Search

We study solutions of the reflection equation related to the quantum affine algebra U q ( s l n ^) . First, we explain how to construct a family of stochastic integrable vertex models with fixed boundary conditions. Then, we construct upper- and lower-triangular solutions of the reflection equation related to symmetric tensor representations of U q ( s l n ^) with arbitrary spin. We also prove the star–star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained. [ABSTRACT FROM AUTHOR]

The integrable structure of the one-dimensional Hubbard model is based on Shastry's R-matrix and the Yangian of a centrally extended sl(2|2) superalgebra. Alcaraz and Bariev have shown that the model admits an integrable deformation whose R-matrix has recently been found. This R-matrix is of trigonometric type and here we derive its underlying exceptional quantum affine algebra. We also show how the algebra reduces to the above-mentioned Yangian and to the conventional quantum affine sl(2|2) algebra in two special limits. [ABSTRACT FROM AUTHOR]

We analyze the construction and origin of unitary operators describing the structure of the space of continuous wavelet transforms inside the space L2(G, dνL) of all square-integrable functions on the affine groupG with respect to the left-invariant Haar measure from the viewpoint of induced representations of G. We show that these operators are, in fact, intertwining operators among pairs of induced representations of the affine group G. A characterization of the space of wavelet transforms using the Cauchy-Riemann-type equations is given. [ABSTRACT FROM AUTHOR]

In this paper, we study an infinite-dimensional Lie algebra of Virasoro-type which is realized as an affinization of a two-dimensional Novikov algebra. It is a special deformation of the Lie algebra of differential operators on a circle of order at most 1. There is an explicit construction of a vertex algebra associated with the Lie algebra. We determine all derivations of this Lie algebra in terms of some derivations and centroids of the corresponding Novikov algebra. The universal central extension of this Lie algebra is also determined. [ABSTRACT FROM AUTHOR]

We study the Akgeneralized model of the O(1) loop model on a cylinder. The affine Hecke algebra associated with the model is characterized by a vanishing condition, the cylindric relation. We present two representations of the algebra: the first one is the spin representation, and the other is in the vector space of states of the Akgeneralized model. A state of the model is a natural generalization of a link pattern. We propose a new graphical way of dealing with the Yang-Baxter equation and q-symmetrizers by the use of the rhombus tiling. The relation between two representations and the meaning of the cylindric relations are clarified through the rhombus tiling. The sum rule for this model is obtained by solving the q-KZ equation at the Razumov-Stroganov point. [ABSTRACT FROM AUTHOR]

We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra in the fundamental representations and in the spin representations. They reproduce the reflection K matrices obtained recently by the matrix product construction connected to the three dimensional integrability. In particular the present approach provides the first proof of the reflection equation for the non type A cases. [ABSTRACT FROM AUTHOR]

We study solutions of the reflection equation associated with the quantum affine algebra and obtain diagonal K-operators in terms of the Cartan elements of a quotient of Uq(gl(N)). We also consider intertwining relations for these K-operators and find an augmented q-Onsager algebra like symmetry behind them. [ABSTRACT FROM AUTHOR]

In this paper we construct a new factorized representation of the R-matrix related to the affine algebra for symmetric tensor representations with arbitrary weights. Using the 3D approach we obtain explicit formulas for the matrix elements of the R-matrix and give a simple proof that a ‘twisted’ R-matrix is stochastic. We also discuss symmetries of the R-matrix, its degenerations and compare our formulas with other results available in the literature. [ABSTRACT FROM AUTHOR]

We identify the algorithm for constructing steady states of the n-species totally asymmetric simple exclusion process (TASEP) on an L site periodic chain by Ferrari and Martin with a composition of combinatorial R for the quantum affine algebra in crystal base theory. Based on this connection and the factorized form of the R matrix derived recently from the tetrahedron equation, we establish a new matrix product formula for the steady state of the TASEP, which is expressed in terms of corner transfer matrices of the q-oscillator valued five-vertex model at q = 0. [ABSTRACT FROM AUTHOR]

The concept of fusing rules for defects in the affine Toda field theories is introduced in the classical and quantum contexts. This idea is employed in finding a new transmission matrix for the theory which obeys the triangle relations and the crossing-unitarity relations. [ABSTRACT FROM AUTHOR]

It was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them; at a given level, this also applies to the fusion multiplicities of affine algebras. Here, we show that, in the case of SU(3), the lists of multiplicities, in the tensor products and , are identical up to permutations. This latter property does not hold in general for other Lie algebras. We conjecture that the same property should hold for the fusion product of the affine algebra of SU(3) at finite levels, but this is not investigated in the present paper. [ABSTRACT FROM AUTHOR]

We generalize the concept of unbiasedness from bases to measurements. We show that mutually unbiased (MU) measurements, which are not necessarily von Neumann measurements, exist in all finite-dimensional Hilbert spaces. We study the geometrical relation between these measurements and symmetric informationally complete measurements. In particular we find that these two kinds of measurements are related to each other as points and lines in finite affine plane geometries. This relation provides a natural way to define discrete phase-space functions based on MU measurements and symmetric informationally complete measurements. [ABSTRACT FROM AUTHOR]

We consider boundary scattering for a semi-infinite one-dimensional deformed Hubbard chain with boundary conditions of the same type as for the Y = 0 giant graviton in the AdS/CFT correspondence. We show that the recently constructed quantum affine algebra of the deformed Hubbard chain has a co-ideal subalgebra which is consistent with the reflection (boundary Yang-Baxter) equation. We derive the corresponding reflection matrix and furthermore show that the aforementioned algebra in the rational limit specializes to the (generalized) twisted Yangian of the Y = 0 giant graviton. [ABSTRACT FROM AUTHOR]

A procedure to construct K-matrices from the generalized q-Onsager algebra $\mathcal O_{q}(\widehat{g})$ 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 q-Onsager algebras. These dynamical K-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 $\mathcal O_{q}\big(a^{(2)}_{2}\big)$ and $\mathcal O_{q}\big(a^{(1)}_{2}\big)$ are treated in detail. [ABSTRACT FROM AUTHOR]

The construction of a nonautonomous mixed mKdV/sine-Gordon model is proposed by employing an infinite-dimensional affine Lie algebraic structure within the zero-curvature representation. A systematic construction of soliton solutions is provided by an adaptation of the dressing method which takes into account arbitrary time-dependent functions. A particular choice of those arbitrary functions provides an interesting solution describing the transition of a pure mKdV system into a pure sine-Gordon soliton. [ABSTRACT FROM AUTHOR]

We construct a realization of the elliptic quantum algebra U_{q,p}(\widehat{sl_N}) for any given level k in terms of free boson fields and their twisted partners. It can be considered as the elliptic deformation of the Wakimoto realization of the quantum affine algebra U_{q}(\widehat{sl_N}) . We also construct a family of screening currents, which commute with the currents of U_{q,p}(\widehat{sl_N}) up to total q-differences. And we give explicit twisted expressions for the type I and type II vertex operators of U_{q,p}(\widehat{sl_N}) by twisting the known results of the type I vertex operators of the quantum affine algebra U_{q}(\widehat{sl_N}) and the new results of the type II vertex operators of U_{q}(\widehat{sl_N}) we obtained in this paper. [ABSTRACT FROM AUTHOR]

We classify regular subalgebras of Kac-Moody algebras in terms of their root systems. In the process, we establish that a root system of a subalgebra is always an intersection of the root system of the algebra with a sublattice of its root lattice. We also discuss applications to investigations of regular subalgebras of hyperbolic Kac-Moody algebras and conformally invariant subalgebras of affine Kac-Moody algebras. In particular, we provide explicit formulae for determining all Virasoro charges in coset constructions that involve regular subalgebras. [ABSTRACT FROM AUTHOR]

This paper develops the theory of affine Lie-Poisson reduction and applies this process to Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids. As a consequence of this approach, the associated Poisson brackets are obtained by reduction of a canonical cotangent bundle. A Kelvin-Noether circulation theorem is presented and is applied to these examples. [ABSTRACT FROM AUTHOR]

We generalized a recent observation (Khoroshkin and Pakuliak 2005 Theor. Math. Phys. 145 1373) that the partition function of the six-vertex model with domain wall boundary conditions can be obtained from a calculation of projections of the product of total currents in the quantum affine algebra U_{q}(\widehat{\mathfrak{sl}}_{2}) in its current realization. A generalization is done for the elliptic current algebra (Enriquez and Felder 1998 Commun. Math. Phys. 195 651, Enriquez and Rubtsov 1997 Ann. Sci. Ecole Norm. Sup. 30 821). The projections of the product of total currents in this case are calculated explicitly and are presented as integral transforms of a product of the total currents. It is proved that the integral kernel of this transform is proportional to the partition function of the SOS model with domain wall boundary conditions. [ABSTRACT FROM AUTHOR]

Within the quantum affine algebra representation theory, we construct linear covariant operators that generate the Askey-Wilson algebra. It has the property of a coideal subalgebra, which can be interpreted as the boundary symmetry algebra of a model with quantum affine symmetry in the bulk. The generators of the Askey-Wilson algebra are implemented to construct an operator-valued K-matrix, a solution of a spectral-dependent reflection equation. We consider the open driven diffusive system where the Askey-Wilson algebra arises as a boundary symmetry and can be used for an exact solution of the model in the stationary state. We discuss the possibility of a solution beyond the stationary state on the basis of the proposed relation of the Askey-Wilson algebra to the reflection equation. [ABSTRACT FROM AUTHOR]