On Z-gradations of twisted loop Lie algebras of complex simple Lie algebras

We define the twisted loop Lie algebra of a finite dimensional Lie algebra $\mathfrak g$ as the Fr\'echet space of all twisted periodic smooth mappings from $\mathbb R$ to $\mathfrak g$. Here the Lie algebra operation is continuous. We call such Lie algebras Fr\'echet Lie algebras. We introduce the notion of an integrable $\mathbb Z$-gradation of a Fr\'echet Lie algebra, and find all inequivalent integrable $\mathbb Z$-gradations with finite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.Comment: 26 page

On Z-graded loop Lie algebras, loop groups, and Toda equations

Toda equations associated with twisted loop groups are considered. Such equations are specified by Z-gradations of the corresponding twisted loop Lie algebras. The classification of Toda equations related to twisted loop Lie algebras with integrable Z-gradations is discussed.Comment: 24 pages, talk given at the Workshop "Classical and Quantum Integrable Systems" (Dubna, January, 2007

Names of Residential Complexes: between Urbanonyms and Advertising Names

The question of the “borderline” status in the onomastic space of the new for the Russian language category of proper names - names of residential complexes - is considered. The novelty of the study is seen in the fact that for the first time the onomastic status of the names of residential complexes is determined, and a comparative analysis of the two corps of such names collected in Yaroslavl and Yekaterinburg is carried out. The existing definitions are analyzed, possible terminological definitions of such proper names are developed. It is noted that these proper names are in field of meanings of term oykodomonym , but it is indicated that the names of residential complexes occupy an intermediate position in onomastic system: on the one hand, they fit into the category of urbanonyms, on the other hand, they are a form of advertising names. A new approach to the analysis of the names of residential complexes based on onomasiological grounds is proposed: the authors divide groups of names according to the degree of increasing the signs of proper name from descriptive designations to conditionally-symbolic nominations. The authors come to the conclusion that in the system of considered oykodomonyms two trends were formed: (1) embedding the name in an existing urbanisation given its identity and (2) creation of names that are not related to existing urban onomastic space. The analysis of the corps of Yekaterinburg and Yaroslavl names confirmed the border status of the considered proper names

Urbanonymic Terminology: Systems and Problems

The article is focused on the problem of ordering and adaptation of modern Russian onomastic terminology to its foreign, especially Western European versions. The material for the analysis was the terms and their definitions from the Dictionary of Russian Onomastic Terminology by N. V. Podolskaya, the list of basic terms of ICOS (International Council of Onomastic Sciences) and lists of basic terms and their definitions in English, French and German. The article shows the history of the Russian urbanonymic terminology, noting its systematic nature. At the same time, its shortcomings are pointed out: emergence of new meanings of the term urbanonym that are contrary to the recommendations of the dictionaries and are not included in the conceptual field of the toponym, which is a hypernym for it. The authors note the difficulties in adapting the Russian terminology under Western European, its inconsistency: the term odonym actually acts as a synonym of the Russian term urbanonym . At the same time, it can be argued that in the Western European tradition, in contrast to the Russian one, the complete concept of urbanonym and its hypero-hyponymic relations is missing. The revision of the list of basic ICOS onomastic terms is needed, introduction of a term similar to the Russian urbanonym , its precise definition, definition of the scope of the specific terms and its components

Ground-state properties of a supersymmetric fermion chain

We analyze the ground state of a strongly interacting fermion chain with a supersymmetry. We conjecture a number of exact results, such as a hidden duality between weak and strong couplings. By exploiting a scale free property of the perturbative expansions, we find exact expressions for the order parameters, yielding the critical exponents. We show that the ground state of this fermion chain and another model in the same universality class, the XYZ chain along a line of couplings, are both written in terms of the same polynomials. We demonstrate this explicitly for up to N = 24 sites, and provide consistency checks for large N. These polynomials satisfy a recursion relation related to the Painlev\'e VI differential equation, and using a scale-free property of these polynomials, we derive a simple and exact formula for their limit as N goes to infinity.Comment: v2: added more information on scaling function, fixed typo

Bethe roots and refined enumeration of alternating-sign matrices

The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to -1/2 and an odd number of sites is considered. Some linear combinations of the components of the considered state, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that those polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M+1 divided by the total number of the alternating-sign matrices of order M, for the chain of length 2M+1.Comment: LaTeX 2e, 12 pages, minor corrections, references adde

Temperley-Lieb Stochastic Processes

We discuss one-dimensional stochastic processes defined through the Temperley-Lieb algebra related to the Q=1 Potts model. For various boundary conditions, we formulate a conjecture relating the probability distribution which describes the stationary state, to the enumeration of a symmetry class of alternating sign matrices, objects that have received much attention in combinatorics.Comment: 9 pages LaTeX, 11 Postscript figures, minor change

Dependent coordinates in path integral measure factorization

The transformation of the path integral measure under the reduction procedure in the dynamical systems with a symmetry is considered. The investigation is carried out in the case of the Wiener--type path integrals that are used for description of the diffusion on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple unimodular Lie group. The transformation of the path integral, which factorizes the path integral measure, is based on the application of the optimal nonlinear filtering equation from the stochastic theory. The integral relation between the kernels of the original and reduced semigroup are obtained.Comment: LaTeX2e, 28 page

Non-local space-time supersymmetry on the lattice

We show that several well-known one-dimensional quantum systems possess a hidden nonlocal supersymmetry. The simplest example is the open XXZ spin chain with \Delta=-1/2. We use the supersymmetry to place lower bounds on the ground state energy with various boundary conditions. For an odd number of sites in the periodic chain, and with a particular boundary magnetic field in the open chain, we can derive the ground state energy exactly. The supersymmetry thus explains why it is possible to solve the Bethe equations for the ground state in these cases. We also show that a similar space-time supersymmetry holds for the t-J model at its integrable ferromagnetic point, where the space-time supersymmetry and the Hamiltonian it yields coexist with a global u(1|2) graded Lie algebra symmetry. Possible generalizations to other algebras are discussed.Comment: 12 page

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 probability 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 Pascal's triangle (which gives solutions to linear relations in terms of integer numbers), to an hexagon, one obtains integer solutions of bilinear relations. These solutions give not only the weights of the various configurations in the three models but also give an insight to the connections between the probability distributions in the stationary states of the three models. Interestingly enough, Pascal's hexagon also gives solutions to a Hirota's difference equation.Comment: 33 pages, an abstract and an introduction are rewritten, few references are adde