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In order to assess possible observable effects of noncommutativity indeformations of quantum mechanics, all irreducible representations of thenoncommutative Heisenberg algebra and Weyl-Heisenberg group on the two-torusare constructed. This analysis extends the well known situation for thenoncommutative torus based on the algebra of the noncommuting positionoperators only. When considering the dynamics of a free particle for any of theidentified representations, no observable effect of noncommutativity isimplied.

The Weyl-Heisenberg Group on the Noncommutative Two-Torus: A Zoo of Representations