publications

2022

1. Deformation and Hochschild cohomology of coisotropic algebras

   M. Dippell, C. Esposito, and S. Waldmann

   Ann. Mat. Pura Appl. (4), 2022

   Abs arXiv Website

   Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.

2. A Serre-Swan theorem for coisotropic algebras

   M. Dippell, F. Menke, and S. Waldmann

   Pacific J. Math., 2022

   Abs arXiv Website

   Coisotropic algebras are used to formalize coisotropic reduction in Poisson geometry as well as in deformation quantization and find applications in various other fields as well. In this paper we prove a Serre-Swan Theorem relating the regular projective modules over the coisotropic algebra built out of a manifold $M, a submanifold C and an integrable smooth distribution D ⊆TC$ with vector bundles over this geometric situation and show an equivalence of categories for the case of a simple distribution.

2019

1. Coisotropic Triples, Reduction and Classical Limit.

   M. Dippell, C. Esposito, and S. Waldmann

   Doc. Math., 2019

   Abs arXiv Website

   Coisotropic reduction from Poisson geometry and deformation quantization is cast into a general and unifying algebraic framework: we introduce the notion of coisotropic triples of algebras for which a reduction can be defined. This allows to construct also a notion of bimodules for such triples leading to bicategories of bimodules for which we have a reduction functor as well. Morita equivalence of coisotropic triples of algebras is defined as isomorphism in the ambient bicategory and characterized explicitly. Finally, we investigate the classical limit of coisotropic triples of algebras and their bimodules and show that classical limit commutes with reduction in the bicategory sense.

2023

1. Constraint Vector Bundles and Reduction of Lie (Bi-)Algebroids

   Marvin Dippell, and David Kern

   2023

   Abs arXiv

   We present a framework for the reduction of various geometric structures extending the classical coisotropic Poisson reduction. For this we introduce constraint manifolds and constraint vector bundles. A constraint Serre-Swan theorem is proven, identifying constraint vector bundles with certain finitely generated projective modules, and a Cartan calculus for constraint differentiable forms and multivector fields is introduced. All of these constructions will be shown to be compatible with reduction. Finally, we apply this to obtain a reduction procedure for Lie (bi-)algebroids and Dirac manifolds.