Abstract We initiate the program of extending to higher-rank graphs ( k -graphs) geometric classification directed graph $C^*$ -algebras, as completed in Eilers et al. (2016, Preprint). To be precise, we identify four “moves,” or modifications, one can perform on a -graph $\Lambda $ , which leave invariant Morita equivalence class its -algebra $C^*(\Lambda )$ . These moves—in-splitting, delay, ...

Using a stable equivalence due to Rouquier, we prove that Broué’s abelian defect group conjecture holds for 3-blocks of defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules. The proof makes use of the fact, also due to Rouquier, that a stable equivalence of Morita type between self-injective algebras induces an isomorphism between the connected components of the ...

We use C-algebras to study complex algebraic curves. Our approach is based on the representation of an algebraic curve of genus g by the interval exchange transformation due to H. Masur, W. Veech et al. We study the C-algebra Oλ connected to such transformation. The main result says that the algebra Oλ, taken up to Morita equivalence, defines the curve C, up to conformal equivalence. The first ...

Owing to the difference in K-theory, an example by Dugger and Shipley implies that equivalence of stable categories Gorenstein projective modules should not be a Quillen equivalence. We give sufficient necessary condition for Frobenius pair faithful functors between two abelian equivalence, which is also equivalent induce mutually inverse equivalences objects. show category objects Waldhausen c...

We study gauge theories on noncommutative tori. It was proved in [5] that Morita equivalence of noncommutative tori leads to a physical equivalence (SO(d, d|Z)-duality) of the corresponding gauge theories. We calculate the energy spectrum of maximally supersymmetric BPS states in these theories and show that this spectrum agrees with the SO(d, d|Z)-duality. The relation of our results with thos...

We show how to extend a classic Morita Equivalence Result of Green’s to the C∗-algebras of Fell bundles over transitive groupoids. Specifically, we show that if p : B → G is a saturated Fell bundle over a transitive groupoid G with stability group H = G(u) at u ∈ G(0), then C∗(G,B) is Morita equivalent to C∗(H,C ), where C = B|H . As an application, we show that if p : B → G is a Fell bundle ov...

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring C. This allows to extend (weak and...