Fuzzy torus and q-deformed Lie algebra

It will be shown that the defining relations for fuzzy torus and deformed (squashed) sphere proposed by [J. Arnlind, M. Bordemann, L. Hofer, J. Hoppe, H. Shimada, hep-th/0602290] (ABHHS) can be rewritten as a new algebra which contains q-deformed commutators. The quantum parameter q (|q|=1) is a function of ℏ. It is shown that the q→1 limit of the algebra with the parameter μ<0 describes fuzzy S2 and that the squashed S2 with q≠1 and μ<0 can be regarded as a new kind of quantum S2. Throughout the Letter 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 μ. 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 (qN=1) and contains undetermined parameters. The 'string solution' exists for generic values of q (qN≠1). In this Letter we will explicitly construct the representation of the q-deformed algebra for generic values of q (qN≠1) and it is shown that the allowed range of the value of q+q−1 must be restricted for each fixed N.