Representations of quantum bicrossproduct algebras

Abstract

We present a method to construct induced representations of quantum algebras having the structure of bicrossproduct. We apply this procedure to some quantum kinematical algebras in (1 + 1) dimensions with this kind of structure: null-plane quantum Poincaré algebra, non-standard quantum Galilei algebra and quantum kappa Galilei algebra.

Acceleration-enlarged symmetries in nonrelativistic space-time with a cosmological constant

“…The two nonrelativistic Newton-Hooke (NH) cosmological groups were introduced by Bacry and Leblond in [2] who classified all kinematical groups in D = (3 + 1). The NH symmetries can be obtained, in the limit c → ∞, from the de-Sitter and anti-de-Sitter geometries [2][3][4][5][6] and they describe, respectively, the nonrelativistic expanding (with symmetry described by the NH + algebra) and oscillating (with symmetry described by the NH − algebra) universes. The cosmological constant describes the time scale τ determining the rate of expansion or the period of oscillation of the universe.…”

“…The plan of our presentation is as follows. First, in Section 2 we recall some known results [5,6,8,11] on the NH algebras with central extensions. We also add the constant acceleration generators, consider the most general central extensions and describe the Casimirs of our new algebras.…”

Induced representations of quantum kinematical algebras and quantum mechanics

“…It has been proved in Ref. [1] that dual bases and * -structures over bicrossproduct Hopf algebras may be constructed when the corresponding bases and * -structures of the bicrossproduct the factors are known. Thus, let H = K⊲◭ L be a bicrossproduct Hopf algebra and (K, K * , ·, · 1 ) and (L, L * , ·, · 2 ) nondegenerate triplets, then the expression kl, κλ = k, κ 1 l, λ 2 .…”

“…The main results about the theory of induced representations for quantum bicrossproduct algebras will be summarized in this Section (see Refs. [1,4,5,6] for a complete description of induced representations of quantum algebras).…”

“…One can classify the elementary systems [2] according to them or to obtain their corresponding Schrödinger equations using local representations [3]. In the last years we are involved in a program for constructing a theory of induced representations for quantum algebras [1,4,5,6]. Our aim is to obtain a quantum counterpart of the program that Wigner started in 1939 for Lie groups [2] that has been so fruitful in quantum physics.…”