We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians Y (gl(m|n) in terms of the current generators of the Yangian double DY (gl(m|n)). We use the method of projections onto intersections of different type Borel subalgebras in this infinite dimensional algebra to construct the Bethe vectors. Calculating these projection the supersymmetric Bethe vectors can be expressed through matrix elements of the universal monodromy matrix elements. Using two different but isomorphic current realizations of the Yangian double DY (gl(m|n)) we obtain two different presentations for the Bethe vectors. These Bethe vectors are also shown to obey some recursion relations which prove their equivalence.

We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra U q ( gl m ). We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, whose factors are characterized by two highest coefficients. We provide different recursions for these highest coefficients.In addition, we show that when the Bethe vectors are on-shell, their norm takes the form of a Gaudin determinant.

We study scalar products of Bethe vectors in the models solvable by the nested algebraic Bethe ansatz and described by gl(m|n) superalgebra. Using coproduct properties of the Bethe vectors we obtain a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of Bethe parameters. We also obtain recursions for the Bethe vectors. This allows us to find recursions for the highest coefficient of the scalar product.

Dedicated to the memory of P.P. Kulish AbstractWe study integrable models with gl(2|1) symmetry and solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for scalar products of Bethe vectors, when the Bethe parameters obey some relations weaker than the Bethe equations. This representation allows us to find the norms of on-shell Bethe vectors and obtain determinant formulas for form factors of the diagonal entries of the monodromy matrix.

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl 3 -invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(N )-invariant R-matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of the Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products of the Bethe vectors. The q-deformed case, as well as the superalgebra case, are also evoked in the conclusion. 1 a.

Dedicated to the memory of P.P. Kulish AbstractWe study scalar products of Bethe vectors in integrable models solvable by nested algebraic Bethe ansatz and possessing gl(2|1) symmetry. Using explicit formulas of the monodromy matrix entries multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors.As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors in integrable models with gl(1|1) symmetry.

Abstract. We study gl(2|1) symmetric integrable models solvable by the nested algebraic Bethe ansatz. Using explicit formulas for the Bethe vectors we derive the actions of the monodromy matrix entries onto these vectors. We show that the result of these actions is a finite linear combination of Bethe vectors. The obtained formulas open a way for studying scalar products of Bethe vectors.