Introduction to the nested algebraic Bethe ansatz

We give a detailed description of the nested algebraic Bethe ansatz. We consider integrable models with a $\mathfrak{gl}_3$-invariant $R$-matrix as the basic example, however, we also describe possible generalizations. We give recursions and explicit formulas for the Bethe vectors. We also give a representation for the Bethe vectors in the form of a trace formula.


Introduction
Algebraic Bethe ansatz (ABA) is a part of the Quantum Inverse Scattering Method (QISM), that emerged in the late 70's in the works of the Leningrad School [1,2]. Almost simultaneously with this method, a nested algebraic Bethe ansatz (NABA) was developed in [3][4][5][6]. NABA is a method that allows us to find the spectrum of quantum integrable models describing systems with several types of excitations. It is an algebraic interpretation of the approach proposed in the works [7][8][9]. These notes are based on a series of lectures given by the author at the Les Houches summer school 2018 Integrability in Atomic and Condensed Matter Physics. We introduce the reader to the basic principles of NABA. The presentation follows the classical scheme described in the papers [3][4][5][6]. However, we give much more details and illustrate the general principles with concrete calculations. To simplify the discussion, we will mainly confine ourselves to the case of models, which are described by a gl 3 -invariant R-matrix. A number of statements, however, are formulated for a fairly general case. We also give a number of comments on how one can generalize the results obtained for the gl 3 case to the models with symmetries of higher ranks.
We also describe a method developed in the works [10][11][12]. In contrast to the classical NABA scheme, which allows to construct Bethe vectors recursively, this approach immediately sets explicit formulas for these vectors. We present a proof of the equivalence of this approach to the NABA.
We focus on the mathematical aspects of the NABA, and we do not consider physical applications of the models solvable by this method. Let us mention, however, that these physical applications are very wide. They include condensed matter physics (spin chains, t-J model, Hubbard model), systems of ultracold atoms (multicomponent one-dimensional Fermi and Bose gases), supersymmetric gauge theories, etc.
To conclude this short introduction, we would like to mention that the NABA is a generalization of the ABA and uses basically the same concepts. Some techniques are also borrowed from the ABA. Therefore, to understand the stuff, the reader must possess the basic principles and techniques of the ABA. In addition to the original works mentioned above, they can be found in [13][14][15][16].

Reminder of the algebraic Bethe ansatz
A key equation of the ABA is an RT T -relation [1,2,[13][14][15][16] R(u, v) T (u) ⊗ I I ⊗ T (v) = I ⊗ T (v) T (u) ⊗ I R(u, v). (1.1) Here T (u) is a monodromy matrix T (u) = A(u) B(u) C(u) D(u) , (1.2) whose matrix elements act in some Hilbert space H. The monodromy matrix also acts in the space C 2 which is called an auxiliary space. The R-matrix R(u, v) acts in C 2 ⊗ C 2 . Another commonly used form of equation (1.1) is Here the subscripts show in which of the two auxiliary spaces C 2 the T -matrices act nontrivially. The R-matrix R 12 (u, v) acts in both spaces C 2 .
The RT T -relation immediately yields Choosing one of I k as a Hamiltonian of a quantum model we automatically obtain many (generically, infinitely many) integrals of motion. Thus, we have a chance to build an integrable model.
It is assumed within the framework of the ABA that the Hilbert space H of the model contains a vacuum vector |0 with the following properties:

Possible generalization
A question arises: can we generalize this construction to the case of the N × N monodromy matrix whose auxiliary space would be C N ? Namely, we still want to have the RT T -relation (1.1). Then, the transfer matrix T ii (u) (1.9) satisfies the commutation relation (1.4). Thus, we can obtain a Hamiltonian and other integrals of motion via (1.5). In order to construct the Hamiltonian eigenstates we assume that the Hilbert space of the model has a vacuum vector |0 with the properties analogous to (1.6): T ii (u)|0 = λ i (u)|0 , i = 1, . . . , N, (1.10) Here λ i (u) are some functions dependent on the particular model.

Examples of R-matrices
The first problem is to find an R-matrix acting in C N ⊗ C N . The R-matrix should satisfy the Yang-Baxter equation R 12 (u 1 , u 2 )R 13 (u 1 , u 3 )R 23 (u 2 , u 3 ) = R 23 (u 2 , u 3 )R 13 (u 1 , u 3 )R 12 (u 1 , u 2 ), (1.11) in order to provide compatibility of the RT T -relation. The first example of the non-trivial R-matrix has exactly the same form as in the case of the C 2 auxiliary space: R(u, v) = I + g(u, v)P, g(u, v) = c u − v . (1.12) Here I is the identity operator in C N ⊗ C N , P is the permutation operator in the same space, and c is a constant. The permutation operator has the form where (E ij ) lk = δ il δ jk , i, j, l, k = 1, ..., N are N × N matrices with unit in the intersection of ith row and jth column and zeros elsewhere. Another solution to the Yang-Baxter equation is given by the q-deformation of the R-matrix (1.12): (1.14) where (1. 15) Pay attention that this R-matrix is not a complete analog of the well known 4 × 4 trigonometric R-matrix acting in C 2 ⊗ C 2 . Indeed, the latter has the following form 2 : (1. 16) One would expect that an analog of R trig (u, v) in the case C N ⊗ C N is However, the R-matrix (1.17) does not satisfy the Yang-Baxter equation except the case N = 2. The matter is that the solutions to the Yang-Baxter equation have some ambiguity.
also solve the Yang-Baxter equation.
The proof of this proposition is given in appendix A. It is easy to check that [R Therefore, both R (q) (u, v) and R trig (u, v) satisfy the Yang-Baxter equation for N = 2. However, for N > 2, the R-matrix (1.14) cannot be reduced to the R-matrix (1.17) by transformation (1.19). There exist, of course, other R-matrices acting in C N ⊗ C N and satisfying the Yang-Baxter equation, for example, Belavin elliptic R-matrix [17][18][19][20]. However, we will restrict our selves with consideration of the simplest R-matrix (1.12) only. Furthermore, the main part of these lectures will be devoted to the case N = 3. We will see that even in this simplest case one should solve several non-trivial problems.
The R-matrix (1.12) is called gl N -invariant due to the property for any G ∈ gl N .

Examples of monodromy matrices
The first example of the monodromy matrix is completely analogous to the gl 2 case This is the monodromy matrix of the SU (N )-invariant inhomogeneous XXX Heisenberg chain. The parameters ξ i are inhomogeneities. Each R-matrix R 0i (u, ξ i ) acts in the tensor product This quantum space has a vacuum vector of the form (1.24) Another example describes a system of bosons. For simplicity we give explicit formulas for the gl 3 case [4] (generalization to gl N is quite obvious). An L-operator of this system has the following form: where 1 is the identity operator and Here m is a complex number and ρ = a † 1 a 1 + a † 2 a 2 . The operators a k and a † k (k = 1, 2) act in a Fock space with the Fock vacuum |0 : a k |0 = 0. They have standard commutation relations of the Heisenberg algebra [a i , a † k ] = δ ik . In order to construct the monodromy matrix we replace the original a k and a † k operators with a k (n) and a † k (n) so that [a i (n), a † k (m)] = δ ik δ nm . Then . (1.27) This monodromy matrix describes a chain in each site of which there may be a particle of one of two sorts. The vacuum vector coincides with the Fock vacuum. This system admits the continuum limit. Then it turns into the system of the two-component Bose-gas with δ-function interaction [7][8][9]21].

Remark about RT T -algebra
RT T -algebra (1.1) with the R-matrix (1.12) is closely related to the concept of Yangian Y (gl N ) (see [22,23] and references therein). Sometimes in the literature it is called the Yangian. In fact, the RT T -algebra with the gl N -invariant R-matrix is somewhat wider. In the case of the Yangian, we must impose an additional condition on the asymptotic behavior of the monodromy matrix elements T ij (u) at u → ∞ where K is a diagonal c-number matrix, then the new matrix KT (u) will also satisfy the RT Trelation. The properties of the vacuum vector will also be preserved. However, the KT (u) matrix no longer has expansion (1.28), since it does not begin with the identity operator. This type of transformation (twist transformation) can be done with any of the L-operators entering the definition of the monodromy matrix. As a result, in some cases the monodromy matrix satisfying the RT T -relation may have essential singularity at infinity (see e.g. continuum models of one-dimensional Bose and Fermi gases [13,21]). Below we denote by R N the RT T -algebra with the gl N -invariant R-matrix (1.12), where N indicates the size of the monodromy matrix. Starting from this point, we consider only such algebras, unless otherwise specified.

Automorphism
Let us define a linear mapping Mapping (1.29) is an automorphism of the R N algebra [23]. Indeed, the RT T -relation implies (see appendix B) The second equation (1.30) can be obtained from the first one via simultaneous replacements i ↔ k, j ↔ l, and u ↔ v. Consider commutation relations of the operatorsT ij . We have Thus, the matrix elementsT ij (−u) satisfy the same commutation relations as T ij (u).

Coloring
In physical models, the vectors of the space H describe states with different types of particles (excitations). We now introduce a notion of coloring, in which particles of different types also appear. To distinguish them from physical particles, we will call them quasiparticles, and their different types are colors.
The space H as generated by the states of the form where i p < j p for p = 1, . . . , n. This means that T ip,jp (u p ) are creation operators. We say that an operator T ij with i < j creates quasiparticles with the colors i, . . . , j − 1, one quasiparticle of each color. In particular, the operator T i,i+1 creates one quasiparticle of the color i, the operator T 1N creates N − 1 quasiparticles of N − 1 different colors. Thus, in gl N based models quasiparticles may have N − 1 colors. Let {a 1 , . . . , a N −1 } be a set of non-negative integers. We say that a state has coloring {a k } ≡ {a 1 , . . . , a N −1 }, if it contains a k quasiparticles of the color k. In other words, we introduce a mapping Col(|Ψ ) that maps |Ψ to its coloring {a k }: Here θ(k) is a step function of integer argument such that θ(k) = 1 for k > 0 and θ(k) = 0 otherwise. (1.34) Observe that the coloring does not depend on the arguments of the operators T ij (u) and on the order of these operators. Assuming that the null-vector 3 has arbitrary coloring we extend the mapping (1.33) to some linear combinations of the states (1.32) as well as to the states containing neutral operators (i.e. T ii (u)) and annihilation operators (i.e. T ij (u) with i > j). Namely, if Col(|Ψ 1 ) = Col(|Ψ 2 ), then linear combinations of these states also have the same coloring: where α and β are complex numbers. Then the states with the same coloring generate a subspace H {a k } of the space H. The latter then can be presented as a direct sum of the subspaces with fixed coloring: Let us consider the states of the form (1.32), but now suppose that among T ip,jp (u p ) there can be neutral operators and annihilation operators. The coloring of these states is defined by the same formula (1.33). Then the integers a k may take negative values. Proof. Suppose that |Ψ = 0. Then we can normal order all the operators T ij , that is, we can move all neutral and annihilation operators to the extreme right position using commutation relations (1.30). Observe that the coloring mapping is compatible with these commutation relations. Thus, at any step of the normal ordering we deal with the state of the initial coloring. After the normal ordering is completed, the state |Ψ depends on creation operators only. Then due to (1.33) a k ≥ 0 for all k = 1, . . . , N − 1. We arrive at the contradiction, hence, |Ψ = 0. Proposition 1.2 allows us in some cases to quickly calculate the action of annihilation operators on the states without use of commutation relations (1.30). For example, we can immediately say that The reader can convince himself that the use of commutation relations (1.30) gives the same result, however it takes much more time and efforts. Generically, if j − 1 < k < i, and the annihilation operator T ij acts on a state in which there is no quasiparticles of the color k, then this action vanishes, like in (1.37). Proof. Obviously, it is enough to consider monomials (1.32) consisting of creation operators only. Otherwise, we always can get rid of the neutral and annihilation operators by normal ordering them. Since this monomial does not contain quasiparticles of the color 1, we conclude that i p > 1 and j p > 1. Then, due to commutation relations (1.30) we have (1. 40) We see that T 11 (z) either commutes with any T ip,jp (u p ) or produces the annihilation operator T ip,1 . The action of the latter on a state without quasiparticles of the first color gives zero. Thus, the operator T 11 (z) goes through all the creation operators T ip,jp (u p ) to the extreme right position, where it acts on the vacuum vector and gives λ 1 (z). Applying the automorphism (1.29) to (1.39) we find that in the R N algebra In conclusion, we note that the coloring mapping can also be introduced for models with the q-deformed R-matrix (1.14).

Bethe vectors
We have already introduced in section 1.1 notions of Bethe vector and on-shell Bethe vector in gl 2 based models. Recall that on-shell Bethe vectors are eigenvectors of the transfer matrix. To solve the spectral problem, they are only needed. However, in computing the correlation functions, we also have to deal with off-shell Bethe vectors. Indeed, a typical problem arising in calculating correlation functions is to compute a matrix element of an operatorÔ of the following form: Here |Ψ is an on-shell Bethe vector, and Ψ ′ | is an on-shell Bethe vector in the dual space (dual on-shell Bethe vector). If the operatorÔ does not commute with the transfer matrix, then |Ψ is not eigenvector ofÔ. Therefore, genericallyÔ|Ψ = |Φ , where |Φ is a linear combination of off-shell Bethe vectors. Thus, the expectation value (2.1) reduces to the scalar product where we have off-shell Bethe vectors in the state |Φ .

Bethe vectors in gl 2 based models
In the gl 2 based models the on-shell Bethe vectors have the form (1.8), provided the parameters u = {u 1 , . . . , u n } satisfy a system of Bethe equations [1,2,[13][14][15][16] where |Φ is an arbitrary vector. It is easy to see that if the setū enjoys the Bethe equations Hence, the coefficient of the vector |Φ vanishes, and the vector (2.4) becomes on-shell. Thus, the combination (2.4) also can be called the Bethe vector, because it turns into the on-shell Bethe vector as soon as the Bethe parameters satisfy Bethe equations. It is clear that we can invent plenty of combinations of this type. Therefore, strictly speaking, the definition of the off-shell Bethe vector is ambiguous.
Among possible definitions, equation (1.8) looks the most simple. However, there exist over presentations for the Bethe vectors, which also have rather simple form. For instance, let where K is a 2 × 2 c-number invertible matrix such that K 11 = 0. It is clear that the new operatorB(u) is a linear combination of the original A, B, C, and D. Nevertheless, a state is the on-shell Bethe vector provided the system (2.3) is fulfilled [24,25]. We suggest the reader to check this statement. Anyway, presentation (2.7) looks as simple as the original formula (1.8).
Thus, we have a big freedom in the definition of the Bethe vectors. Nevertheless, following the tradition we define them by equation (1.8), in particular, for the reasons of simplicity.

Bethe vectors in gl 3 based models
The problem of Bethe vectors in the gl 3 (and higher rank) based models is much more sophisticated than in the case considered above. The ambiguity of their definition still exists, however, now the form of the on-shell Bethe vectors is much more complex than (1.8). Therefore, we cannot use even the reasons of simplicity for choosing an appropriate definition. The matter is that there is only one creation operator in the gl 2 case (T 12 ), while there are three creation operators in the gl 3 case (T 12 , T 13 , T 23 ).
Consider a simple example that will allow us to feel the difference between the Bethe vectors in the gl 2 and gl 3 based models. For this we will try to construct simple on-shell Bethe vectors in these two cases.
Consider the commutation relations (1.30). We see that generically we have different operators in the l.h.s. (T ij and T kl ) and in the r.h.s. (T il and T kj ). However, if the operators in the l.h.s. belong to the same row (column), then we obtain the same operators in the r.h.s. In particular, we have Let us try to find an on-shell Bethe vector in a model described by the R 2 algebra. This means that we are looking for the eigenvectors of the transfer matrix T 11 (z) + T 22 (z). Let us test a vector T 12 (u)|0 . Then due to (2.8) we obtain We see that we still deal with the vectors of the type T 12 (·)|0 . Indeed, using T ii (u)|0 = λ i (u)|0 (where λ 1 (u) = a(u) and λ 2 (u) = d(u) (1.6)) we obtain Thus, the result of the action of the transfer matrix T 11 (z) + T 22 (z) gives two vectors: T 12 (u)|0 and T 12 (z)|0 . The first one is the same as in the l.h.s., while the second is different. Traditionally this second term is called unwanted term. We will call it unwanted term of the first type. It is still given as the action of T 12 on the vacuum (as in the l.h.s.), but the operator T 12 has new argument. This unwanted term can be killed, if we choose an appropriate u = u 0 , namely, such that a(u 0 ) = d(u 0 ). Observe that this condition coincides with Bethe equations (2.3) at n = 1. Then the vector T 12 (u 0 )|0 becomes the eigenvector of the transfer matrix.
It is easy to see that generically, if we test a vector of the form T 12 (u 1 ) . . . T 12 (u n )|0 , then the action of the transfer matrix produces unwanted terms of the first type only: we still obtain the products of the operators T 12 applied to |0 , but some of these operators may have new arguments. Now let us consider the R 3 algebra. Let us test the vector T 13 (u)|0 . We should act with the transfer matrix onto this vector We see immediately a principle difference with the case considered above. Namely, the operators T 22 and T 13 do not belong to the same row or column. Due to the commutation relations we have and the action of the transfer matrix is We obtain the vectors of a new type T 12 (u)T 23 (z)|0 and T 12 (z)T 23 (u)|0 , that we call unwanted terms of the second type. These unwanted terms generically cannot be killed by an appropriate choice of the original argument u. Remark 1. Strictly speaking, the term in the third line of (2.14) vanishes at u → ∞. In some models (for example, SU (3)-invariant XXX chain) Bethe equations have infinite roots, and then the corresponding contribution in (2.14) vanishes.
Remark 2. In some models the operator T 23 (u) actually plays the role of the annihilation operator: T 23 (u)|0 = 0 for all u. Then the unwanted terms of the second type in (2.14) automatically vanish, and the vector T 13 (u)|0 becomes the on-shell Bethe vector for u = u 0 such that λ 1 (u 0 ) = λ 3 (u 0 ).
Summarizing the above considerations we conclude that we deal with unwanted terms of two types: • first type: the operators are the same as in the l.h.s., but some of them accept new arguments; • second type: the operators in the r.h.s. are different form the ones in the l.h.s. In this case, they can either keep their original arguments or accept new arguments.
In the case of the R 2 algebra we obtain the first type of unwanted terms only. For the R 3 algebra (and higher) we necessarily obtain both types of unwanted terms. Therefore the structure of the Bethe vectors in the gl 3 case generically cannot be so simple as in the gl 2 case. The above example shows that not every combination of the creation operators applied to the vacuum has a chance to become an eigenvector of the transfer matrix even for some specific values of the Bethe parameters. In particular, in the general case, the vector T 13 (u)|0 cannot be an on-shell Bethe vector for any values of u. We will see below, that in order to obtain a Bethe vector, one should take the following combination of the terms T 13 (u)|0 and T 12 (u)T 23 (v)|0 : If the parameters u and v enjoy the system of equations There are several ways to construct on-shell Bethe vectors in the models with the gl Ninvariant R-matrix. In addition to NABA, it is also worth mentioning the approach associated with the so called trace formula [10][11][12], as well as the method based on the use of a special current algebra to describe the RT T -relation [26][27][28][29]. It is remarkable that all the three methods listed above give eventually the same expression, not only for on-shell, but also for off-shell Bethe vectors. And this is despite the initial ambiguity in defining the off-shell Bethe vectors. Therefore, we will adopt this formula as the definition of the Bethe vector. Details will be described later.

Notation
• Rational functions. We have introduced already two rational functions g(x, y) and f (x, y).
Recall that Observe that g(x, y) = −g(y, x). Below we will permanently use these functions.
• Sets of variables. We denote sets of variables by a bar:x,ū,v etc. Individual elements of the sets are denoted by the subscripts: v j , u k etc. A notationū i , meansū \ u i etc. Instead of the standard notationū ∪v we use braces {ū,v} for the union of sets.
• Shorthand notation for products.
In order to make formulas more compact we use a shorthand notation for the products of commuting operators or functions depending on one or two variables. Namely, if the functions λ i , g, f , as well as the operators T ij depend on sets of variables, this means that one should take the product over the corresponding set. For example, Observe that [T ij (u), T ij (v)] = 0 due to the commutation relations (1.30). Therefore, the product T ij (ū) is well defined. By definition, any product over the empty set is equal to 1. A double product is equal to 1 if at least one of the sets is empty.
We will extend this convention for new functions that will appear later on. For the moment, let us show how this convention works in particular examples. Equation (1.8) takes the following form If necessary, we should add a special comment on the cardinality of the setū. The system of Bethe equations (2.3) reads (2.20)

Nested algebraic Bethe ansatz
In this section we find a representation for on-shell and off-shell Bethe vectors in the models with the gl 3 -invariant R-matrix. The presentation follows the works [3-6].

Basic notions
Consider a model with the gl 3 -invariant R-matrix (1.12) where the identity matrix I and the permutation matrix P are 9 × 9 matrices: We will need also the gl 2 -invariant R-matrix, which we denote by r(u, v): Here the identity matrix 1 and the permutation matrix p are 4 × 4 matrices: It satisfies RT T -relation (1.1). This relation implies the set of commutation relations (1.30) for the operators T ij .
We will also use one more parametrization of the monodromy matrix Here in the intermediate formula the T -matrix is presented as a 2 × 2 block-matrix. The block A has the size 1 × 1, the block B has the size 1 × 2, the block C has the size 2 × 1, and the block D has the size 2 × 2.
Remark. One can also consider another embedding where now the block A ′ is a 2 × 2 matrix, while the block D ′ has the size 1 × 1. These two embeddings are equivalent due to the automorphism (1.29). For definiteness, we consider in details parametrization (3.6) and give short comments about parametrization (3.7).

Particular case of gl 3 invariant models
In this section we consider a particular case of the Bethe vectors. It will give us an idea of their construction in the general case. Generically, the operators T ij with i < j are the creation operators. However, in some models the operator T 23 annihilates the vacuum vector 4 : T 23 (u)|0 = 0. Actually, we very often deal with this situation in the models of physical application (SU (3)-invariant Heisenberg chain, two-component Bose gas, t − J model). Therefore this particular case is rather important.
Consider the simplest example of such monodromy matrix: T (u) = R(u, ξ), where ξ is a fixed complex number. This is the monodromy matrix of the XXX chain consisting of one site. Then T 23 (u) = g(u, ξ)E 32 . Obviously Consider now the chain with L sites. Then the monodromy matrix is given by (1.22) where Hence, the operator T where T (L−1) ij (u) and T (1) ij (u) are the entries of the monodromy matrices respectively corresponding to the sub-chains of the lengths L − 1 and 1. Since T kl (v)] = 0 for arbitrary subscripts and arbitrary arguments.
We know that T (u)|0 = 0 due to the induction assumption. This method also allows one to find the vacuum eigenvalues λ i (u) of the diagonal entries T ii (u): Observe that we have λ 2 (u) = λ 3 (u). This is the direct consequence of the property Applying this equation to |0 we obtain Hence, the ratio λ 2 (u)/λ 3 (u) does not depend on u.

Action of the operators D ξα
Let we have a model such that T 23 (u)|0 = 0, however, we do not specify the Hilbert space H in which the operators T ij (u) act. Let the monodromy matrix be normalized in such a way that λ 3 (u) = 1. Then λ 2 (u) must be a constant. For simplicity we assume that λ 2 (u) = 1, like in (3.12). More general case will be considered later.
In the case under consideration we have only two creation operators: (3.6)). We can try to check a monomial as a candidate for the transfer matrix eigenvector. Here every β i is equal to either 1 or 2. We should act with the transfer matrix onto this vector. We start our consideration with the action of the operators D αα (z). It follows from (1.30) that We see that acting with D ξα (z) onto the vector (3.15) we may have unwanted terms of the second type. Indeed, the operator D 11 acting on B 2 gives contributions with B 1 , and the operator D 22 acting on B 1 gives contributions with B 2 . Thus, the operator structure in the vector (3.15) is not invariant under the action of D 11 (z) + D 22 (z). At the same time the action of the operator A(z) does not produce unwanted terms of the second type (see section 3.4). Thus, the monomial (3.15) generically is not invariant under the action of tr T (z). Therefore, it is quite natural to replace the monomial (3.15) by a polylinear combination Here F β 1 ,...,βa are some numerical coefficients. The sum is taken over every β i ∈ {β 1 , . . . , β a }.
Each β i takes the values β i = 1, 2. Let us write down (3.18) in the form of a scalar product. For this we introduce a two- This is a 2 a -component vector-row. Then we can write down the vector |Ψ a (ū) as the scalar product where F(ū) is a vector belonging to the space The commutation relations (3.17) can be written as follows where p 01 is 4 × 4 permutation matrix. Here the matrix D 0 acts in the auxiliary space V 0 ∼ C 2 and the vector B belongs to the auxiliary space V 1 ∼ C 2 . The r-matrix r 01 (z, u) and the matrix p 01 act in the tensor product V 0 ⊗ V 1 . We call the first term in the r.h.s. of (3.22) the first scheme of commutation. The second term in the r.h.s. of (3.22) is called the second scheme of commutation.
It follows from the commutation relations (1.30) that Hence, the matrix D(u) satisfies the RT T -relation with the R-matrix r(u, v) Therefore, D(u) can be treated as the monodromy matrix of a model with the gl 2 -invariant R-matrix. Let us act with tr D(z) onto |Ψ a (ū) . We have Here we have stressed that D(z) acts in V 0 , which is different from V 1 , . . . , V a . Permuting D 0 (z) and B 1 (u 1 ) we obtain We have two contributions. The second one definitely is unwanted, as it contains the operators B β (z) (in the vector-row B 1 (z)). Let us leave this term for some time and only deal with the wanted contributions. In other words, we use the first scheme of commutation only. In the case of the R 2 algebra the use of the first scheme would necessarily give us wanted terms only. However, in the case of the R 3 algebra we still may have unwanted terms of the second type. Our first goal is to get read of them. We have where Z denotes unwanted contributions. Clearly, the R-matrix r 01 (z, u 1 ) can be moved to the right and the subscript 0 stresses that the auxiliary space of this matrix is V 0 . Recall that the matrix D 0 (z) can be treated is the monodromy matrix satisfying the R 2 algebra due to (3.24). Its matrix elements act in the original Hilbert space H as follows: The matrix T  If we present T (a) (z) as Thus, T (a) 0 (z) is the monodromy matrix of the R 2 algebra, being the product of two monodromy matrices whose entries act in different spaces. It remains to act with tr 0 T (a) 0 (z) on the vector F(ū)|0 . Due to (3.31) we have Thus, if we do not want to have unwanted terms of the second type in the action of tr D(z), we should require that F(ū) be an eigenvector of the transfer matrix tr T (a) (z). Hence, F(ū) has the form Recall that here we use the shorthand notation (2.18) for the products of the f -functions.
Observe also that the number of the operators B (a) (v b ) cannot exceed the number of the sites of the chain a. Hence, b ≤ a. Thus, we have obtained that the vector F(ū) should be the eigenvector of the transfer matrix of the inhomogeneous XXX chain with the inhomogeneitiesū. This is the main idea of the nested algebraic Bethe ansatz. Namely, the on-shell Bethe vector of the model with the gl 3invariant R-matrix is expressed in terms of the on-shell Bethe vector of the model with the We see that F(ū) depends on the set of auxiliary parametersv, that is F(ū) = F(ū;v). Hence, the vector |Ψ a (ū) also depends on these parameters: |Ψ a (ū) = |Ψ a,b (ū;v) . By construction, this vector is symmetric over the variablesv. It turns out that it is also symmetric over the variablesū, however, this symmetry is far from the evident. We postpone the corresponding proof till section 4.2. For the moment we assume that |Ψ a,b (ū;v) is symmetric overū.
Thus, we obtain where Up to now we used the first scheme of commutation only. Let us take into account the second scheme.
It is clear that if we use at least once the second scheme, then the operators D ij (z) and B β k (u l ) exchange their arguments. Therefore, after moving the matrix D 0 (z) through the product of B j (u j ) to the right it will have an argument u k ∈ū. At the same time one of the operator-valued vectors B j 0 will have the argument z. Other operator-valued vectors will have arguments u j such that u j = u k .
Due to the symmetry of |Ψ a,b (ū;v) overū it is enough to consider the case when B 1 (u 1 ) looses its argument and absorbs the argument z, while D 0 (z) arrives at the extreme right position with the argument u 1 . Then at the first step we should use the second scheme in (3.26), otherwise D 0 (z) never absorbs the argument u 1 . We have where Z now denotes all the terms which do not give contributions to the desired result. Obviously, we can move p 01 to the right Moving further D 0 (u 1 ) to the right we should keep its argument, therefore, now we can use only the first scheme. Then we obtain It is easy to see that D 0 (u 1 )r 0a (u 1 , u a ) . . . r 02 (u 1 , u 2 )p 01 = 1 c Res D 0 (z)r 0a (z, u a ) . . . r 02 (z, u 2 )r 01 (z, u 1 ) Hence, we obtain where we used (3.36). Since F(ū;v) is the eigenvector of tr T (a) 0 (z) for any z we find where the eigenvalue τ D (z|ū;v) is given by (3.40). Substituting this eigenvalue into (3.46) we eventually arrive at Thus, using the symmetry of |Ψ a,b (ū;v) overū we find the total action of tr D(z) = T 22 (z)+ T 33 (z) on the vector |Ψ a,b (ū;v) . It is given by (3.50)

Action of A(z)
We did not consider yet the action of the operator T 11 (z) = A(z) on the vector |Ψ a,b (ū;v) . This action is relatively simple and reminds the action of the operator A in the case of the R 2 algebra. Indeed, the commutation relation of A(z) and B β (u) is Therefore, the action of A(z) does not produce unwanted terms of the second type. It is easy to see that the result should have the following form: where τ A (z|ū;v) and Λ k are some numerical coefficients. They can be found exactly in the same manner as in the R 2 case. Recall this procedure. As usual, let us call the first term in the r.h.s. of (3.51) the first scheme of commutation. Respectively, the second term in the r.h.s. of (3.51) is called the second scheme of commutation. In the first scheme both A and B β keep their original arguments, while in the second scheme they exchange them. Obviously, in order to obtain the contribution proportional to τ A (z|ū;v) we should use the first scheme of commutation only. We obtain where Z denotes all unwanted terms. Acting with A(z) on |0 we immediately obtain and thus, this coefficient actually does not depend on the setv. In order to find the coefficients Λ k it is enough to find one of them, say, Λ 1 . Here we use the symmetry of |Ψ a,b (ū;v) overū. Then at the first step we must use the second scheme of commutation, and after this we must use only the first scheme of commutation. This gives us where now Z denotes all the terms that do not give contributions to the desired result. From this we find Λ 1 = λ 1 (u 1 )g(z, u 1 )f (ū 1 , u 1 ).
(3.56) Thus, we have computed the action of the transfer matrix tr T (z) on the vector |Ψ a,b (ū;v) . It is given by the sum of (3.49) and (3.52): The coefficients M k are It is clear that |Ψ a,b (ū;v) becomes on-shell Bethe vector, if we set M k = 0 for k = 1, . . . , a, and for all complex z. This leads us to a new system of equations

General case
All the consideration above concerned the particular case T 23 (u)|0 = 0. What should be done in the general case? Remarkably, almost the entire scheme remains the same. We should only assume that the vector F(ū;v) in the expression has operator-valued components, which depend on the operators D αβ (in particular, on T 23 ). In other words, the vector F(ū;v)|0 belongs to the tensor product H ⊗ H (a) (because the action of T 23 on the vacuum |0 gives a vector in the space H).
We also should take into account that and now we have no the restriction λ 2 (z)/λ 3 (z) = const. We start with representation (3.61) and act on this vector with tr D(z). Using the first scheme of commutation only we arrive at (3.29): (3.65) Thus, the eigenvectors of tr T (a) (z) have the form similar to (3.37) provided the setv satisfies Bethe equations Observe that now the matrix T (a) (z) is no longer the monodromy matrix of the XXX chain. It is the product of two monodromy matrices D(z) and T (a) (z). Therefore, there is no restriction on the number of the operators B (a) in (3.66). Thus, in this case we do not have the constraint b ≤ a as it was previously.
Thus, we obtain Consideration of the unwanted terms of the first type produced by the action of tr D(z) can be done exactly in the same manner as before. This leads us to the analog of (3.47) where |Φ a,b (z, u 1 ;ū;v) is still given by (3.48). The only natural difference between (3.70) and (3.47) is that we have the additional factor λ 2 (u 1 ). Previously this factor was equal to 1. Thus, the total action of tr D(z) on the vector |Ψ a,b (ū;v) has the form where |Φ a,b (z, u k ;ū;v) is given by (3.50). Recall that this result is obtained under assumption that |Ψ a,b (ū;v) is symmetric over the setū. This symmetry still should be proved. Considering the action of A(z) on |Ψ a,b (ū;v) we deal with only one new problem. Namely, we should prove that A(z)F(ū;v)|0 = λ 1 (z)F(ū;v)|0 . This property was obvious in the previous case, because F(ū;v) did not belong to the space H, in which the operator A(z) acted. Now F(ū;v)|0 ∈ H ⊗ H (a) , therefore, the property (3.72) should be proved. However, the proof immediately follows from proposition 1.3. Indeed, since the components of F(ū;v)|0 depend on the operators D αβ , they only contain quasiparticles of the second color. Thus, the action of A(z) on each of these components reduces to multiplication by λ 1 (z). In all other respects the action of A(z) on |Ψ a,b (ū;v) can be derived via the same lines leading eventually to equation (3.52), where τ A (z|ū;v) and Λ k respectively are given by (3.54) and (3.56).
Thus, the action of the transfer matrix tr T (z) on the vector |Ψ a,b (ū;v) reads and the coefficients M k have the form Setting M k = 0 for k = 1, . . . , a we obtain a system of equations

Definition of Bethe vectors
At this point we can turn back to the problem of off-shell Bethe vectors (or simply Bethe vectors). Now we are able to give their definition at least for the gl 3 based models.  Remark. Note that when we looked for the on-shell Bethe vectors we required the vector F(ū;v)|0 to be the eigenvector of the transfer matrix tr T (a) (z) (3.64). This requirement led us to the set of equations (3.67). Now we do not impose this constraint. Thus, F(ū;v)|0 is not necessarily the eigenvector of tr T (a) (z), but it has the form (3.66).
If the parametersū andv satisfy the system of Bethe equations (3.67) and (3.76), that is then the vector (3.77) becomes the on-shell Bethe vector. Formally, definition 3.1 uniquely fixes the Bethe vector as a polynomial in the creation operators 5 T ij with i < j acting on the vacuum vector. However, these operators, generally speaking, do not commute with each other. Therefore, their reordering leads to new representations for the Bethe vectors. Formula (3.77) is one of such representations.
Unfortunately, equation (3.77) does not give an explicit dependence of the Bethe vector on the creation operators. We will derive such explicit dependence later. In the meantime, as a example, consider a couple of the simplest cases.
The most simple case is a = 0, that isū = ∅. Then T (a) (z) = D(z), and hence, B (a) (z) = T 23 (z). We obtain We see that in this case the R 3 Bethe vector reduces to the R 2 Bethe vector. This is not surprising, because the state |Ψ 0,b (∅;v) has quasiparticles of the color 2 only. Thus, it should coincide with the Bethe vector of the gl 2 based models. Let now a = b = 1. Then where |Ω (1) = ( 1 0 ). The matrix T (1) (v) is given by (3.30) at z = v and a = 1, that is and we stressed by the subscript 0 that the auxiliary space of this matrix is V 0 . Thus, and hence, the action of B (1) (v) on |Ω (1) is given by Thus, we obtain the explicit expression for the Bethe vector |Ψ 1,1 (u; v) : what coincides with (2.15). This Bethe vector becomes on-shell, if u and v satisfy the system (3.78). It is easy to see that for a = b = 1 this system turns into (2.16).

Remarks about different embedding
Let us say few words about parametrization (3.7). This parametrization also can be used for constructing the Bethe vectors. The general strategy is the same in this case, however, several minor details are different. We recommend that the reader obtain himself the formula for the Bethe vector using parametrization (3.7). We restrict ourselves with several comments. In the case of embedding (3.7) we deal with a 2 × 2 matrix A ′ and a two-component vectorcolumn Instead of (3.61) we use the following ansatz for the Bethe vectors: Here the superscript T means transposition in the space The commutation relations between the operator-valued matrix A ′ and the operator-valued vector-column B ′ have the form Here, in distinction of (3.22), the R-matrix r 01 (v, z) and the permutation matrix p 01 act on other matrices from the left. Therefore, moving A ′ 0 (z) through the product of B ′ j (v j ) we obtain the product of the R-matrices in the extreme left position. However, after the transposition we obtain the product of the R-matrices to the right from the product of the B ′ j (v j )-operators. Then we require that the vector F ′ (ū;v)|0 would be an eigenvector of the monodromy matrix where t k means the transposition in the space V k . These matrices appear when we take the transposition of the product r 01 . . . r 0b in the space V 1 ⊗ · · · ⊗ V b . We leave to the reader to prove that r ′ (u, v) satisfies the RT T -relation with the R-matrix r(u, v): Thus, the product r ′ 01 (z, v 1 ) . . . r ′ 0b (z, v b ) also satisfies the RT T -relation with the R-matrix r(u, v).

The entries of T (b) (z) act in the space
has the following vacuum vector: | If we set then the vector F ′ (ū;v)|0 has the following form: We will see below that formulas for the Bethe vectors based on the embeddings (3.6) and (3.7) look very different. First, they have different ordering of the creation operators. Second, some of those operators have different arguments. Nevertheless, these different representations describe the same Bethe vector |Ψ a,b (ū;v) .

Remarks about gl N Bethe vectors
The scheme described above does not change in the case of the models with gl N -invariant Rmatrix or its q-deformed analog (1.14). We present the N × N monodromy matrix as a 2 × 2 block-matrix Otherwise, all the arguments remain unchanged. They lead us to the conclusion that the vector F|0 must be an eigenvector of the transfer matrix of the model with gl N −1 -invariant R-matrix.
Of course, an explicit expression for this vector is no longer given by (3.66), but is much more complicated. It is clear that using this method we obtain Bethe vectors depending on N − 1 sets of variables 6t = {t 1 , . . . ,t N −1 }. In its turn, every sett k consists of individual elementst k = {t k 1 , . . . , t k a k }, where a k = #t k . Then we can refine formula (3.97) as whereā is a multi-index consisting of the cardinalitiesā = {a 1 , . . . , a N −1 }. In this formula, F(t)|0 is the Bethe vector of the monodromy matrix where now the R-matrix r(u, v) acts in C N −1 ⊗ C N −1 . Equation (3.99) can be taken as the definition of the off-shell Bethe vector. This vector becomes on-shell if the Bethe parameterst satisfy a system of Bethe equations. It has the following form: Here we set by definitiont 0 =t N = ∅ and used the shorthand notation for the products of the f -functions over the setst k .

Bethe vector via trace formula
In this section we obtain one more representation for off-shell Bethe vectors [10][11][12]. Consider several copies of the space C 3 . Let and Here every T j acts in We would like to draw attention of the reader to the ordering of the R-matrices in the double product (4.2). There the index i changes in the standard increasing direction, while the index j changes in the decreasing direction. For example, for a = b = 2, the product (4.2) reads Proposition 4.1. The off-shell Bethe vectors of the gl 3 -invariant models have the following form: The trace is taken over all the spaces V k 1 , . . . , V ka , V n 1 , . . . , V n b . The matrices E 21 k j and E 32 n j are the elementary units that respectively act in the spaces V k j and V n j . In distinction of the previous section we use here superscripts for different elementary units.

Equation (4.4) is known as a trace formula.
We will prove that the trace formula is equivalent to the representation obtained in the previous section.
Proof. Let us present all the monodromy matrices in (4.4) as Substituting this into the trace formula we obtain where we have used cyclicity of the trace. The sum is taken over all i s , j s (with s = 1, . . . , a) and all α p , β p (with p = 1, . . . , b). Taking the product of the E-matrices via E ab E cd = δ bc E ad we obtain that all i s = 1 and all α p = 2. Then To calculate the remaining trace we present the product of the R-matrices Rn ,k (v,ū) as where r λ 1 µ 1 ,...,pa,qa (v,ū) are numeric coefficients, and the sum is taken over allλ = {λ 1 , . . . , λ b }, µ = {µ 1 , . . . , µ b },p = {p 1 , . . . , p a }, andq = {q 1 , . . . , q a }. Then we obtain (see appendix C for more details) Now we should compute the coefficients rβ ,j (v,ū). For this, it is convenient to use a diagram technique [16,30]. We present a single R-matrix as a vertex (see Fig. 1). Observe that R αβ;ij (v, u) = 0, if either α = β and i = j or α = j and i = β. Thus, the index, which enters the vertex from the north, can go further to the south or turn to the west. Respectively, the index, which enters the vertex from the east, can go further to the west or turn to the south (see Fig. 2).
The product of the R-matrices R np,ka (v p , u a ) . . . R np,k 1 (v p , u 1 ) (4.10) is given by the horizontal line (see Fig. 3). Respectively, the total product Rn ,k (v,ū) looks as  Figure 3: Product of the R-matrices. it is shown on the Fig 4. Finally, the matrix element rβ ,j (v,ū) has a graphical representation shown on Fig. 5.
Generically, the indices on the edges of the lattice on Fig. 4 can take three values: 1, 2, 3. However, in the case of the lattice on Fig. 5 the value 1 is forbidden. Indeed, we have seen that moving through any vertex, every index goes in the direction from the north-east to the south-west. Thus, any index of an arbitrary edge has its source either on the northern or eastern lattice boundary. But all the indices on those boundaries take the values 2 or 3. Thus, there is Figure 5: Graphical interpretation of the matrix element rβ ,j (v,ū).
Vn 1 ✛ ✛ Figure 6: Line of the constant index no the index 1 on the edges of the lattice on the Fig. 6.
The above consideration shows that the original gl 3 -invariant R-matrix R αβ;ij (u, v) turns into the gl 2 -invariant R-matrix r αβ;ij (u, v), where all the indices take values 2 and 3.
Let T (a) (v p |ū) = r np,ka (v p , u a ) . . . r np,k 1 (v p , u 1 ). (4.11) It is easy to see that T (a) (v p |ū) coincides with the monodromy matrix introduced by (3.30). Then and we obtain Observe that here we have changed the summation limits forj andβ. When taking the sum overj one should remember that the monodromy matrix T If we introduce and then (4.13) takes the form Now it becomes obvious that this formula coincides with (3.61), where F(ū;v)|0 is given by (3.66).
Concluding this section, we note that the trace formula has a fairly obvious generalization to the case of models with gl N -invariant R-matrix. We do not give the explicit formula, since this would require the introduction of a large number of new notations. The reader, however, can find this formula in [11].

Symmetry overū
Trace formula (4.4) allows us to prove the long standing problem of the symmetry of the Bethe vectors over the setū. For this we first consider some properties of the matrix R(u, v)P : (4.17) Let j < 3. Then Indeed, using (4.17) we have, for example, Consider now the right action of the matrix R 2,1 (u 2 , u 1 )P 2,1 on the product of the R-matrices R n,2 (v, u 2 )R n,1 (v, u 1 ), where the subscript n refers to some space V n which is different from V 1 and V 2 . Due to the Yang-Baxter equation we have R n,2 (v, u 2 )R n,1 (v, u 1 ) R 2,1 (u 2 , u 1 )P 2,1 = R 2,1 (u 2 , u 1 ) R n,1 (v, u 1 )R n,2 (v, u 2 ) P 2,1 . (4.20) Then moving the permutation matrix to the left we exchange the spaces V 1 and V 2 : R n,2 (v, u 2 )R n,1 (v, u 1 ) R 2,1 (u 2 , u 1 )P 2,1 = R 2,1 (u 2 , u 1 )P 2,1 R n,2 (v, u 1 )R n,1 (v, u 2 ). (4.21) Thus, acting form the right on R n,2 (v, u 2 )R n,1 (v, u 1 ), the matrix R 2,1 (u 2 , u 1 )P 2,1 in fact makes the replacement u 1 ↔ u 2 . Similarly R 2,1 (u 2 , u 1 )P 2,1 acts on the product T 1 (u 1 )T 2 (u 2 ). For this we first write down the RT T -relation and multiply it from both sides by R 2,1 (u 2 , u 1 ): Here we used the fact that R 1,2 (u 1 , u 2 )R 2,1 (u 2 , u 1 ) = f (u 1 , u 2 )f (u 2 , u 1 )I. Consider now how the permutation matrix acts on T 1 (u 1 )T 2 (u 2 ). We have Then Thus, using (4.23) and (4.25) we obtain T 1 (u 1 )T 2 (u 2 ) R 2,1 (u 2 , u 1 )P 1,2 = R 2,1 (u 2 , u 1 ) T 2 (u 2 )T 1 (u 1 ) P 1,2 = R 2,1 (u 2 , u 1 )P 1,2 T 1 (u 2 )T 2 (u 1 ). (4.26) Hence, here we also deal with the replacement u 1 ↔ u 2 . Now everything is ready for the proof of the symmetry of the Bethe vector |Ψ a,b (ū;v) over the parametersū. Consider a vector The matrix R k i+1 ,k i (u i+1 , u i )P k i+1 ,k i first can be moved to the left through the products of the matrices E 32 n 1 . . . E 32 n b and E 21 k 1 . . . E 21 ka . Then, moving through the R-matrices Rn ,k we should exchange it with the combinations R ns,k i+1 (v ns , u i+1 )R ns,k i (v ns , u i ). Due to (4.21) this leads to the replacement u i ↔ u i+1 in Rn ,k . After this we should move R k i+1 ,k i (u i+1 , u i )P k i+1 ,k i through the product of the T -matrices Tk(ū). Here we meat a combination T k i (u i )T k i+1 (u i+1 ). And again we obtain the replacement u i ↔ u i+1 in Tk(ū). Thus, we arrive at Using cyclicity of the trace we move R k i+1 ,k i (u i+1 , u i )P k i+1 ,k i to its original position Formally, one can prove the symmetry over the setv using exactly the same way. However, this symmetry is obvious in the representations (3.61), (3.66).

Recursion for the Bethe vectors
In this section we derive a relation between Bethe vectors |Ψ a,b , |Ψ a−1,b , and |Ψ a−1,b−1 . For this we use representation (3.61) where we equipped the vector F (a) with the additional superscript (a). This superscript stresses that F (a) |0 belongs to the space H ⊗ H (a) , where H (a) is the tensor product of a spaces C 2 (see (3.32)). Our goal is to express this vector in terms of vectors F (a−1) |0 , which belong to the space H ⊗ H (a−1) , where H (a−1) is the tensor product of a − 1 spaces C 2 . This is a typical problem of a composite model [13,15,16,31]. Recall that representation (5.1) is equivalent to the following sum: We know that the vector F (a) |0 has the form (3.66), where the operator B (a) is the matrix element of the monodromy matrix T (a) (3.64). We can present this monodromy matrix as follows: The entries of T It is easy to see that and we recall thatū 1 =ū \ u 1 . In its turn, the matrix r 01 (z, u 1 ) can be considered as the monodromy matrix of the XXX chain consisting of one site. We dealt already with this monodromy matrix in section 3.6 (see (3.82)). Recall that the auxiliary space of this matrix is V 0 ∼ C 2 , the quantum space is V 1 ∼ C 2 . It can be presented as a 2 × 2 matrix in the space V 0 where the entries act in the space V 1 ∼ C 2 with the vacuum vector ( 1 0 ). Obviously A peculiarity of this monodromy matrix is that b n (z) = 0 for n > 1, because b(z) = g(z, u 1 )E 21 1 . We can treat T (a) (z) as the monodromy matrix of the composite model (see appendix D) with T (2) (z) = T (a−1) (z) and T (1) (z) = r 01 (z, u 1 ). In this model Here we have extended the convention on the shorthand notation (2.18) to the products of the commuting operators B (a) (v), B (a−1) (v), and b(v).
Formally, the sum in (5.11) is taken over all possible partitions of the setv into two disjoint subsetsv I andv II . However, due to the property b n (z) = 0 for n > 1 we conclude that either #v I = 0 or #v I = 1. In the first case we havev I = ∅ andv II =v. In the second case we can set v I = v ℓ andv II =v ℓ , where ℓ = 1, . . . , b. Thus, we find where the first term corresponds tov I = ∅ and the sum over ℓ corresponds to the partitions v I = v ℓ . From this equation we find the components F (5.14) Then we recognise the Bethe vector |Ψ a−1,b (ū 1 ;v) in the first line of (5.14) and the sum of the Bethe vectors |Ψ a−1,b−1 (ū 1 ;v ℓ ) in the second line: Since B 1 (u) = T 12 (u) and B 2 (u) = T 13 (u), we recast (5.15) as follows: Recursion (5.16) allows one to build the Bethe vectors successively, starting from the case a = 0. Indeed, for a = 0 we have |Ψ 0,b (∅;v) = T 23 (v)|0 (see (3.79)). Then we immediately find an explicit expression for the Bethe vector |Ψ 1,b (u;v) : To conclude this section we note that it follows from recursion (5.16) and initial condition (3.79) that the Bethe vectors are the states of fixed coloring Col |Ψ a,b (ū;v) = {a, b}. (5.18) This statement can be easily proved via induction over a.

Second recursion for Bethe vectors
Using representation for the Bethe vectors described in section 3.7 one can obtain another recursion This recursion is derived via the composite model in the same way as recursion (5.16). We provide the readers with the opportunity to do this themselves. Recursion (5.19) allows one to build the Bethe vectors starting from |Ψ a,0 (ū; ∅) = T 12 (ū)|0 . (5.20)

Explicit form of Bethe vector
Successive application of recursion (5.16) allows us to guess a general explicit formula for the Bethe vector. In this formula a partition function of the six-vertex model with domain wall boundary conditions (DWPF) appears. Therefore, we give its brief description (for more details see [13,15,16,32]).

Partition function with domain wall boundary conditions
We denote the DWPF by K n (v|ū). It depends on two sets of variablesv andū; the subscript indicates that #v = #ū = n. The function K n has the following determinant representation where ∆ ′ n (v) and ∆ n (ū) are More explicitly .
Obviously, K n (v|ū) is a symmetric function ofv and a symmetric function ofū. It decreases as 1/v 1 (resp. as 1/u 1 ), if v 1 → ∞ (resp. u 1 → ∞) and all other variables are fixed. It has simple poles at v j = u k , j, k = 1, . . . , n. The residues in the poles are proportional to K n−1 : where reg stays for the terms which remain regular at u n → v n . To prove (6.4) it is enough to observe that the determinant in (6.3) becomes singular at v n → u n due to the pole of the matrix element at the intersection of the nth row and the nth column. Then the determinant (6.3) reduces to the product of this matrix element and the corresponding minor, leading eventually to (6.4).
Using this property one can decompose K n (v|ū) over the poles at u n = v i , i = 1, . . . , n, as follows: (6.5) In particular, where we used K 1 (v|u) = g(v, u). (6.7) 6.2 Bethe vector as a sum over partitions Proposition 6.1. Bethe vectors of the R 3 algebra have the following form: Here K n (v I |ū I ) is the DWPF (6.1). The sum is taken over partitionsū → {ū I ,ū II } andv → {v I ,v II } such that #ū I = #v I = n and n = 0, 1, . . . , min(a, b). Everywhere the convention on the shorthand notation (2.18) is used.
Remark. Observe that the r.h.s. of (6.8) is obviously symmetric overū and symmetric over v.
The proof of proposition 6.1 can be found in [33]. Actually, it is enough to show that (6.8) satisfies recursion (5.16). However, the proof is rather bulky, therefore we do not give it here. Instead we illustrate representation (6.8) by several examples for a small and b arbitrary.
In concluding this section, we note that explicit formulas for the R N Bethe vectors in terms of polynomials in the creation operators acting on the vacuum vector were obtained in [34].

Alternative representation for Bethe vectors
Along with the representation (6.8), there is another representation for the Bethe vectors: Here the sum is taken over partitionsū → {ū I ,ū II } andv → {v I ,v II } like in (6.8).
In order to prove (6.29) we first prove the following proposition. where | Ψ b,a (−v; −ū) is the Bethe vector corresponding to the monodromy matrixT (z).

Commutation relations and Bethe vectors
Comparing representations (6.8) and (6.29) we see that we deal with different ordering of the operators in these formulas. Nevertheless, these formulas are equivalent.
On the one hand, acting with (6.46) on |0 we immediately prove the equivalence of the representations (6.8) and (6.29). On the other hand, (6.46) can be considered as a multiple commutation relation between the products T 23 (v) and T 12 (ū). Indeed, extracting explicitly the term n = 0 we recast (6.46) as follows: Thus, we see that the explicit representations for the Bethe vectors in the gl 3 based models are closely related to the multiple commutation relations. Whether this correspondence exists in the models with higher rank of symmetry is an opened question.

Summary
We have considered the basic principles of NABA. The main example for us were models with the gl 3 -invariant R-matrix. However, in more general cases, the principle scheme for obtaining off-shell and on-shell vector Beta persists. In particular, this scheme works in models with the q-deformed R-matrix (1.14), as well as in the models based on graded algebras [12,34,36,37]. Further development of NABA is associated with the application of this method to the calculation of the matrix elements of local operators and correlation functions. In these matters, however, there are still many unsolved problems. Basically these problems are of a technical nature and involve very non-trivial representations for the Bethe vectors. Partially these problems are solved in the models with the gl 3 -invariant R-matrix, as well as its graded and q-deformed analogues [38][39][40].
Hilbert space H only, while the elementary units act in the auxiliary spaces C N . Recall also that Now we simply substitute (B.2) and (B.3) into (B.1). Then we obtain in the lhs In the rhs, we have g(u, v) T 2 (v)T 1 (u)P 12 − P 12 T 1 (u)T 2 (v) = g(u, v) Now let X be a matrix acting in the tensor product V 1 ⊗ · · · ⊗ V m , where V j ∼ C n . Then

D Composite model
Let us recall the basic formulas of the R 2 composite model. More detailed description can be found in [13,15,16,31]. In the R 2 composite model the full monodromy matrix is equal to the product of partial monodromy matrices We mark the entries of the matrices T (1) (v) and T (2) (v) with a superscript In particular, it is easy to see that Each of the matrices T (j) (v) possesses a vector |0 (j) , such that |0 = |0 (2) ⊗ |0 (1) . Formulas of the actions of the operators onto vacuum vectors are similar to the formulas of the action of the full monodromy matrix We define partial Bethe vectors in a standard way where we used the shorthand notation for the product of the operators B (j) . Then a full Bethe vector can be expressed in terms of the partial ones as follows: Here the sum is taken over all possible partitions of the setv into two subsetsv I andv II . We have also extended our convention on the shorthand notation to the products of the functions a (j) and d (j) . For example, d (1) (v II ) means the product of d (1) (v i ) over v i ∈v II .