v | u) = d(a, u, K I ) where we include the following normalization factor d(a, u, K I ) = (−1) a The matrix T a,1 in an anti-symmetric irrep is obtained from T sl (2) (v | u) through the replacement T a,1 ≡ T sl(2) | x ± →x ∓ ,x ± i →x ∓ We take the auxiliary space 0 to be the one corresponding to the long representation. This transfer matrix acts on the tensor product It is convenient to identify the long representation V 0 (u, q) as the tensor product of two short V a (u a (u, q)) ⊗ V b (u b (u, q)). Under this identification we have S 0i = S ai S bi and this allows for the determination of the transfer matrix S bi (5.27) in terms of short representations. In the last formula the factorization property of the supertrace operation has been used. Obviously, the right hand side of the (5.27) coincides with the product T 1,1 (u a | v)T 1,1 (u b | v). This factorization happens for any q. On the left hand side we indicate the long representations which for generic q are irreducible and can be written as the tensor product of lower dimensional irreps. They do not have a description in terms of the Young tableaux and for special values of q become reducible but indecomposable. All the representations appearing on the right hand side of the Hirota equations, like, for instance T 16|2 2,2 , have an associated Young tableaux. Concrete transfer matrices in long representations can be obtained by using the Smatrix we have constructed in this paper. When trying to check the above with this concrete realization, however, one has to take into account an extra degree of freedom corresponding to the normalization of the T-functions. This normalization comes on top of the one chosen for the S-matrix, which we fix in the paper to be the canonical normalization.
Although we have not studied the normalization issue in detail, we have checked in several cases that solving for transfer matrices of long representations from some of the Hirota equations and plugging them in the remaining equations leads to the consistency conditions on the transfer matrices of short representations which are indeed satisfied.