Shape-Invariance and Many-Body Physics

Recent developments in the study of shape-invariant Hamiltonians are briefly summarized. Relations between certain exactly solvable problems in many-body physics and shape-invariance are explored. Connection between Gaudin algebras and supersymmetric quantum mechanics is pointed out.Comment: To appear in the proceedings of "Symmetries in Nuclear Structure, Ettore Majorana Centre, Erice - Sicily, Italy, March 23-29, 2003"; 10 pages of Latex outpu

Holomorphic deformation of Hopf algebras and applications to quantum groups

In this article we propose a new and so-called holomorphic deformation scheme for locally convex algebras and Hopf algebras. Essentially we regard converging power series expansion of a deformed product on a locally convex algebra, thus giving the means to actually insert complex values for the deformation parameter. Moreover we establish a topological duality theory for locally convex Hopf algebras. Examples coming from the theory of quantum groups are reconsidered within our holomorphic deformation scheme and topological duality theory. It is shown that all the standard quantum groups comprise holomorphic deformations. Furthermore we show that quantizing the function algebra of a (Poisson) Lie group and quantizing its universal enveloping algebra are topologically dual procedures indeed. Thus holomorphic deformation theory seems to be the appropriate language in which to describe quantum groups as deformed Lie groups or Lie algebras.Comment: 40 page

Topological Hopf algebras, quantum groups and deformation quantization

After a presentation of the context and a brief reminder of deformation quantization, we indicate how the introduction of natural topological vector space topologies on Hopf algebras associated with Poisson Lie groups, Lie bialgebras and their doubles explains their dualities and provides a comprehensive framework. Relations with deformation quantization and applications to the deformation quantization of symmetric spaces are describedComment: 13 pages, to appear in the proceedings of the conference "Hopf algebras in noncommutative geometry and physics" (VUB, Brussels, May 2002

On the Existence of Local Observables in Theories With a Factorizing S-Matrix

A recently proposed criterion for the existence of local quantum fields with a prescribed factorizing scattering matrix is verified in a non-trivial model, thereby establishing a new constructive approach to quantum field theory in a particular example. The existence proof is accomplished by analyzing nuclearity properties of certain specific subsets of Fermionic Fock spaces.Comment: 13 pages, no figures, comment in sect. 3 adde