Braided algebras and their applications to Noncommutative Geometry - Archive ouverte HAL Access content directly
Journal Articles Advances in Applied Mathematics Year : 2013

Braided algebras and their applications to Noncommutative Geometry

(1) , (2, 3)


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 Poincaré–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.
Fichier principal
Vignette du fichier
1211.5506.pdf (300.35 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-03155708 , version 1 (05-07-2022)



Dimitri Gurevich, Pavel Saponov. Braided algebras and their applications to Noncommutative Geometry. Advances in Applied Mathematics, 2013, 51 (2), pp.228-253. ⟨10.1016/j.aam.2013.02.004⟩. ⟨hal-03155708⟩
14 View
4 Download



Gmail Facebook Twitter LinkedIn More