I work entirely over a field of characteristic $0$, in case it matters.

Recall that a *Poisson algebra* is a commutative algebra $A$ with a bracket $\lbrace,\rbrace : A^{\wedge 2} \to A$ which is (1) a Lie bracket, i.e. it satisfies a Jacobi identity, and (2) a derivation in each variable. Or, maybe even better is that $(A,\lbrace,\rbrace: A^{\wedge 2} \to A)$ is a *Poisson algebra* if $A$ is a commutative algebra and $\forall a\in A$ $\lbrace a,-\rbrace : A \to A$ is a derivation of $(A,\lbrace,\rbrace)$.

A *coisotrope* in $A$ is a vector subspace $I \subseteq A$ which is (1) an ideal for the multiplication on $A$ and (2) Lie subalgebra for $\lbrace,\rbrace$. In particular, I do not demand that $I$ be an ideal for the bracket. The most important examples of Poisson algebras are $A = \mathcal C^\infty(M)$, where $M$ is a symplectic manifold; then an embedded submanifold $N \hookrightarrow M$ is *coisotropic* ($\mathrm T^\perp N \subseteq \mathrm T N$) iff the vanishing ideal of $N$ is a coisotrope. For more details and equivalent characterizations, see:

- Alan Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan Volume 40, Number 4 (1988), 705-727. http://projecteuclid.org/euclid.jmsj/1230129807

One reason to invent Poisson algebras is that the arise naturally as "deformation" or "quantization" problems: a Poisson algebra is the linear or infinitesimal data for a noncommutative algebra. In the most studied symplectic case, the Poisson algebra is in an important sense "maximally Poisson-noncommutative": the Poisson-center of $A$ consists only of (locally) constant functions. After quantization, the corresponding algebras are maximally noncommutative, and so should be algebras of (bounded) operators on some Hilbert space. In this sense, the quantization of a symplectic manifold $M$ *is* some Hilbert space $H$, or maybe its projectivization $\mathbb P H = H / \mathbb C^\times$.

Then the general dictionary says that *lagrangian*, i.e. minimal coisotropic, submanifolds of a symplectic manifold $M$ should correspond to elements of $H$ (or maybe elements of $\mathbb P H$, i.e. lines in $H$). My question is to understand a generalization of this that relaxes two things:

I am interested in algebras that are Poisson but not symplectic, and so might have center; then I do not expect to have as good a "Hilbert-space" description of the quanization. Rather, my quantizations of Poisson algebras $(A,\lbrace,\rbrace)$ are nothing more nor less than (flat) deformations of $A$ in the $\lbrace,\rbrace$ direction.

In the non-symplectic setting, one loses a good theory of "lagrangian" submanifold, and the best stand-ins are the coisotropes.

My question is then something along the following:

Suppose I have a Poisson algebra $(A,\lbrace,\rbrace)$ with a coisotrope $I \subseteq A$, and I have a reasonably good quantization of $A$ in the $\lbrace,\rbrace$-direction. What should I expect/hope to see at the quantum level that corresponds to $I$?

Asking the same question in the opposite direction:

Suppose I have an associative algebra $B$ with a formal parameter $\hbar$ (and satisfying some strong flatness/topological freeness conditions), such that $B/(\hbar B)$ is commutative. Then the associated graded algebra $A = \bigoplus (\hbar B)^n / (\hbar B)^{n+1}$ is Poisson. What structures (e.g. ideals, left-modules, etc.) on $B$ become coisotropes in $A$ upon taking associated-graded?