Quantum Double-Torus

A symmetry extending the $T^2$-symmetry of the noncommutative torus $T^2_q$ is studied in the category of quantum groups. This extended symmetry is given by the quantum double-torus defined as a compact matrix quantum group consisting of the disjoint union of $T^2$ and $T^2_{q^2}$. The bicross-product structure of the polynomial Hopf algebra of the quantum double-torus is computed. The Haar measure and the complete list of unitary irreducible representations of the quantum double-torus are determined explicitly.