My question is prompted by Ben Webster's answer to this question.

**Is there a notion of tensor product for representations of a finite W-algebra?**

I thought about this question years ago in the context of the infinite-dimensional W-algebras in conformal field theory, but failed to reach a satisfactory conclusion. I was hoping that now that the professional representation theorists have started to study finite W-algebras an answer might be forthcoming.

**Edit** (Added in response to Ben's answer below.)

My (basic) intuition about finite W-algebras is that they are more akin to universal enveloping algebras than to Lie algebras, so one way to rephrase this question is whether perhaps there is some additional structure (coproduct or what have you) on finite W-algebras which would allow one to "fuse" representations in some way.

I think I understand the remark about the Slodowy slice not possessing enough of an algebraic structure, but there is a different description of finite W-algebras besides the one coming from the Slodowy slice and which perhaps suggests that there may be some more structure. This is the interpretation of finite W-algebras as the quantisation of the Poisson algebra of casimirs. Interpreting the symmetric algebra $\mathfrak{S} := \mathrm{Sym}(\mathfrak{g})$ as the polynomial functions on $\mathfrak{g}^*$, it becomes a Poisson algebra à la Kirillov-Kostant. The adjoint Lie group $G$ acts on $\mathfrak{S}$ via automorphisms and the invariant subalgebra is a Poisson subalgebra, which is generated by (the image in the symmetric algebra of) the centre of the universal enveloping algebra: the so-called casimirs.

Another version of the question is whether anything of the Hopf algebra structure present in the universal enveloping algebra survives this procedure.

**Edit** (in response to comments below)

Here's one more stab at motivating why one would expect some "fusing" of representations to exist. The homological construction of the W-algebras actually gives you more: it gives you a machine (I have not checked if it's a functor) to which you feed it a representation of the Lie algebra $\mathfrak{g}$ and you get a representation of the W-algebra. Let us call this machine $H$. If $R,S$ are representations of $\mathfrak{g}$ and $H(R)$ and $H(S)$ the corresponding representations of the W-algebra, one could define a sort of product $\boxtimes$ by $$H(R) \boxtimes H(S) := H(R \otimes S)$$ Is this an interesting notion of tensor product?

I guess I should have mentioned that the reason I was interested in this question was as a first step in the definition of W-covariant operator product expansion. This would allow one to determine the correlators of any fields in a conformal field theory with chiral algebra some W-algebra, in terms of the correlators of the W-primaries. This is all by analogy with the case of superconformal field theories, where one can work in a manifestly supersymmetric formalism.