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.
In recent years, a new algebraic structure called the q-Onsager algebra (or equivalently the tridiagonal algebra) has emerged in different problems of mathematical physics.
On one side, it appears in the mathematical litterature of P -and Q-polynomial association schemes and their relationship with the Askey scheme of orthogonal polynomials [Zhed,ITTer,GruHa,Ter1,Ter2], related Jacobi matrices and, more generally, certain families of symmetric functions of one variable and related block tridiagonal matrices (see e.g. [Ter3,Bas3]).
On the other side, this algebra appears in several quantum integrable systems. Playing a crucial role at q = 1 in the exact solution of the planar Ising [Ons] and superintegrable Potts model [voGR], it also finds applications in solving the XXZ open spin chain with non-diagonal boundary parameters and generic deformation parameter q. Indeed, the transfer matrix of this model has been shown to admit an expansion in terms of the elements of the q-Onsager algebra [BasK0,BasK1] acting on some finite dimensional representation. As a consequence, the solution of the model i.e. the complete spectrum and eigenstates can be derived using solely its representation theory, bypassing the Bethe ansatz approach which does not apply in the generic regime of parameters [BasK2]. Appart from lattice models, in quantum field theory the q-Onsager algebra is known to be the hidden non-Abelian symmetry of the boundary sine-Gordon model [Bas1,Bas2].
By definition, the q-Onsager algebra is an associative algebra with unity generated by two elements (called the standard generators), say A 0 , A 1 . Introducing the q-commutator 1 X, Y q = XY -qY X, the fundamental (sometimes called q-Dolan-Grady) relations take the form
where q is a deformation parameter (assumed to be not a root of unity) and ρ 0 , ρ 1 are fixed scalars. Note that for ρ 0 = ρ 1 = 0 this algebra reduces to the q-Serre relations of U q ( sl 2 ), and for q = 1, ρ 0 = ρ 1 = 16 it leads to the Onsager algebra [Ons, Per] defined by the Dolan-Grady relations [DoG].
1 For further convenience, definitions for the parameter q and the q-commutator chosen here differ compared to [Bas3,BasK0,BasK1,BasK2].
Similarly to the well-established relationship between the Onsager algebra and the affine Lie algebra sl 2 [Dav, DaRo], the q-Onsager algebra (1.1) is actually closely related with the U q ( sl 2 ) algebra, a fact that may be also expected from the structure of the l.h.s. of (1.1) compared with the q-Serre relations of U q ( sl 2 ). Indeed, examples of algebra homomorphisms for the standard generators A 0 , A 1 have been proposed for ρ 0 = 0, ρ 1 = 0, and related finite dimensional representations studied in details. We refer the reader to [ITer1,Bas2,AlCu,ITer2] for details. In particular, the following realization immediately follows from [Bas2]:
where 2 {h i , e i , f i } denote the generators of U q ( sl 2 ) and one identifies ρ i = c i c i (q + q -1 ) 2 for i = 0, 1. Thanks to the Hopf algebra structure of U q ( sl 2 ), finite dimensional representations have been studied in details (see for instance [Bas3,ITer2]). In addition, a new type of current algebra has been recently derived [BasS1] which rigorously establishes the isomorphism between the reflection equation algebra associated with U q ( sl 2 ) R-matrices and the q-Onsager algebra (1.1).
In the context of quantum integrable systems, the elements A 0 , A 1 take the form of non local operators on the lattice or continuum. According to the model and objective considered, they are used either to eventually derive second order difference equations fixing the spectrum of the model [BasK2], or the complete set of scattering amplitudes of the fundamental particles [MN98,DeM,BasK3].
In view of all these results, finding an analogue of the deformed relations (1.1) that may be related to higher rank affine Lie algebras in a similar manner, as well as considering potential implications for quantum integrable systems with extended symmetries seems to be a rather interesting problem. In the undeformed case q = 1, a step towards this direction has been made by Uglov and Ivanov who introduced the so-called sl n -Onsager’s algebra for n ≥ 2. However, to our knowledge since these results no further progress in this direction were ever published.
In the present letter, we remedy this situation. Namely, to each affine Lie algebra (of classical or exceptional type) g we associate a q-Onsager algebra denoted O q ( g). Then, by analogy with the sl 2 case, we propose an algebra homomorphism from O q ( g) to the coideal subalgebra 3 of U q ( g) generalizing (1.2). Applications to boundary quantum affine Toda field theories introduced in [FrK, BCDRS] -with soliton non-preserving boundary conditions -are then considered. Despite of the fact that defining relations of the underlying hidden symmetry in these models were not known up to now (except for the sine-Gordon model [Bas1,Bas2]), the explicit knowle
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