Braided affine geometry and q-analogs of wave operators

TL;DR

This paper reviews various approaches to affine braided geometry related to Hecke type braidings, focusing on quantum groups, q-analogs of vector bundles, and q-differential calculus, including applications to wave operators.

Contribution

It compares different methods of constructing affine braided geometry, introduces q-analogs of derivatives and bundles, and develops q-analogs of wave operators within this framework.

Findings

Comparison of Poisson and quantum varieties

Construction of q-vector bundles as projective modules

Development of q-differential calculus and wave operators

Abstract

The main goal of this review is to compare different approaches to constructing geometry associated with a Hecke type braiding (in particular, with that related to the quantum group U_q(sl(n))). We make an emphasis on affine braided geometry related to the so-called Reflection Equation Algebra (REA). All objects of such type geometry are defined in the spirit of affine algebraic geometry via polynomial relations on generators. We begin with comparing the Poisson counterparts of "quantum varieties" and describe different approaches to their quantization. Also, we exhibit two approaches to introducing q-analogs of vector bundles and defining the Chern-Connes index for them on quantum spheres. In accordance with the Serre-Swan approach, the q-vector bundles are treated as finitely generated projective modules over the corresponding quantum algebras. Besides, we describe the basic properties of the REA used in this construction and compare different ways of defining q-analogs of partial derivatives and differentials on the REA and algebras close to them. In particular, we present a way of introducing a q-differential calculus via Koszul type complexes. The lements of the q-calculus are applied to defining q-analogs of some relativistic wave operators.