arXiv subjects
Publications and source records attributed to Sayan Chakraborty.
We compute the $K$-theory of crossed products of rotation algebras $\mathcal{A}_θ$, for any real angle $θ$, by matrices in $\mathrm{SL}(2,\mathbb{Z})$ with infinite order. Using techniques of continuous fields, we show that the canonical inclusion of $\mathcal{A}_θ$ into the crossed products is injective at the level of $K_0$-groups. We then give an explicit set of generators for the $K_0$-groups and compute the tracial ranges concretely.
In this article we describe extensions of some K-theory classes of Heisenberg modules over higher-dimensional noncommutative tori to projective modules over crossed products of noncommutative tori by finite cyclic groups, aka noncommutative orbifolds. The two dimensional case was treated by Echterhoff, Lück, Phillips and Walters. Our approach is based on the theory of metaplectic transformations of the representation theory of the Heisenberg group. We also describe the generators of the K-groups of the crossed products of flip actions by $\mathbb{Z}_2$ on 3-dimensional noncommutative tori.
Let $θ, θ'$ be irrational numbers and $A, B$ be matrices in $SL_2(\mathbb{Z})$ of infinite order. We compute the $K$-theory of the crossed product $\mathcal{A}_θ\rtimes_A \mathbb{Z}$ and show that $\mathcal{A}_θ \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{θ'} \rtimes_B \mathbb{Z}$ are $*$-isomorphic if and only if $θ= \pmθ' \pmod{\mathbb{Z}}$ and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Combining this result and an explicit construction of equivariant bimodules, we show that $\mathcal{A}_θ \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{θ'} \rtimes_B \mathbb{Z}$ are Morita equivalent if and only if $θ$ and $θ'$ are in the same $GL_2(\mathbb{Z})$ orbit and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Finally, we determine the Morita equivalence class of $\mathcal{A}_θ \rtimes F$ for any finite subgroup $F$ of $SL_2(\mathbb{Z})$.
We give a description of cyclic cohomology and its pairing with K-groups for 2-cocycle deformation of algebras graded over discrete groups. The proof relies on a realization of monodromy for the Gauss-Manin connection on periodic cyclic cohomology in terms of the cup product action of group cohomology.