Multiple Actions of the Monodromy Matrix in gl(2|1)-Invariant Integrable Models

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

New symmetries of gl(N)\mathfrak{gl}(N)-invariant Bethe vectors

International audienceWe consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing -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 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

Bethe vectors for orthogonal integrable models

International audienceWe consider quantum integrable models associated with the $\mathfrak{so}_3$ algebr

Norm of Bethe vectors in models with gl(m|n) symmetry

We study quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(m|n)-invariant R-matrix. We compute the norm of the Hamiltonian eigenstates. Using the notion of a generalized model we show that the square of the norm obeys a number of properties that uniquely fix it. We also show that a Jacobian of the system of Bethe equations obeys the same properties. In this way we prove a generalized Gaudin hypothesis for the norm of the Hamiltonian eigenstates

Current presentation for the double super-Yangian DY(gl(m|n)) and Bethe vectors

International audienceWe find Bethe vectors for quantum integrable models associated with thesupersymmetric Yangians $Y(\mathfrak{gl}(m|n)$ in terms of the currentgenerators of the Yangian double $DY(\mathfrak{gl}(m|n))$. More specifically,we use the method of projections onto intersections of different type Borelsubalgebras in this infinite dimensional algebra to construct the Bethevectors. Calculating these projection the supersymmetric Bethe vectors can beexpressed through matrix elements of the universal monodromy matrix elements.Using two different but isomorphic current realizations of the Yangian double$DY(\mathfrak{gl}(m|n))$ we obtain two different presentations for the Bethevectors. These Bethe vectors are also shown to obey some recursion relationswhich prove their equivalence

Scalar products of Bethe vectors in the models with gl(m∣n)\mathfrak{gl}(m|n) symmetry

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

Actions of the monodromy matrix elements onto gl(m∣n)\mathfrak{gl}(m|n)-invariant Bethe vectors

International audienceMultiple actions of the monodromy matrix elements onto off-shell Bethe vectors in the -invariant quantum integrable models are calculated. These actions are used to describe recursions for the highest coefficients in the sum formula for the scalar product. For simplicity, detailed proofs are given for the case. The results for the supersymmetric case can be obtained similarly and are formulated without proofs