Searcharxiv⌕ Search

arXiv subjects

P. A. Saponov

Publications and source records attributed to P. A. Saponov.

10 recordsLinked to original sources
arXiv2010

Quantization of pencils with a gl-type Poisson center and braided geometry

In the algebra Sym(gl(m)) we consider Poisson pencils generated by the linear Poisson-Lie bracket {,}_{gl(m)} and that corresponding to the so-called Reflection Equation Algebra. Each bracket of such a pencil has the Poisson center coinciding with that of the bracket {,}_{gl(m)}. Consequently, any bracket from this pencil can be restricted to a generic GL(m)-orbit O in gl(m)*. Quantization of such a restricted bracket can be done in the frameworks of braided affine geometry. In the paper we consider these Poisson structures, their super-analogs as well as their quantum (braided) counterparts. Also, we exhibit some detailed examples.

math.QA↗
arXiv2010

Generic super-orbits in gl(m|n)* and their braided counterparts

We introduce some braided varieties -- braided orbits -- by considering quotients of the so-called Reflection Equation Algebras associated with Hecke symmetries (i.e. special type solutions of the quantum Yang-Baxter equation). Such a braided variety is called regular if there exists a projective module on it, which is a counterpart of the cotangent bundle on a generic orbit O in gl(m)* in the framework of the Serre approach. We give a criterium of regularity of a braided orbit in terms of roots of the Cayley-Hamilton identity valid for the generating matrix of the Reflection Equation Algebra in question. By specializing our general construction we get super-orbits in gl(m|n)* and a criterium of their regularity.

math.QA↗
arXiv2003

Weyl approach to representation theory of reflection equation algebra

The present paper deals with the representation theory of the reflection equation algebra, connected with a Hecke type R-matrix. Up to some reasonable additional conditions the R-matrix is arbitrary (not necessary originated from quantum groups). We suggest a universal method of constructing finite dimensional irreducible non-commutative representations in the framework of the Weyl approach well known in the representation theory of classical Lie groups and algebras. With this method a series of irreducible modules is constructed which are parametrized by Young diagrams. The spectrum of central elements s(k)=Tr_q(L^k) is calculated in the single-row and single-column representations. A rule for the decomposition of the tensor product of modules into the direct sum of irreducible components is also suggested.

math.QA↗
arXiv1997

The four-fermion interaction in D=2,3,4: a nonperturbative treatment

A new nonperturbative approach is used to investigate the Gross-Neveu model of four fermion interaction in the space-time dimensions 2, 3 and 4, the number $N$ of inner degrees of freedom being a fixed integer. The spontaneous symmetry breaking is shown to exist in $D=2,3$ and the running coupling constant is calculated. The four dimensional theory seems to be trivial.

hep-th↗
arXiv1996

Hecke symmetries and characteristic relations on Reflection Equation algebras

We discuss how properties of Hecke symmetry (i.e., Hecke type R-matrix) influence the algebraic structure of the corresponding Reflection Equation (RE) algebra. Analogues of the Newton relations and Cayley-Hamilton theorem for the matrix of generators of the RE algebra related to a finite rank even Hecke symmetry are derived.

q-alg↗
arXiv1995

Singular Perturbations in Quantum Field Theory

In this talk we discuss a new approximation scheme for non-perturbative calculations in a quantum field theory which is based on the fact that the Schwinger equation of a quantum field model belongs to the class of singularly perturbed equations. The self-interacting scalar field and the Gross-Neveu model are taken as the examples and some non-perturbative solutions of an equation for the propagator are found for these models. The application to QCD is also discussed.

hep-th↗
arXiv1995

Schwinger Equation as Singularly Perturbed Equation

A new approximation scheme for non-perturbative calculations in a quantum field theory is proposed. The scheme is based on investigation of solutions of the Schwinger equation with its singular character taken into account. As a necessary supplementary boundary condition the Green functions' connected structures correspondence principle is used. Besides the usual perturbation theory expansion which is always available as a particular solution of our scheme some non-perturbative solutions of an equation for the propagator are found in the model of a self-interacting scalar field.

hep-th↗