arXiv subjects

C. Schwiebert

Publications and source records attributed to C. Schwiebert.

4 recordsLinked to original sources

arXiv1995

Generalized Quantum Inverse Scattering

A generalization of the quantum inverse scattering method is proposed replacing the quantum group $RLL$ commutation relations of Lax operators by reflection equation type $RLRL$ commutation relations. Under some natural assumptions the most general algebra of this type allowing to construct the neccessary integrals of motion is found. It serves to describe Lax operators with completely non-ultralocal commutation relations. An example of this new formalism is an integrable model on monodromies of flat connections on a Riemann surface which is related to the XXZ quantum spin chain.

hep-th↗

arXiv1994

Extended reflection equation algebras, the braid group on a handlebody and associated link polynomials

The correspondence of the braid group on a handlebody of arbitrary genus to the algebra of Yang-Baxter and extended reflection equation operators is shown. Representations of the infinite dimensional extended reflection equation algebra in terms of direct products of quantum algebra generators are derived, they lead to a representation of this braid group in terms of $R$-matrices. Restriction to the reflection equation operators only gives the coloured braid group. The reflection equation operators, describing the effect of handles attached to a 3-ball, satisfy characteristic equations which give rise to additional skein relations and thereby invariants of links on handlebodies. The origin of the skein relations is explained and they are derived from an adequately adapted handlebody version of the Jones polynomial. Relevance of these results to the construction of link polynomials on closed 3-manifolds via Heegard splitting and surgery is indicated.

hep-th↗

arXiv1993

Reflection equation and link polynomials for arbitrary genus solid tori

The correspondence between the braid group on a solid torus of arbitrary genus and the algebra of Yang-Baxter and reflection equation operators is shown. A representation of this braid group in terms of $R$-matrices is given. The characteristic equation of the reflection equation matrix is considered as an additional skein relation. This could lead to an intrinsic definition of invariant link polynomials on solid tori and, via Heegaard splitting, to invariant link polynomials on arbitrary three-manifolds without boundary.

hep-th↗

arXiv1992

Constant Solutions of Reflection Equations and Quantum Groups

To the Yang-Baxter equation an additional relation can be added. This is the reflection equation which appears in various places, with or without spectral parameter. For example, in factorizable scattering on a half-line, integrable lattice models with non-periodic boundary conditions, non-commutative differential geometry on quantum groups, etc. We study two forms of spectral parameter independent reflection equations, chosen by the requirement that their solutions be comodules with respect to the quantum group coaction leaving invariant the reflection equations. For a variety of known solutions of the Yang-Baxter equation we give the constant solutions of the reflection equations. Various quadratic algebras defined by the reflection equations are also given explicitly.

hep-th↗