Morita Equivalence of Multidimensional Noncommutative Tori

Abstract

One can describe an n-dimensional noncommutative torus by means of an antisymmetric n × n matrix θ. We construct an action of the group SO(n, n|Z) on the space of n × n antisymmetric matrices and show that, generically, matrices belonging to the same orbit of this group give Morita equivalent tori. Some applications to physics are sketched. By definition [R5], an n-dimensional noncommutative torus is an associative algebra with involution having unitary generators U 1 , ..., U n obeying the relationswhere e(t) = e 2πit and θ is an antisymmetric matrix. The same name is used for different completions of this algebra. In particular, we can consider the noncommutative torus as a C ⋆ -algebra A θ (the universal C ⋆ -algebra generated by n unitary operators satisfying (1) ). Noncommutative tori are important in many problems of mathematics and physics. It was shown recently that they are essential in consideration of compactifications of M(atrix) theory ([CDS]; for further development see [T]). The results of the present paper also have application to physics. If two algebras A and are Morita equivalent (see the definition below), then for every A-module R one can construct anÂ-moduleR in such a way that the correspondence R →R is an equivalence of categories of A-modules andÂ-modules. * M. A. R.

Instantons on the Quantum 4-pheres S4q

“…Most of the literature concentrated so far on noncommutative tori or Moyal deformations of R 4 [17,18,14,10,11].…”

A note on the BPS spectrum of the matrix model

“…Equation (3) was first written in [12]. A version of equation (4) in [10], where also the transformation (4) was identified as a chiral spinor transformation.…”

Invariants for normal completely positive maps on the hyperfinite II1 factor

“…Thus, the C * algebras B, C(T 2 ) ⋊ Z 2 and C * (U l , V l , U r , V r )(⊆ B(H τ )) are all canonically isomorphic, and we shall identify them whenever there is no chance of confusion, and shall call this C * algebra the 'second order irrational rotation algebra', and denote it by A (2) θ . It is actually the 4-dimensional noncommutative torus in the sense of Rieffel and Schwarz [12], corresponding to the skew-symmetric 4 × 4 matrix A = ((a ij )), with a 12 = a 34 = θ, a 21 = a 43 = −θ, and all other entries are zero. Moreover, from the proof of the above Lemma, it is clear that the map η gives an algebraic isomorphism between A fin and the * -algebra generated by W i , i = 1, ..., 4.…”