On the Notion of Noncommutative Submanifold

TL;DR

This paper explores the concept of submanifold algebras within associative algebras, examining their properties, examples, and obstructions, especially in the context of smooth functions and deformation quantizations of symplectic manifolds.

Contribution

It reviews the notion of submanifold algebra, discusses its properties, and identifies topological obstructions in deformation quantizations, providing new insights into noncommutative geometry.

Findings

Every quotient algebra of smooth functions on manifolds is a submanifold algebra.

Topological obstructions exist for submanifold algebras in deformation quantizations.

Provides examples and counterexamples in both commutative and noncommutative settings.

Abstract

We review the notion of submanifold algebra, as introduced by T. Masson, and discuss some properties and examples. A submanifold algebra of an associative algebra $A$ is a quotient algebra $B$ such that all derivations of $B$ can be lifted to $A$. We will argue that in the case of smooth functions on manifolds every quotient algebra is a submanifold algebra, derive a topological obstruction when the algebras are deformation quantizations of symplectic manifolds, present some (commutative and noncommutative) examples and counterexamples.