Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 050, 21 pages      arXiv:1912.01225      https://doi.org/10.3842/SIGMA.2020.050

On the Notion of Noncommutative Submanifold

Francesco D'Andrea
Università di Napoli ''Federico II'' and I.N.F.N. Sezione di Napoli, Complesso MSA, Via Cintia, 80126 Napoli, Italy

Received January 11, 2020, in final form May 30, 2020; Published online June 09, 2020

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.

Key words: submanifold algebras; tangential star products; coisotropic reduction.

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