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arXiv · hep-th/0604010

Fuzzy Torus and q-Deformed Lie Algebra

Ryuichi Nakayama
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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 μ<0 describes fuzzy S^2 and that the squashed S^2 with q \neq 1 and μ<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 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 (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.

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Ryuichi Nakayama. 2006-05-18. Fuzzy Torus and q-Deformed Lie Algebra. https://doi.org/10.1016/j.physletb.2006.05.052

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