On algebras and groups of formal series over a groupoid and application to some spaces of cobordism

TL;DR

This paper introduces a framework for deformed algebras and their associated groups, illustrating their structure with examples involving pseudo-differential operators and cobordisms, and explores their applications to stochastic cosurfaces.

Contribution

It generalizes the concept of deformed algebras over groupoids, demonstrating their realization as regular Lie groups and applying this to spaces of cobordisms and stochastic cosurfaces.

Findings

Deformed algebras can produce regular Fr"olicher and Fr"echet Lie groups.

Examples include deformations of pseudo-differential operator groups.

Application to cobordism-related groupoids and stochastic cosurfaces.

Abstract

We develop here a concept of deformed algebras and their related groups through two examples. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of deformed algebra used in \cite{Ma2013} is only an example among others, and how they often give rise to regular Fr\"olicher Lie groups, and sometimes to Fr\'echet Lie groups, that are regular. The first example, indexed by $\mathbb{N},$ is obtained by deformations of the group of bounded classical pseudo-differential operators $Cl^{0,*}$ by algebras of (maybe unbounded) classical pseudo-differential operators. In the second one, the set of indexes is a $\N-$graded groupo\"id, which is made of manifolds with boundary that are understood as morphisms of cobordisms. Here again, we get regular Lie groups, and we show how this setting applies to a class of examples that are derived of so-called stochastic cosurfaces.