Fourier--Mukai transforms for non-commutative complex tori

Let $X$ be a complex torus of dimension $g$ and $\widehat{X}$ be the dual torus. For any $g(g-1)\mathrm{/}2$-tuple $\lambda$ of complex numbers of absolute value $1$, we define a non-commutative complex torus $X_{\lambda }$ as a sheaf of algebras on a real torus of dimension $g$. We prove that if all components of $\lambda$ are roots of unity, then the category of coherent sheaves on $X_{\lambda }$ is abelian and derived-equivalent to the category of coherent sheaves on $\widehat{X}$ twisted by an element of the Brauer group of $\widehat{X}$ determined by $\lambda$.