Nonlinear Superposition Formulas Based on Lie Group SO(n+1,n)

Systems of nonlinear ordinary differential equations are constructed, for which the general solution is algebraically expressed in terms of a finite number of particular solutions. Expressions of that type are called the nonlinear superposition formulas. These systems are connected with local Lie groups tranformations on their homogeneous spaces. In the presented work the nonlinear superposition formulas are constructed for the case of the SO(3,2) group and some aspects in the general case of SO(n+1,n) are studied.Comment: 11 pages, LaTe

Auxiliary representations of Lie algebras and the BRST constructions

The method of construction of auxiliary representations for a given Lie algebra is discussed in the framework of the BRST approach. The corresponding BRST charge turns out to be non -- hermitian. This problem is solved by the introduction of the additional kernel operator in the definition of the scalar product in the Fock space. The existence of the kernel operator is proven for any Lie algebra.Comment: 11 pages, LaTe

Remarks towards the spectrum of the Heisenberg spin chain type models

The integrable close and open chain models can be formulated in terms of generators of the Hecke algebras. In this review paper, we describe in detail the Bethe ansatz for the XXX and the XXZ integrable close chain models. We find the Bethe vectors for two--component and inhomogeneous models. We also find the Bethe vectors for the fermionic realization of the integrable XXX and XXZ close chain models by means of the algebraic and coordinate Bethe ansatz. Special modification of the XXZ closed spin chain model ("small polaron model") is consedered. Finally, we discuss some questions relating to the general open Hecke chain models.Comment: 50 pages, small corrections in the Section 1

On Lagrangian formulations for arbitrary bosonic HS fields on Minkowski backgrounds

We review the details of unconstrained Lagrangian formulations for Bose particles propagated on an arbitrary dimensional flat space-time and described by the unitary irreducible integer higher-spin representations of the Poincare group subject to Young tableaux $Y(s_1,...,s_k)$ with $k$ rows. The procedure is based on the construction of Verma modules and finding auxiliary oscillator realizations for the symplectic $sp(2k)$ algebra which encodes the second-class operator constraints subsystem in the HS symmetry algebra. Application of an universal BRST approach reproduces gauge-invariant Lagrangians with reducible gauge symmetries describing the free dynamics of both massless and massive bosonic fields of any spin with appropriate number of auxiliary fields.Comment: 8 pages, no figures, extended Contribution to the Proceedings of the International Workshop "Supersymmetry and Quantum Symmetries" (SQS'2011, July 18- July 23, 2011, Dubna, Russia), v.2: 9 pages, 2 references with comments in Introduction adde

Interactions of a massless tensor field with the mixed symmetry of the Riemann tensor. No-go results

Non-trivial, consistent interactions of a free, massless tensor field t_{\mu \nu |\alpha \beta} with the mixed symmetry of the Riemann tensor are studied in the following cases: self-couplings, cross-interactions with a Pauli-Fierz field and cross-couplings with purely matter theories. The main results, obtained from BRST cohomological techniques under the assumptions on smoothness, locality, Lorentz covariance and Poincar\'{e} invariance of the deformations, combined with the requirement that the interacting Lagrangian is at most second-order derivative, can be synthesized into: no consistent self-couplings exist, but a cosmological-like term; no cross-interactions with the Pauli-Fierz field can be added; no non-trivial consistent cross-couplings with the matter theories such that the matter fields gain gauge transformations are allowed.Comment: for version 3: 45 pages, uses amssymb; shortened version, the three appendices from version 2 can be found in hep-th/040209

Quantum differential forms

Formalism of differential forms is developed for a variety of Quantum and noncommutative situations