Search results for "nonlinear"
showing 10 items of 3684 documents
Contractions of Filippov algebras
2010
We introduce in this paper the contractions $\mathfrak{G}_c$ of $n$-Lie (or Filippov) algebras $\mathfrak{G}$ and show that they have a semidirect structure as their $n=2$ Lie algebra counterparts. As an example, we compute the non-trivial contractions of the simple $A_{n+1}$ Filippov algebras. By using the \.In\"on\"u-Wigner and the generalized Weimar-Woods contractions of ordinary Lie algebras, we compare (in the $\mathfrak{G}=A_{n+1}$ simple case) the Lie algebras Lie$\,\mathfrak{G}_c$ (the Lie algebra of inner endomorphisms of $\mathfrak{G}_c$) with certain contractions $(\mathrm{Lie}\,\mathfrak{G})_{IW}$ and $(\mathrm{Lie}\,\mathfrak{G})_{W-W}$ of the Lie algebra Lie$\,\mathfrak{G}$ as…
The Minkowski and conformal superspaces
2006
We define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, based on the real conformal supergroup and its Lie superalgebra.
Central extensions of the families of quasi-unitary Lie algebras
1998
The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras su(p,q) of the Cartan series A_l and the pseudo-unitary algebras u(p,q), are completely determined and classified for arbitrary p,q. In addition to the su(p,q) and u({p,q}) algebras, whose second cohomology group is well known to be trivial, each family includes many non-semisimple algebras; their central extensions, which are explicitly given, can be classified into three types as far as their properties under contraction are involved. A closed expression for the dimension of the second cohomology group of any member of these families …
Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy
2021
We consider the Dubrovin--Frobenius manifold of rank $2$ whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck's dessins d'enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin--Frobenius manifold is a tau-function of the extended nonlinear Schr\"odinger hierarchy, an extension of a particular rational reduction of the Kadomtsev--Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental--M…
On the computation of intersection numbers for twisted cocycles
2020
Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example square roots), although the final result may be expressed without algebraic extensions. In this article I present an improvement of this algorith…
Spin Chains with Non-Diagonal Boundaries and Trigonometric SOS Model with Reflecting End
2011
In this paper we consider two a priori very different problems: construction of the eigenstates of the spin chains with non parallel boundary magnetic fields and computation of the partition function for the trigonometric solid-on-solid (SOS) model with one reflecting end and domain wall boundary conditions. We show that these two problems are related through a gauge transformation (so-called vertex-face transformation) and can be solved using the same dynamical reflection algebras.
Entanglement in continuous-variable systems: recent advances and current perspectives
2007
We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures, and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillabil…
SOV approach for integrable quantum models associated to general representations on spin-1/2 chains of the 8-vertex reflection algebra
2013
The analysis of the transfer matrices associated to the most general representations of the 8-vertex reflection algebra on spin-1/2 chains is here implemented by introducing a quantum separation of variables (SOV) method which generalizes to these integrable quantum models the method first introduced by Sklyanin. More in detail, for the representations reproducing in their homogeneous limits the open XYZ spin-1/2 quantum chains with the most general integrable boundary conditions, we explicitly construct representations of the 8-vertex reflection algebras for which the transfer matrix spectral problem is separated. Then, in these SOV representations we get the complete characterization of t…
Low-temperature large-distance asymptotics of the transversal two-point functions of the XXZ chain
2014
We derive the low-temperature large-distance asymptotics of the transversal two-point functions of the XXZ chain by summing up the asymptotically dominant terms of their expansion into form factors of the quantum transfer matrix. Our asymptotic formulae are numerically efficient and match well with known results for vanishing magnetic field and for short distances and magnetic fields below the saturation field.
Ostrogradsky's Hamilton formalism and quantum corrections
2010
By means of a simple scalar field theory it is demonstrated that the Lagrange formalism and Ostrogradsky's Hamilton formalism in the presence of higher derivatives, in general, do not lead to the same results. While the two approaches are equivalent at the classical level, differences appear due to the quantum corrections.