微分几何中的协变Picard广群

形变量子化中的Picard广群

The Picard groupoid in deformation quantization

本文综述了星积代数的表示论的最新进展。特别地，我们将环C = R（i）上的 *-代数的 *-表示论与环\mathbb{C}上的 *-代数的 *-表示论联系起来，并指出了Picard广群与强Morita等价概念相对应的一些性质.一些森田不变量被解释为产生于这个广群的行动。

H-äquivariante Morita-Äquivalenz und Deformationsquantisierung

English abstract: This work contains of five chapters: The first one deals with Morita equivalence of star algebras. In particular star algebras which are equipped with a symmetry given by a Hopf (star-) algebra. In the second chapter we describe the equivariant Picard group and the equivariant Picard groupoid, which are a direct consequence of the results from the first chapter. We calculated the kernel and the image of the groupoid morphism from the equivariant Picard groupoid into the normal one. We give furthermore a formalized version of Morita invariance: given by a functor from the Pica…

Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds

In this paper we investigate equivariant Morita theory for algebras with momentum maps and compute the equivariant Picard groupoid in terms of the Picard groupoid explicitly. We consider three types of Morita theory: ring-theoretic equivalence, *-equivalence and strong equivalence. Then we apply these general considerations to star product algebras over symplectic manifolds with a Lie algebra symmetry. We obtain the full classification up to equivariant Morita equivalence.

Galois differential algebras and categorical discretization of dynamical systems

A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and suitable categories of abstract dynamical systems. The integrable maps obtained share with their continuous counterparts a large class of solutions and, in the linear case, the Picard-Vessiot group.

The duality covariant geometry and DSZ quantization of abelian gauge theory

We develop the Dirac-Schwinger-Zwanziger (DSZ) quantization of classical abelian gauge theories with general duality structure on oriented Lorentzian four-manifolds $(M,g)$ of arbitrary topology, obtaining, as a result, the duality-covariant geometric formulation of such theories through connections on principal bundles. We implement the DSZ condition by restricting the field strengths of the theory to those which define classes originating in the degree-two cohomology of a local system valued in the groupoid of integral symplectic spaces. We prove that such field strengths are curvatures of c…