Fuzzy Torus and q-Deformed Lie Algebra

TL;DR

This paper reformulates the algebraic structure of fuzzy torus and deformed spheres using q-deformed commutators, exploring their representations and limits, and introduces a new perspective on quantum S^2 surfaces.

Contribution

It provides explicit constructions of q-deformed algebra representations for generic q and analyzes the restrictions on q for fixed representation dimensions.

Findings

The q -> 1 limit recovers fuzzy S^2.

Squashed S^2 with q ≠ 1 is a new quantum surface.

Allowed q values depend on the representation dimension N.

Abstract

It will be shown that the defining relations for fuzzy torus and deformed (squashed) sphere proposed by J. Arnlind, et al (hep-th/0602290) (ABHHS) can be rewriten as a new algebra which contains q-deformed commutators. The quantum parameter q (|q|=1) is a function of \hbar. It is shown that the q -> 1 limit of the algebra with the parameter \mu <0 describes fuzzy S^2 and that the squashed S^2 with q \neq 1 and \mu <0 can be regarded as a new kind of quantum S^2. Throughout the paper the value of the invariant of the algebra, which defines the constraint for the surfaces, is not restricted to be 1. This allows the parameter q to be treated as independent of N (the dimension of the representation) and \mu. It was shown by ABHHS that there are two types of representations for the algebra, ``string solution'' and ``loop solution''. The ``loop solution'' exists only for q a root of unity (q^N=1) and contains undetermined parameters. The 'string solution' exists for generic values of q (q^N \neq 1). In this paper we will explicitly construct the representation of the q-deformed algebra for generic values of q (q^N \neq 1) and it is shown that the allowed range of the value of q+q^{-1} must be restricted for each fixed N.