On the Notion of Noncommutative Submanifold

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.