Siberian Federal University

The rotation algebra A is the universal C -algebra generated by unitary operators U; V satisfying the commutation relation UV = !V U where ! = e2 i : They are rational if = p=q with 1 6 p 6 q1; othewise irrational. Operators in these algebras relate to the quantum Hall effect [2,26,30], kicked quantum systems [22,34], and the spectacular solution of the Ten Martini problem [1]. Brabanter [4] and Yin [38] classified rational rotation C -algebras up to -isomorphism. Stacey [31] constructed their automorphism groups. They used methods known to experts: cocycles, crossed products, Dixmier- Douady classes, ergodic actions, K–theory, and Morita equivalence. This expository paper defines Ap=q as a C -algebra generated by two operators on a Hilbert space and uses linear algebra, Fourier series and the Gelfand–Naimark–Segal construction [16] to prove its universality. It then represents it as the algebra of sections of a matrix algebra bundle over a torus to compute its isomorphism class. The remarks section relates these concepts to general operator algebra theory. We write for mathematicians who are not C -algebra experts