arXiv:2607.00594

Yangian Doubles and off-Shell Bethe Vectors

Andrii Nikolaevich Liashyk, S. Pakuliak, E. Ragoucy

Published Jul 1, 2026• 0 citations

Abstract

Off-shell Bethe vectors for a generic $\fg$ invariant integrable model are constructed through the currents of the Yangian doubles of the classical series. These off-shell Bethe vectors are shown to satisfy the defining properties which were used in \cite{LPR-RR} to prove the rectangular recurrence relations and verify the eigenvalue property of the on-shell Bethe vectors in $\ggo$-invariant integrable models.

DOI: 10.48550/arxiv.2607.00594

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arXiv:2607.00594

Yangian Doubles and off-Shell Bethe Vectors

Andrii Nikolaevich Liashyk, S. Pakuliak, E. Ragoucy

Published Jul 1, 2026• 0 citations

arXiv HTML PDF

Abstract

Off-shell Bethe vectors for a generic

\fg

invariant integrable model are constructed through the currents of the Yangian doubles of the classical series. These off-shell Bethe vectors are shown to satisfy the defining properties which were used in \cite{LPR-RR} to prove the rectangular recurrence relations and verify the eigenvalue property of the on-shell Bethe vectors in

\ggo

-invariant integrable models.

DOI: 10.48550/arxiv.2607.00594

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Andrii Nikolaevich Liashyk, Stanislav Zdislavovich Pakuliak, E. Ragoucy +1 more

We consider quantum integrable models associated with the $\mathfrak{so}_3$ algebra and describe Bethe vectors of these models in terms of the current generators of the $\mathcal{D}Y(\mathfrak{so}_3)$ algebra. To implement this program, we use an isomorphism between the R-matrix and the Drinfeld current realizations of the Yangians and their doubles for classical type B-, C-, and D-series algebras. Using these results, we derive the actions of the monodromy matrix elements on off-shell Bethe vectors. We obtain recurrence relations for off-shell Bethe vectors and Bethe equations for on-shell Bethe vectors. The formulas for the action of the monodromy matrix elements can also be used to calculate scalar products in the models associated with the $\mathfrak{so}_3$ algebra.We consider quantum integrable models associated with the $$\mathfrak{so}_3$$ algebra and describe Bethe vectors of these models in terms of the current generators of the $$\mathcal{D}Y(\mathfrak{so}_3)$$ algebra. To implement this program, we use an isomorphism between the R-matrix and the Drinfeld current realizations of the Yangians and their doubles for classical type B-, C-, and D-series algebras. Using these results, we derive the actions of the monodromy matrix elements on off-shell Bethe vectors. We obtain recurrence relations for off-shell Bethe vectors and Bethe equations for on-shell Bethe vectors. The formulas for the action of the monodromy matrix elements can also be used to calculate scalar products in the models associated with the $$\mathfrak{so}_3$$ algebra.

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A new method is introduced to derive general recurrence relations for off-shell Bethe vectors in quantum integrable models with either type $\mathfrak{gl}_n$ or type $\mathfrak{o}_{2n+1}$ symmetries. These recurrence relations describe how to add a single parameter $z$ to specific subsets of Bethe parameters, expressing the resulting Bethe vector as a linear combination of monodromy matrix entries that act on Bethe vectors which do not depend on $z$ . We refer to these recurrence relations as rectangular because the monodromy matrix entries involved are drawn from the upper-right rectangular part of the matrix. This construction is achieved within the framework of the zero mode method.A new method is introduced to derive general recurrence relations for off-shell Bethe vectors in quantum integrable models with either type $\mathfrak{gl}_n$ or type $\mathfrak{o}_{2n+1}$ symmetries. These recurrence relations describe how to add a single parameter $z$ to specific subsets of Bethe parameters, expressing the resulting Bethe vector as a linear combination of monodromy matrix entries that act on Bethe vectors which do not depend on $z$. We refer to these recurrence relations as rectangular because the monodromy matrix entries involved are drawn from the upper-right rectangular part of the matrix. This construction is achieved within the framework of the zero mode method.

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arXiv:2008.03664Theoretical and Mathematical PhysicsJan 1, 20218 citations

Off-shell Bethe vectors for a generic

\fg

\ggo

-invariant integrable models.

We consider quantum integrable models associated with the

\mathfrak{so}_3

algebra and describe Bethe vectors of these models in terms of the current generators of the

\mathcal{D}Y(\mathfrak{so}_3)

algebra. To implement this program, we use an isomorphism between the R-matrix and the Drinfeld current realizations of the Yangians and their doubles for classical type B-, C-, and D-series algebras. Using these results, we derive the actions of the monodromy matrix elements on off-shell Bethe vectors. We obtain recurrence relations for off-shell Bethe vectors and Bethe equations for on-shell Bethe vectors. The formulas for the action of the monodromy matrix elements can also be used to calculate scalar products in the models associated with the

\mathfrak{so}_3

algebra.

Rectangular Recurrence Relations in

\mathfrak{gl}_{n}

and

\mathfrak{o}_{2n+1}

Invariant Integrable Models

A new method is introduced to derive general recurrence relations for off-shell Bethe vectors in quantum integrable models with either type

\mathfrak{gl}_n

or type

\mathfrak{o}_{2n+1}

symmetries. These recurrence relations describe how to add a single parameter

z

to specific subsets of Bethe parameters, expressing the resulting Bethe vector as a linear combination of monodromy matrix entries that act on Bethe vectors which do not depend on

z

. We refer to these recurrence relations as rectangular because the monodromy matrix entries involved are drawn from the upper-right rectangular part of the matrix. This construction is achieved within the framework of the zero mode method.

Bethe vectors in integrable models based on the super-Yangian

Y(\mathfrak{gl}(m|n))

We find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians

Y(\mathfrak{gl}(m|n)

in terms of the current generators of the Yangian double

DY(\mathfrak{gl}(m|n))

. More specifically, 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(\mathfrak{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 find Bethe vectors for quantum integrable models associated with the supersymmetric Yangians

Y(\mathfrak{gl}(m|n)

in terms of the current generators of the Yangian double

DY(\mathfrak{gl}(m|n))

DY(\mathfrak{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.

Algebraic Bethe ansatz for

\mathfrak o_{2n+1}

-invariant integrable models

We study the class of

\mathfrak{o} _{2n+1}

-invariant quantum integrable models in the framework of the algebraic Bethe ansatz and propose a construction of the

\mathfrak{o} _{2n+1}

-invariant Bethe vector in terms of the Drinfeld currents for the Yangian double

\mathcal{D}Y ( \mathfrak{o} _{2n+1})

. We calculate the action of the monodromy matrix elements on the off-shell Bethe vectors for these models and obtain recurrence relations for these vectors. The action formulas can be used to investigate scalar products of Bethe vectors in

\mathfrak{o} _{2n+1}

-invariant models.