Braided algebras and their applications to Noncommutative Geometry

TL;DR

This paper introduces braided algebras, explores their properties, develops a differential calculus on related algebras, and applies these concepts to quantum deformations of classical spaces, with potential physical implications.

Contribution

It defines braided algebras with rational Poincare-Hilbert series, develops a new differential calculus on Reflection Equation algebras, and applies these to quantum deformations of geometric spaces.

Findings

Proved the mountain property for Poincare-Hilbert series of braided algebras.

Developed a Leibniz rule for derivatives on modified Reflection Equation algebras.

Computed quantum radii and Laplace operators on deformed spaces.

Abstract

We introduce the notion of a braided algebra and study some examples of these. In particular, R-symmetric and R-skew-symmetric algebras of a linear space V equipped with a skew-invertible Hecke symmetry R are braided algebras. We prove the "mountain property" for the numerators and denominators of their Poincare-Hilbert series (which are always rational functions). Also, we further develop a differential calculus on modified Reflection Equation algebras. Thus, we exhibit a new form of the Leibniz rule for partial derivatives on such algebras related to involutive symmetries R. In particular, we present this rule for the algebra U(gl(m)). The case of the algebra U(gl(2)) and its compact form U(u(2)) (which can be treated as a deformation of the Minkowski space algebra) is considered in detail. On the algebra U(u(2)) we introduce the notion of the quantum radius, which is a deformation of the usual radius, and compute the action of rotationally invariant operators and in particular of the Laplace operator. This enables us to define analogs of the Laplace-Beltrami operators corresponding to certain Schwarzschild-type metrics and to compute their actions on the algebra U(u(2)) and its central extension. Some "physical" consequences of our considerations are presented.