Deformation and Hochschild Cohomology of Coisotropic Algebras

Abstract

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 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.

A Serre-Swan Theorem for Coisotropic Algebras

“…Remark 2.8 Instead of keeping the underlying set of the 0-component of a given coisotropic module we could also use the equivalence relation on the N-component induced by the 0-component. This leads to the notion of coisotropic sets as introduced in [8]. These two concepts do not agree, but for our purpose coisotropic index sets will be more useful.…”

“…In this preliminary section we introduce coisotropic algebras and their modules following [7,8]. For this we will first need to consider coisotropic -modules as the fundamental algebraic structure underlying coisotropic algebras and their modules.…”

“…Remark 2.28 As noted in Remark 2.8, the concepts of coisotropic index sets Set 3 and coisotropic sets C 3 Set as in [8] do not agree. However, it can be shown that regular projective coisotropic modules can equivalently be described using the corresponding notion of C 3 Set inj -free coisotropic modules: Again they can be viewed as direct summands or images of idempotents on such free coisotropic modules and also a version of a dual basis exists.…”

“…In [7,8] a more algebraic and conceptual approach was proposed: the notion of coisotropic (triples of) algebras is designed to provide an algebraic formulation for the above reduction scheme. One considers a total algebra A tot together with another algebra A N and an algebra homomorphism ι : A N → A tot .…”

“…While [7,8] focus on general algebraic features, the aim of this paper is to give a first contact to the underlying geometry for a particular class of coisotropic algebras by formulating a Serre-Swan Theorem in this context.…”