arXiv subjects

S. Belliard

Publications and source records attributed to S. Belliard.

23 records · Page 2Linked to original sources

arXiv2010

A note on the $O_q(\hat{sl_2})$ algebra

An explicit homomorphism that relates the elements of the infinite dimensional non-Abelian algebra generating $O_q(\hat{sl_2})$ currents and the standard generators of the $q-$Onsager algebra is proposed. Two straightforward applications of the result are then considered: First, for the class of quantum integrable models which integrability condition originates in the $q-$Onsager spectrum generating algebra, the infinite $q-$deformed Dolan-Grady hierarchy is derived - bypassing the transfer matrix formalism. Secondly, higher Askey-Wilson relations that arise in the study of symmetric special functions generalizing the Askey-Wilson $q-$orthogonal polynomials are proposed.

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arXiv2010

Generalized q-Onsager algebras and boundary affine Toda field theories

Generalizations of the q-Onsager algebra are introduced and studied. In one of the simplest case and q=1, the algebra reduces to the one proposed by Uglov-Ivanov. In the general case and $q\neq 1$, an explicit algebra homomorphism associated with coideal subalgebras of quantum affine Lie algebras (simply and non-simply laced) is exhibited. Boundary (soliton non-preserving) integrable quantum Toda field theories are then considered in light of these results. For the first time, all defining relations for the underlying non-Abelian symmetry algebra are explicitely obtained. As a consequence, based on purely algebraic arguments all integrable (fixed or dynamical) boundary conditions are classified.

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arXiv2009

Scattering matrix for a general gl(2) spin chain

We study the general L_0-regular gl(2) spin chain, i.e. a chain where the sites {i,i+L_0,i+2L_0,...} carry the same arbitrary representation (spin) of gl(2). The basic example of such chain is obtained for L_0=2, where we recover the alternating spin chain. Firstly, we review different known results about their integrability and their spectrum. Secondly, we give an interpretation in terms of particles and conjecture the scattering matrix between them.

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arXiv2009

Nested Bethe ansatz for `all' open spin chains with diagonal boundary conditions

We present in an unified and detailed way the nested Bethe ansatz for open spin chains based on Y(gl(\fn)), Y(gl(\fm|\fn)), U_{q}(gl(\fn)) or U_{q}(gl(\fm|\fn)) (super)algebras, with arbitrary representations (i.e. `spins') on each site of the chain and diagonal boundary matrices (K^+(u),K^-(u)). The nested Bethe anstaz applies for a general K^-(u), but a particular form of the K^+(u) matrix. The construction extends and unifies the results already obtained for open spin chains based on fundamental representation and for some particular super-spin chains. We give the eigenvalues, Bethe equations and the form of the Bethe vectors for the corresponding models. The Bethe vectors are expressed using a trace formula.

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arXiv2008

Nested Bethe ansatz for "all" closed spin chains

We present in an unified and detailed way the Nested Bethe Ansatz for closed spin chains based on Y(gl(n)), Y(gl(m|n)), U_q(gl(n)) or U_q(gl(m|n)) (super)algebras, with arbitrary representations (i.e. `spins') on each site of the chain. In particular, the case of indecomposable representations of superalgebras is studied. The construction extends and unifies the results already obtained for spin chains based on Y(gl(n)) or U_q(gl(n)) and for some particular super-spin chains. We give the Bethe equations and the form of the Bethe vectors. The case of gl(2|1), gl(2|2$ and gl(4|4) superalgebras (that are related to AdS/CFT correspondence) is also detailed.

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