15 research outputs found
A note on the algebra
An explicit homomorphism that relates the elements of the infinite
dimensional non-Abelian algebra generating currents and the
standard generators of the 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 Onsager
spectrum generating algebra, the infinite 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 orthogonal polynomials are proposed.Comment: 11 page
The half-infinite XXZ chain in Onsager's approach
The half-infinite XXZ open spin chain with general integrable boundary
conditions is considered within the recently developed `Onsager's approach'.
Inspired by the finite size case, for any type of integrable boundary
conditions it is shown that the transfer matrix is simply expressed in terms of
the elements of a new type of current algebra recently introduced. In the
massive regime , level one infinite dimensional representation
(vertex operators) of the new current algebra are constructed in order to
diagonalize the transfer matrix. For diagonal boundary conditions, known
results of Jimbo {\it et al.} are recovered. For upper (or lower) non-diagonal
boundary conditions, a solution is proposed. Vacuum and excited states are
formulated within the representation theory of the current algebra using
bosons, opening the way for the calculation of integral representations of
correlation functions for a non-diagonal boundary. Finally, for generic the
long standing question of the hidden non-Abelian symmetry of the Hamiltonian is
solved: it is either associated with the Onsager algebra (generic
non-diagonal case) or the augmented Onsager algebra (generic diagonal
case).Comment: 28 pages; Presentation improved; misprints corrected; to appear in
Nucl. Phys.
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 , 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.Comment: 13 pages; to appear in Lett. Math. Phy
Central extension of the reflection equations and an analog of Miki's formula
Two different types of centrally extended quantum reflection algebras are
introduced. Realizations in terms of the elements of the central extension of
the Yang-Baxter algebra are exhibited. A coaction map is identified. For the
special case of , a realization in terms of elements
satisfying the Zamolodchikov-Faddeev algebra - a `boundary' analog of Miki's
formula - is also proposed, providing a free field realization of
(q-Onsager) currents.Comment: 11 pages; two references added; to appear in J. Phys.
Generalized q-Onsager Algebras and Dynamical K-matrices
A procedure to construct -matrices from the generalized -Onsager
algebra \cO_{q}(\hat{g}) is proposed. This procedure extends the intertwiner
techniques used to obtain scalar (c-number) solutions of the reflection
equation to dynamical (non-c-number) solutions. It shows the relation between
soliton non-preserving reflection equations or twisted reflection equations and
the generalized -Onsager algebras. These dynamical -matrices are
important to quantum integrable models with extra degrees of freedom located at
the boundaries: for instance, in the quantum affine Toda field theories on the
half-line they yield the boundary amplitudes. As examples, the cases of
\cO_{q}(a^{(2)}_{2}) and \cO_{q}(a^{(1)}_{2}) are treated in details
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.Comment: 40 pages; examples of Bethe vectors added; Bethe equations for
U_q(gl(2/2)) added; misprints correcte