Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$. Define $\mathcal Z(\mathfrak g)$ to be the center of the universal enveloping algebra $\mathcal U\mathfrak g$, and define $(\mathcal S\mathfrak g)^{\mathfrak g}$ to be the ring of invariant elements of the symmetric algebra $\mathcal S\mathfrak g$ under the induced adjoint action of $\mathfrak g$. (Clearly $\mathcal Z(\mathfrak g) = (\mathcal U\mathfrak g)^{\mathfrak g}$ via the adjoint action.) The *Duflo isomorphism* is an isomorphism of algebras $\mathcal Z(\mathfrak g) \cong (\mathcal S\mathfrak g)^{\mathfrak g}$. At the level of vector spaces, the trick is to realize that the PBW map $\mathcal U\mathfrak g \to \mathcal S \mathfrak g$ is an isomorphism of $\mathfrak g$-modules. For the isomorphism of algebras in the semisimple case, see for example [my unedited notes on the class by V. Serganova](http://math.berkeley.edu/~theojf/QuantumGroups10.pdf).


(I read [here](http://www.sfb45.de/events/the-duflo-isomorphism-and-its-relatives) that this isomorphism can be realized as a composition of the PBW vector-space isomorphism $\mathcal U\mathfrak g \to \mathcal S\mathfrak g$ with the map $\mathcal S\mathfrak g \to \mathcal S\mathfrak g$ given by $x \mapsto \sinh(x/2)/(x/2)$. But it's not at all obvious that this composition is even well-defined or linear when restricted to $\mathcal Z(\mathfrak g)$. I should mention that $\mathcal Z$ is not a functor, I think. The PBW isomorphism is non-canonical, although a canonical version can be given via the symmetrization map, and I guess on the center it is canonical.)


When $\mathfrak g$ is semisimple of rank $n$, at least, one can further show that $(\mathcal S\mathfrak g)^{\mathfrak g} \cong \mathbb C[x\_1,\dots,x\_n]$, although you have some choice about how to make this isomorphism. Thus, at least when $\mathfrak g$ is semisimple, $\mathcal Z(\mathfrak g)$ is a polynomial ring.


But any polynomial ring can be given a Hopf structure. By choosing an isomorphism with $\mathbb C[x\_1,\dots,x\_n]$, we can take the Hopf structure generated by $\Delta: x\_i \mapsto x\_i \otimes 1 + 1 \otimes x\_i$. In fact, this structure doesn't depend quite on the full choice of isomorphism. A Hopf structure on a commutative algebra $R$ is by definition the same as an algebraic group structure on $\text{Spec}(R)$. But $\text{Spec}(\mathbb C[x\_1,\dots,x\_n])$ is $n$-dimensional affine space — the algebra isomorphisms of $\mathbb C[x\_1,\dots,x\_n])$ are precisely the affine maps — so picking a commutative group structure is the same as picking an origin. (For certain values of $n$ there are also non-commutative group structures on affine $n$-space, and so non-cocommutative Hopf structures on the polynomial ring. For example, the group of upper-triangular matrices with $1$s on the diagonal is affine.)


My question is whether this Hopf structure can be picked out canonically.



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> **Question:** If $\mathfrak g$ is a finite-dimensional Lie algebra over $\mathbb C$, can the center $\mathcal Z(\mathfrak g)$ of the universal enveloping algebra be given a canonical (cocommutative) Hopf algebra structure? If no, how much extra structure on $\mathfrak g$ is needed?
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Here by "canonical" I of course don't mean that there is a unique one, so you may make choices once and for all. But there should be some definition/construction that does not require the user to make any choices to implement it. By "extra structure" I mean either extra structure (an invariant metric, for example) or extra properties (semisimplicity, for example).


My suspicion is that the answer is "yes" for a *metric Lie algebra*, which is a Lie algebra $\mathfrak g$ along with a choice of an invariant nondegenerate metric, i.e. a chosen isomorphism of $\mathfrak g$-modules $\mathfrak g \cong \mathfrak g^\*$. Metric Lie algebras include the semisimples and the abelians, and certain extensions of these (in fact, I believe that there is a structure theorem that any metric Lie algebra is a metric extension of semisimples and abelians, but don't quote me), but generally there are many choices of metric (e.g. any metric on an abelian Lie algebra $\mathfrak a$ is invariant, so there are $\mathfrak{gl}(\dim \mathfrak a)$ many choices).


The motivation for my question is this: by studying Vassiliev invariant and/or perturbative Chern-Simons theory, Bar Natan and others have defined a certain commutative and cocommutative Hopf algebra $A$ of "diagrams". Any choice of metric Lie algebra $\mathfrak g$ determines an algebra homomorphism $A \to \mathcal Z(\mathfrak g)$. I would like to know if this can be made into a Hopf algebra homomorphism.



Is there a canonical Hopf structure on the center of a universal enveloping algebra?
In the semisimple case, one has the Harish-Chandra isomorphism between the center ${\mathcal Z}(\mathfrak g)$ and $(S{\mathfrak h})^W$, where ${\mathfrak h}$ is a Cartan subalgebra of ${\mathfrak g}$ and $W$ is the Weyl group. On $S{\mathfrak h}$, there is a natural inner product $(f,g)=f(\partial)g(x)|\_{x=0}$ induced by the natural inner product on ${\mathfrak h}$. 


Let $I\subset (S{\mathfrak h})^W$ be the augmentation ideal. Let $E$ be the orthogonal complement of the ideal $I^2$ in $(S{\mathfrak h})^W$ (the square of $I$), and let $E\_+$ be the positive degree part of $E$. Then $(S{\mathfrak h})^W=SE\_+$, so we can define the Hopf algebra structure on $(S{\mathfrak h})^W$ by declaring that $E\_+$ consists of primitive elements. 


This was explained to me by Kostant. I don't know, however, if it is sufficiently canonical, or helpful in the question related to the Bar-Natan construction. 


