# Distinguishing the Duflo star product

$\newcommand{\g}{\mathfrak g}\newcommand{\h}{\hbar}$ For a finite dimensional Lie algebra $\g$, he Duflo isomorphism is a complicated algebra isomorphism between the $\g$-invariant part $S(\g)^\g$ of the symmetric algebra, and the center $Z(U(\g)$ of the enveloping algebra.

One way to undertsand this result is as follow : $S=S(\g)$ is identified with the algebra of polyonmial functions on $\g^*$. The Lie bracket of $\g$ turns this algebra into a Poisson algebra. Let $U_{\h}$ be the enveloping algebra of $\g_{\h}$, which is $\g[[\h]]$ in which the bracket is multiplied by $\h$. In particular $U_0=S$ and the PBW theorem implies that $U_\h$ is a flat deformation of $S$. Hence there is a linear (in fact $\g$-linear) isomorphism $S[[\h]]\rightarrow U_\h$. The pull-back of the product in $U_\h$ through this isomorphism induces a star product $m_{pbw}$ on $S[[\h]]$ which is more or less by definition a quantization of the Poisson algebra $S(\g)$.

The space $S(\g)^\g$ is the Poisson center of $S(\g)$. Then Duflo theorem can be rephrased by saying that there exists a star-product $m_d$ whose restriction to the Poisson center is undeformed, i.e. identical to the original product on $S(\g)^g$.

Two star-products are said to be equivalent if they are related by an algebra isomorphism which is the identity modulo $\h$. Equivalence classes of star product are essentially classified by the Hochschild cohomology.

It turns out that while $m_d$ satisfies a strong property which $m_{pbw}$ doesn"t, they are equivalent as star products. Hence my question is:

Is there a finer equivalence relation on star-products which distinguishes between those ?

Or perhaps

Is there an appropriate replacemnt for Hochschild cohomology which classify star-products on Poisson algebras respecting the Poisson center ?

A sort of obvious guess would be cyclic cohomology since both cyclic cohomology and the Duflo star-product have very much to do with traces. Yet it seems that cyclic cohomology is relvant only for unimodular Poisson manifolds, and describes so-caled closed star-product respecting a specific trace rather than the space of those.

edit: I found some answers here. They basically give a sufficient condition for a star product on $A$ to be trivial on $C$ in term of the kernel of the restriction map $$CH(A,A)\rightarrow CH(C,A).$$

They also claim without full details that under mild assumption, if there is a star product on $A$ whose restriction to $C$ is commutative then this star product is equivalent to one whose restriction to $C$ is actually trivial. This sound surprising since if it were true in the case I'm interested in it would give a rather easy proof of Duflo's theorem...

• Why " Duflo theorem can be rephrased by saying that there exists a star-product mdmd whose restriction to the Poisson center is undeformed, " ? – Alexander Chervov Feb 16 '16 at 10:06
• Both star products can be specialized at $\hbar=1$, and in both case by universal property the result is isomorphic as an algebra to the enveloping algebra. Hence, for the Duflo one, we get an algebra which is isomorphic to the enveloping algebra and whose subalgebra of invariant is literally equal to $S(\mathfrak g)^{\mathfrak g}$. So the restriction $$(S(\mathfrak g),m_d) \supset S(\mathfrak g)^{\mathfrak g} \cong U(\mathfrak g)^{\mathfrak g} \subset U(\mathfrak g)$$ recover the Duflo isomorphism. – Adrien Feb 16 '16 at 10:15