Abstract. The investigation is continued of the universal weight function for the quantum affine algebra U q ( p gl N ). Two recurrence relations are obtained for the universal weight function with the help of the method of projections. On the level of the evaluation representation of U q ( p gl N ), two recurrence relations are reproduced, which were calculated earlier for the off-shell Bethe vectors by combinatorial methods. One of the results of the paper is a description of two different but isomorphic currents or "new" realizations of the algebra U q ( p gl N ), corresponding to two different Gauss decompositions of the fundamental L-operators. §1. IntroductionThe hierarchical (nested) Bethe ansatz was designed in [13] to construct the eigenvectors of the commuting integrals for quantum integrable models associated with the Lie algebra gl N . It is based on the inductive procedure that relates gl N and gl N −1 Bethe vectors. Since the Bethe vectors for models with gl 2 symmetry are known, this hierarchical procedure yields an implicit description of the Bethe vectors of the models with symmetry of higher rank.If the parameters of these vectors satisfy the Bethe equations, then the corresponding vectors are eigenvectors of a commuting set of operators in some quantum integrable model. In the current paper, we consider Bethe vectors with free parameters, the socalled "off-shell Bethe vectors".Further development of the off-shell Bethe vectors theory was achieved in [17], where they were presented as particular matrix elements of monodromy operators. This construction was used in [18] to obtain explicit formulas for the off-shell Bethe vectors on the tensor product of evaluation modules of U q ( p gl N ). Two different recurrence relations for the off-shell Bethe vectors on the evaluation U q ( p gl N )-modules were obtained in [18]. Iteration of these two relations allows us to obtain different explicit formulas for the off-shell Bethe vectors (see examples (2.18) and (2.20) below). The existence of two types of recurrence relations in the nested Bethe ansatz is a consequence of two different ways of embedding. When these algebras are realized in terms of L-operators, the L-operator of U q ( p gl N −1 ) can be placed either into the top-left or into the down-right corner of the U q ( p gl N ) L-operator. Due to applications to the theory of quantized Knizhnik-Zamolodchikov equations, the off-shell Bethe vectors are called weight functions. We use both names for these objects. We say that a weight function is universal if it is defined in an arbitrary U q ( p gl N )-module generated by an arbitrary singular vector.2010 Mathematics Subject Classification. Primary 81R10.
We give a new construction of primitive idempotents of the Hecke algebras associated with the symmetric groups. The idempotents are found as evaluated products of certain rational functions thus providing a new version of the fusion procedure for the Hecke algebras. We show that the normalization factors which occur in the procedure are related to the Ocneanu-Markov trace of the idempotents.
Abstract. We consider the integrable open chain models formulated in terms of generators of the Hecke algebra. The spectrum of the Hamiltonians for the open Hecke chains of finite size with free boundary conditions is deduced for special (corner type) irreducible representations of the Hecke algebra.
We consider integrable open-chain models formulated in terms of generators of Hecke algebra. The algebraic analogs of Sklyanin's commutative transfer matrices are considered for these models. The spectrum of Hamiltonians for open Hecke chains of finite size is deduced for special representations of the Hecke algebra.
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