Noncommutative coverings of quantum tori

Abstract

We introduce a framework for coverings of noncommutative spaces. Moreover, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.

An Atiyah Sequence for Noncommutative Principal Bundles

“…for all finite-dimensional representations σ, π, ρ of G and b ∈ B (see [37,Lemma 4.3]). The triple (H, γ, ω) of the above families is referred to as the factor system of (A, G, α) associated with s(σ), σ ∈ Irr(G), or simply as a factor system of (A, G, α) when no explicit reference to the isometries is needed.…”

“…Additionally, noncommutative principal bundles are becoming increasingly prevalent in various applications of geometry (cf. [22,23,28,37,38]) and mathematical physics (see, e.g., [7,10,13,14,18,21,25,41] and references therein).…”

Lifting spectral triples to noncommutative principal bundles

“…In [19,21] the authors consider spectral triples that are equivariant with respect to a torus action. Given such a spectral triple A key feature of our free C * -dynamical system (A, G, α) is the factor system associated with the isometries s(σ), σ ∈ Irr(G), (see [53,Def. 4.1]), which we now recall for the convenience of the reader.…”

“…In addition, noncommutative principal bundles are becoming increasingly prevalent in various applications of geometry (cf. [34,35,38,53]) and mathematical physics (see, e. g., [6,10,19,20,23,31,36,56] and ref. therein).…”

An Atiyah Sequence for Noncommutative Principal Bundles

“…Additionally, noncommutative principal bundles are becoming increasingly prevalent in various applications of geometry (cf. [22,23,28,37,38]) and mathematical physics (see, e. g., [7,10,13,14,18,21,25,41] and ref. therein).…”