Petr P. KULISH*
St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia
Received date: December 20, 2007; Revised date: March 10, 2008
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Diagonalization of integrable spin chain Hamiltonians by the quantum inverse scattering method gives rise to the connection with representation theory of different (quantum) algebras. Extending the Schur-Weyl duality between sl2 and the symmetric group SN from the case of the isotropic spin 1/2 chain (XXX-model) to a general spin chains related to the Temperley-Lieb algebra TLN(q) one finds a new quantum algebra Uq(n) with the representation ring equivalent to the sl2 one.
The development of the quantum inverse scattering method (QISM) [1, 2, 3, 4] as an approach to construction and solution of quantum integrable systems has lead to the foundations of the theory of quantum groups [5,6, 7, 8, 9].
The theory of representations of quantum groups is naturally connected to the spectral theory of the integrals of motion of quantum systems. In particular, this connection appeared in the combinatorial approach to the question of completeness of the eigenvectors of the XXX Heisenberg spin chain  with the Hamiltonian
where are the Pauli matrices.
Three algebras are connected to this system: the Lie algebra sl2 of rotations, the group algebra of the symmetric group SN and the infinite dimensional algebra the Yangian , with the corresponding R-matrix where is the 4 × 4 permutation matrix flipping the two factors of
The Yangian is the dynamical symmetry algebra which contains all the dynamical observables of the system. It is important to note that the algebras and are related by the Schur- Weyl duality in the representation space This follows from the fact that and are each other’s centralizers in this representation space. As a consequence, since the Hamiltonian commutes with the global generators of : is an element of This can also be seen from the expression of HXXX in terms of the permutation operators, which are the generators of the symmetric group SN
An analogous situation arises in the anisotropic XXZ chain
in the space coincide with the constant R-matrix
As in the case of the XXX spin chain, the Hamiltonian (1.2) can be expressed in terms of the generator (1.3) of the algebra
The dynamical symmetry algebra of the XXZ chain is the quantum affine algebra
The eigenvectors for both models can be constructed by the coordinate Bethe Ansatz (see ) or by an algebraic Bethe Ansatz [1,2,3]. The latter one follows from the main relation of the QISM for the auxiliary L-operator (see Sec. 2)
where the indices a and j refer to the corresponding auxiliary and quantum spaces
The Temperley-Lieb algebra is a quotient of the Hecke algebra (see section 3) and allows for an R-matrix representation in the space for any n = 2, 3, . . .. There is corresponding spectral parameter depending R-matrix obtained by the Yang - Baxterization process. Consequently, it is possible to construct an integrable spin chain . The open spin chain Hamiltonian is the sum of the generators
where Xj act nontrivially on and as the identity matrix on the other factors of H. The aim of this work is to describe the quantum algebra which is the symmetry algebra of such spin system and to show that the structures (categories) of finite dimensional representations of these algebras and coincide. In this case and are each other’s centralizers in the space We consider the general case when the complex parameter is not a root of unity.
Let us note that the relation between TLN(q) and integrable spin chains was actively used in many works and monographs (see for example [13,14,15,16,17,18,19] and the references within). However, the authors used particular realizations of the generators Xj , related to some Lie algebras (or quantum algerbas). Characteristic property of the latter ones was the existence of one-dimensional representation in the decomposition of the tensor product of two fundamental representations Then Xj was proportional to the rank one projector on this subspace, and the symmetry algerba was identified with the choosen algebra. We point out that the symmetry algerba Uq(n) is bigger and its Clebsch - Gordan decomposition of has only two summands similar to the sl(2) case
The dual Hopf algebra was introduced as the quantum group of nondegenerate bilinear form in [20,21]. The categories of co-modules of and their generalisations were studied in [22,23] where it was shown that the categories of co-modules of are equivalent to the category of co-modules of the quantum group SLq(2).
Using the L-operator (1.4) a new set of variables (operators in the space depending on the parameter ¸) is introduced by an ordered product of as 2 × 2 matrices on the auxiliary space according to the QISM -
The entries of the monodromy matrix T(¸) are new variables. The commutation relations of the new operators (A(¸), . . . ,D(¸)) can be obtained from the local relaltion for the L-operator at one site:
where Multiplying the RTT-relation by and taking the trace over two auxiliary spaces one gets commutativity property of transfer matrix t(λ):
The operator B(μ) is a creation operator of the eigenvectors we are looking for. These operators act on a vacuum state (a highest weight vector) Ω. The Hamiltonian is extracted from the transfer matrix t(λ) which is a generating function of mutually commuting integrals of motion. The vector Ω is the tensor product of states corresponding to spin up at each site of the chain:
Using the explicit form of the L-operator and the definition of the monodromy matrix T(λ) it is easy to get the relations
where It follows also from quadratic relation of that
and a similar relation for D(λ) and the product of B(μj ). Sum of these relations acting on the vacuum gives the eigenvector of the transfer matrix t(λ)
under the condition that the parameters μk satisfy the Bethe equations (k = 1, 2, . . . ,M)
The eigenvalue is
Originally these eigenvectors of the XXX spin chain were found by H. Bethe at 1931 as a linear combination of one magnon eigenstates using the local operators
It is easy to see that is an eigenvector of HXXX with the eigenvalue However, the condition of periodicity i.e. the requirement that is also an eigenvector of the shift operator: results in the quantization of z
These yields N − 1 states, m = 1, 2, . . . ,N − 1 (because the state with z = 1 belongs to the vacuum multiplet: Multimagnon states are given by a Bethe sum or (coordinate Bethe Ansatz)
where the coefficients (amplitudes) A(P) are defined by the elements P of permutation group SM, quasimomenta zPj and the two magnon S-matrix 
Constructed eigenstates are highest weight vectors for the global symmetry algebra sl2 with generators
The proof is purely algebraic and it follows from the RTT-relation and the asymptotic of the monodromy matrix 
Both algebras andare quotients of the group algebra of the braid group BN generated by(N − 1) generators their inverses and subject to the relations (see )
The Hecke algebra is obtained by adding to these relations the following characteristic equations obeyed by generators
It is known that is isomorphic to the group algebra Consequently, irreducible representations of the Hecke algebra, as that of SN, are parametrized by Young diagrams. By virtue of (3.2) we can write ˇR using the idempotents
Substituting the expression (3.3) for ˇR in terms of X, into the braid group relations (3.1) one gets relations for
Requiring that each side of (3.4) is zero we obtain the quotient algebra of the Hecke algebra, the Temperley-Lieb algebra It is defined by the generatorsand the relations
The dimension of the Hecke algebra is N!, the same as the dimension of the symmetric group SN, the dimension of is equal to the Catalan number In connection with integrable spin systems we will be interested in representations of on the tensor product space .One representation is defined by an invertible matrix which can also be seen as an n2 dimensional vector . We use the notationand view this matrix also as an n2 dimensional vector The generators Xj can be expressed as
where we explicitely write the indices corresponding to the factors in the tensor product space H. It is easy to see, that the second relation (3.5) is automatically satisfied and the first one determines the parameter
An obvious invariance of the braid group relations (the Yang - Baxter equation) (3.1) in this representation with respect to the transformation of the R-matrix
results in the following transformation of the matrix If one uses an R-matrix depending on a spectral parameter (Yang - Baxterization of
where then relation (3.5) can be written as
In terms of constant R-matrices (generators of this relation has the form
Replacing in (3.9) the expression or in (3.5) substituting X = yields the vanishing of the q-antisymmetriser
Thus the irreducible representations of are parametrized by Young diagrams containing only two rows with N boxes.
The constructed representation (3.3), (3.6) is reducible. The decomposition of this representation into the irreducible ones will be discussed in the next section.
According to the R-matrix approach to the theory of quantum groups , the R-matrix defines relations between the generators of the quantum algebra and its dual Hopf algebra, the quantum group A(R). In this paper the emphasis will be on the quantum algebra and its finite dimensional representations Vk, k = 0, 1, 2, . . . . The generators of can be identified with the L-operator (L-matrix) entries and their exchange relations (commutation relations) follow from the analogue of the Yang-Baxter relations (2.2) (withouht spectral parameter)
where the indices a1 and a2 refer to the representation spaces Va1 and Va2 , respectively, and index q refers to the algebra . Hence the equation (4.1) is given in End
In general the L-operator is defined through the universal R-matrix, where a finite dimensional representation is applied to one of the factors of the universal R-matrix
Thus, choosing the appropriate representation space as the first space, one obtains the co-product of the generators of from the following matrix equation
The case when is of particular interest and it will be presented below in detail. To this end the generators of are denoted byand the L-matrix is given by
Multiplying two L-matrices with entries in the corresponding factors we obtain
or explicitly for the generators
etc. The central element in is obtained from the defining relation (4.1)
However this central element is group-like: It is proportional to the identity in the tensor product of representations. The analogue of the Casimir operator can be obtained according to  using as
c2 is written as
Parameters p and q are related: For the explicit L-operator and its 3 × 3 blocks we get c2 = qIq where Iq is the identity operator on The form of the generators which corresponds to the choice of the b matrix (4.12), follows from the expression for the ˇR and L-matrices where P is the permutation matrix.
For example, we have
If we choose b as in equation (4.12) the R-matrix commutes with where h = diag(1, 0,−1). As a highest spin vector (pseudovacuum of the corresponding integrable spin chain ) we choose [12, 16]. If we act on the tensor product of these vectors with the coproduct of the lowering operators we obtain new vectors. By looking at the explicit forms of the operators in the space we can convince ourselves that the vectors are linearly independent. Together with the vectors and
they span an 8 dimensional subspace. The vector is a linear combination of the vectors The vector spans a one dimensional invariant subspace. Thus we have the following decomposition
This decomposition can also be obtained using the projectors expressed in terms of b matrix (vector) (4.12). Due to the commutativity of the R-matrix with the co-product (4.5), (4.7) the corresponding subspaces are invariant. Similarly, using one can get the decomposition of
This type of decomposition is valid for any n.
The result of applying the co-product Δ, given by (4.5), on the generators of the quantum algebra several times can also be presented in the matrix form
where etc. In general case of the tensor representation of TLN(q), with the spaceat each site, the generators of the algebracommute with the generators (4.16) of the global (diagonal) action of the quantum algebra Uq(n) in the space This follows from the relation
and the possibility due to the co-associativity of the coproduct to write the product of of Laqj as
Thus, the algebras Uq(n) and TLN(q) are each other’s centralizers in the space H. The tensor representation of TLN(q) in H decomposes into irreducible factors whose multiplicities are given by the dimensions of the irreducible representations of the algebra Uq(n), corresponding to the same Young diagrams
In this decomposition the index k parametrizes the Young diagrams with two rows and N boxes and multiplicities are given by the dimensions of the corresponding irreducible representations
of the algebras Uq(n) and TLN(q), respectively. As for the finite dimensional irreducible representations of the Lie algebra sl2, V0(n) = C is the one-dimensional (scalar) representation and the fundamental representation of the algebra Uq(n) is n dimensional, . The dimensions of other representations follow from the trivial multiplicities of the factors in the decomposition of the tensor product of the Vk(n) and the fundamental representation V1(n) into two irreducible factors, as for the ,
Thus, for the dimensions pk(n) = dim Vk(n) the following recurrence relation is valid
with the initial conditions p−1(n) = 0, p0(n) = 1, whose solutions are Chebyshev polynomials of the second kind
The multiplicity or the dimensions of the subspacesin (4.17) is the number of paths that go from the top of the Bratteli diagram to the Young diagram corresponding to the representation Ifis the partition of N, then
The subspaces invariant under the diagonal action of the quantum algebra Uq(n) on the space H, can be obtained using the projectors (idempotents), which can be expressed in terms of the elements of the Temperley-Lieb algebra TLN(q). Using the R-matrix depending on a spectral parameter, the projector P(+) N on the symmetric subspace can be written in the following way [4,27]
This form follows from the intertwiner of four monodromy matrices
where Multiplying by appropriate product of the permutation operators Pk k+1 one can get the expression in terms of the baxterized Hecke generators
Theorem 4.1. Consider the quotient of the Hecke algebra by the ideal I generated by the q-antisymmetrizers P(4) − ,
The tensor product representation of (q) in the space.with the qantisymmetrizers of rank 1 define the quantum algebraas the centralizer algebra of
Let us mention that although the spectrum of the spin chains related to the general Temperley- Lieb R-matrix was found by the fusion procedure and a functional Bethe Ansatz , it would be nice to get the corresponding eigenvectors. Also the subject of reconstructing algebras from their representation ring structure is actively discussed in the literature (see e.g ).
It is a pleasure to thank the organizers for having arranged this nice Baltic-Nordic Workshop. The useful discussions with P. Etingof, A. Mudrov and A. Stolin are highly appreciated. This research was partially supported by RFBR grant 06-01-00451.