Diffeological Morita Equivalence

非阿贝尔Hodge理论的微分学视角

A diffeological perspective on non-Abelian Hodge theory

在紧Kähler流形上构造微分学模叠参数化Higgs丛和平坦丛，并建立子叠间的等价，该等价延拓了粗模空间同胚与范畴等价。

Nijenhuis结构的Morita等价

Morita equivalence of Nijenhuis structures

本文研究Nijenhuis群oids及其微分结构的Morita等价，通过李 functor建立全局与微分的对应关系，并增强对准辛群oids和Dirac结构的Morita等价理解，证明Poisson-Nijenhuis流形的模类在Morita等价下不变。

Poisson geometry and Morita equivalence

These notes discuss various aspect of the ``representation theory'' of Poisson manifolds, with focus on Morita equivalence and Picard groups. We give a brief introduction to Poisson geometry (including Dirac and twisted Poisson structures) and algebraic Morita theory before presenting the geometric Morita theory of Poisson manifolds. We also point out the connections with the theory of symplectic groupoids and hamiltonian actions.

Morita equivalence of Poisson manifold via stack groupoids

This is a condensed exposition of the results of a future work, based on a talk of the second author at the Oberwolfach workshop "Poisson Geometry", April 30--4 May 2007.

移位辛 Lie n-群oid 的 Morita 等价

Morita equivalence of shifted symplectic Lie n-groupoids

本文研究了 Lie n-群oid 上的 m-移位辛结构在 Lie n-群oid 的 Morita 等价下保持不变的性质，提供了严谨的证明。

Deformation Cohomology of Lie Algebroids and Morita Equivalence

Let $A \Rightarrow M$ be a Lie algebroid. In this short note, we prove that a pull-back of $A$ along a fibration with homologically $k$-connected fibers, shares the same deformation cohomology of $A$ up to degree $k$.

Ideals and Formal Morita Equivalence of Algebras

Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of $^*$-algebras over a ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring $\ring R$. The goal of the present paper is twofold. First, we clarify the relationship between formal Morita equivalence, Ara's notion of Morita $^*$-equivalence of rings with involution, and strong Morita equivalence of $C^*$-algebras. Second, in the general setting of $^*$-algebras over $\ring C$, we define `closed' $^*$-ideals as the ones occuring as kernels of $^*$…

Equivalence between the Morita categories of etale Lie groupoids and of locally grouplike Hopf algebroids

Any etale Lie groupoid G is completely determined by its associated convolution algebra C_c(G) equipped with the natural Hopf algebroid structure. We extend this result to the generalized morphisms between etale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c(G)-C_c(H)-bimodule C_c(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C_c gives an equivalence between the Morita category of etale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.

Morita Equivalence and Spectral Triples on Noncommutative Orbifolds

Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical orbifold groupoids, a Morita equivalence for the crossed product spectral triples is developed. Noncommutative orbifolds are Morita equivalence classes of the crossed product spectral triples. As a special case of this Morita theory one can study freeness of the $G$-action on the …

Rational Morita equivalence for holomorphic Poisson modules

We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a meromorphic flat connection or a co-Higgs sheaf. As an application, we study the geometry of rank two meromorphic rank two $\mathfrak{sl}_2$-Poisson modules which can be interpreted as a Poisson analogous to transversally projective structures for codimension one holomorphic foli…

Morita equivalences of vector bundles

We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and use it to prove the Morita invariance of VB-cohomology, with implications to deformation cohomology of Lie groupoids and of classic geometries. We discuss applications of our theory to Poisson geometry, providing a new insight over Marsden-Weinstein reduction and the integration of Dirac structures. We conclude by proving that the derived category of VB-groupo…