Pavel Pyatov

Magic in the spectra of the XXZ quantum chain with boundaries at Δ=0 and Δ=−1/2

Abstract We show that from the spectra of the U q ( sl ( 2 ) ) symmetric XXZ spin- 1 / 2 finite quantum chain at Δ = − 1 / 2 ( q = e π i / 3 ) one can obtain the spectra of certain XXZ quantum chains with diagonal and non-diagonal boundary conditions. Similar observations are made for Δ = 0 ( q = e π i / 2 ). In the finite-size scaling limit the relations among the various spectra are the result of identities satisfied by known character functions. For the finite chains the origin of the remarkable spectral identities can be found in the representation theory of one and two boundaries Temperley–Lieb algebras at exceptional points. Inspired by these observations we have discovered other spectral identities between chains with different boundary conditions.

On quantum matrix algebras satisfying the Cayley - Hamilton - Newton identities

The Cayley - Hamilton - Newton identities which generalize both the characteristic identity and the Newton relations have been recently obtained for the algebras of the RTT-type. We extend this result to a wider class of algebras defined by a pair of compatible solutions of the Yang - Baxter equation. This class includes the RTT algebras as well as the reflection equation algebras.

Generalized Cayley-Hamilton-Newton identities

The q-generalizations of the two fundamental statements of matrix algebra — the Cayley-Hamilton theorem and the Newton relations — to the cases of quantum matrix algebras of RTT and Reflection-equation types have been obtained. We construct a family of matrix identities which we call Cayley-Hamilton-Newton identities and which underlie the characteristic identity as well as the Newton relations for the RTT and Reflection equation algebras, in the sense that both the characteristic identity and the Newton relations are direct consequences of the Cayley-Hamilton-Newton identities.

Characteristic relations for quantum matrices

The general algebraic properties of the algebras of vector fields over the quantum linear groups GLq(N) and SLq(N) are studied. These quantum algebras appear to be quite similar to the classical matrix algebra. In particular, the quantum analogues of the characteristic polynomial and characteristic identity are obtained for them. The q-analogues of the Newton relations connecting two different generating sets of central elements of these algebras (the determinant-like and trace-like ones) are derived. This allows one to express the q-determinant of quantized vector fields in terms of their q-traces.

GLq(N)-covariant quantum algebras and covariant differential calculus

We consider GLq(N)-covariant quantum algebras with generators satisfying quadratic polynomial relations. We show that, up to some inessential arbitrariness, there are only two kinds of such quantum algebras, namely, the algebras with q-deformed commutation and q-deformed anticommutation relations. The connection with the bicovariant differential calculus on the linear quantum groups is disscussed.Abstract We consider GL q ( N )-covariant quantum algebras with generators satisfying quadratic polynomial relations. We show that, up to some inessential arbitrariness, there are only two kinds of such quantum algebras, namely, the algebras with q -deformed commutation and q -deformed anticommutation relations. The connection with the bicovariant differential calculus on the linear quantum groups is discussed.

Quantum matrix algebra for the SU(n) WZNW model

The zero modes of the chiral SU (n) WZNW model give rise to an intertwining quantum matrix algebra A generated by an n x n matrix a = (a i α ), i, α = 1,....,n (with noncommuting entries) and by rational functions of n commuting elements q P 1 satisfying Π n i = 1 q p i = 1, q p i a j α = a j α q p i + δ j i - . We study a generalization of the Fock space (F) representation of A for generic q (q not a root of unity) and demonstrate that it gives rise to a model of the quantum universal enveloping algebra U q = U q (sl n ), with each irreducible representation entering F with multiplicity 1. For an integer su (n) height h (= k + n ≥ n) the complex parameter q is an even root of unity, q h = -1, and the algebra A has an ideal I h such that the factor algebra A n = A/I h is finite dimensional. All physical Uq modules-of shifted weights satisfying p 1 n ≡ p 1 - p n < h-appear in the Fock representation of A h .

Hecke algebraic properties of dynamical R-matrices. Application to related quantum matrix algebras

The quantum dynamical Yang–Baxter (or Gervais–Neveu–Felder) equation defines an R-matrix R(p), where p stands for a set of mutually commuting variables. A family of SL(n)-type solutions of this equation provides a new realization of the Hecke algebra. We define quantum antisymmetrizers, introduce the notion of quantum determinant and compute the inverse quantum matrix for matrix algebras of the type R(p)a1a2=a1a2R. It is pointed out that such a quantum matrix algebra arises in the operator realization of the chiral zero modes of the WZNW model.The quantum dynamical Yang–Baxter (or Gervais–Neveu–Felder) equation defines an R-matrix R̂(p) , where p stands for a set of mutually commuting variables. A family of SL(n)-type solutions of this equation provides a new realization of the Hecke algebra. We define quantum antisymmetrizers, introduce the notion of quantum determinant and compute the inverse quantum matrix for matrix algebras of the type R̂(p)a1a2 = a1a2R̂. It is pointed out that such a quantum matrix algebra arises in the operator realization of the chiral zero modes of the WZNW model. On leave of absence from: Division of Theoretical Physics, Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria; e-mail address: lhadji@inrne.acad.bg On leave of absence from: Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, 141 980 Moscow Region, Russia; e-mail address: isaevap@thsun1.jinr.ru 3 On leave of absence from: P.N. Lebedev Physical Institute, Theoretical Department, 117924 Moscow, Leninsky prospect 53, Russia; e-mail address: oleg@cpt.univ-mrs.fr e-mail address: pyatov@thsun1.jinr.ru On leave of absence from: Division of Theoretical Physics, Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria; e-mail address: todorov@inrne.acad.bg

Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon

The raise and peel model of a one-dimensional fluctuating interface (model A) is extended by considering one source (model B) or two sources (model C) at the boundaries. The Hamiltonians describing the three processes have, in the thermodynamic limit, spectra given by conformal field theory. The probabilities of the different configurations in the stationary states of the three models are not only related but have interesting combinatorial properties. We show that by extending Pascals triangle (which gives solutions to linear relations in terms of integer numbers), to an hexagon, one obtains integer solutions of bilinear relations. These solutions not only give the weights of the various configurations in the three models but also give an insight into the connections between the probability distributions in the stationary states of the three models. Interestingly enough, Pascals hexagon also gives solutions to a Hirotas difference equation.

Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik-Zamolodchikov equation with reflecting boundaries in the Dyck path representation of the Temperley-Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of @t^2-weighted punctured cyclically symmetric transpose complement plane partitions where @t=-(q+q^-^1). In the cases of no or minimal punctures, we prove that these generating functions coincide with @t^2-enumerations of vertically symmetric alternating sign matrices and modifications thereof.

Representation theory of (modified) Reflection Equation Algebra of GL(m|n) type

Let R : V 2 → V 2 be a Hecke type solution of the quantum Yang-Baxter equation (a Hecke symmetry). Then, the Hilbert-Poincre series of the associated R-exterior algebra of the space V is a ratio of two polynomials of degree m (numerator) and n (denominator). Assuming R to be skew-invertible, we define a rigid quasitensor category SW(V(m|n)) of vector spaces, generated by the space V and its dual V � , and compute certain numerical char- acteristics of its objects. Besides, we introduce a braided bialgebra structure in the modified Reflection Equation Algebra, associated with R, and equip objects of the category SW(V(m|n)) with an action of this algebra. In the case related to the quantum group Uq(sl(m)), we con- sider the Poisson counterpart of the modified Reflection Equation Algebra and compute the semiclassical term of the pairing, defined via the categorical (or quantum) trace.