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, et al (hep-th/0602290) can be rewriten as a new algebra which contains q-deformed commutators. The quantum parameter q (|q| = 1) is a function of ~. It is shown that the q → 1 limit of the algebra with the parameter μ < 0 describes fuzzy S and that the squashed S with q 6= 1 and μ < 0 can be regarded as a new kind of quantum S. The N-dim representations of the q-deformed Lie algebra for T and S are derived. 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 and μ. It is shown that the allowed range of the value of q + q must be restricted for each fixed N. The range of q+ q which describes fuzzy T is also obtained. [email protected]