[Paper Review] Coisotropic Triples, Reduction and Classical Limit

Marvin Dippell, Chiara Esposito+1 more|arXiv (Cornell University)|Dec 23, 2018

Advanced Topics in Algebra25 references4 citations

TL;DR

This paper introduces coisotropic triples of algebras as a unifying algebraic framework for coisotropic reduction in Poisson geometry and deformation quantization. It constructs a bicategory of bimodules over such triples, defines a reduction functor that commutes with classical limits, and characterizes Morita equivalence via explicit compatibility conditions between components. The key contribution is a functorial, bicategorical formulation of reduction that preserves Morita equivalence and classical limits.

ABSTRACT

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.

Motivation & Objective

• To provide a general algebraic framework for coisotropic reduction beyond geometric constraints.
• To define a bicategory of bimodules over coisotropic triples to formalize Morita equivalence in this context.
• To construct a reduction functor on this bicategory that preserves isomorphisms and Morita equivalence.
• To investigate the compatibility of classical limits with reduction in the bicategorical setting.
• To characterize Morita equivalence of coisotropic triples via explicit structural conditions on their components.

Proposed method

• Introduce coisotropic triples (Atot, AN, A0) where A0 is a two-sided ideal in a subalgebra AN of Atot, generalizing the normalizer construction in coisotropic reduction.
• Define bimodules over such triples with a well-behaved tensor product, forming a bicategory C3Bimod of coisotropic triples and their bimodules.
• Construct a reduction functor red: C3Bimod → Bimod that maps triples to their reduced algebras and bimodules to their quotients.
• Use the bicategorical framework to define Morita equivalence as isomorphism in C3Bimod, and derive explicit conditions for equivalence.
• Analyze the classical limit of coisotropic triples and their bimodules, proving that the limit functor commutes with reduction in the bicategorical sense.
• Employ pseudofunctors and natural transformations in bicategories to formalize the structure and coherence of the reduction process.

Experimental results

Research questions

• RQ1 How can coisotropic reduction in Poisson geometry be formalized algebraically in a way that preserves categorical structure?
• RQ2 What is the correct notion of bimodule and Morita equivalence for coisotropic triples of algebras?
• RQ3 Can a reduction functor be defined on a bicategory of coisotropic triples and their bimodules that preserves isomorphisms and Morita equivalence?
• RQ4 Does the classical limit of a coisotropic triple commute with the reduction process in the bicategorical setting?
• RQ5 What explicit conditions characterize Morita equivalence of coisotropic triples in terms of their components?

Key findings

• The paper constructs a bicategory C3Bimod of coisotropic triples and their bimodules, with a well-defined tensor product and composition.
• A reduction functor red: C3Bimod → Bimod is defined, which maps isomorphic objects to isomorphic objects and thus preserves Morita equivalence.
• Morita equivalence of coisotropic triples is characterized in Theorem 5.5 as isomorphism in C3Bimod, requiring Morita equivalence of the Atot and AN components and a compatibility condition on A0.
• The classical limit of coisotropic triples and their bimodules commutes with reduction, meaning the limit functor commutes with the reduction functor in the bicategorical sense.
• The framework allows for a systematic treatment of quantum and classical reductions, with explicit compatibility conditions ensuring consistency across quantization and classical limits.
• The construction justifies the use of intermediate subalgebras (like the normalizer) in coisotropic reduction by showing they are essential for defining a consistent bicategorical structure.