Is the category of representations of a finite W-algebra monoidal?

Question

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.

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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.

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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.

Answer 1

I don't think any is known or expected.

Perhaps the simplest heuristic for why one shouldn't expect such a structure is that $\mathfrak{g}^*$ is a Poisson-Lie group (with the usual addition), and the Slodowy slice is not.

Answer 2

This is just to add another point of view to Ben's and David's comments and mainly for your Edit regarding superconformal field theories, as this is more about the affine case than the finite one.

1) There is a formal relation between the finite W-algebra and the affine one: The finite is the Zhu algebra of the affine one (Kac-De Sole '05).

This in particular implies that irreducible modules for the affine algebra are in 1-1 correspondence with irreducible modules for the finite one. From here perhaps your suspicion that if you expect tensor products for modules of the affine ones then you should expect them for the finite one. That article also answer your question regarding a BRST construction of the finite W-algebra (see the appendix, and also the very nice article by Gan-Ginzburg in the finite case)

2) There is a fusion category structure for the affine W-algebra at certain levels, namely, it is proved in some cases when these W-algebras are rational, and it is still conjectured in many others. At any rate, we know of at least a few examples when we indeed have the tensor structure in the categories of modules for the affine W-algebra (See articles by Arakawa and Kac-Wakimoto starting in 2005 up to late '10).

One observation is that the affine W-algebra that is rational is the simple quotient of the affine W-algebra mentioned in 1) above.

3) The functor that you mention is just the "top component" of the homological reduction functor in the affine situation, where you feed a representation for the affine algebra g and you get a representation for the affine W-algebra (the simple one). This functor has been studied by many, there was a conjecture of Frenkel-Kac-Wakimoto regarding the exactness and behaviour of this functor. Arakawa proved (for the principal nilpotent first in Invent. '05 and then in more generality starting in '08, see also Kac-Wakimoto '07) that this functor is exact and it sends irreducible modules to either zero or irreducible modules. He used this to prove existence of modular invariant representations of the W-algebra. In particular, as you mention this gives a way of fusing W-representations by using the fusion of the affine ones.

Actually, Arakawa proved the irreducibility part in the principal nilpotent case, and as far as I know the almost-irreducibility in general (0802.1564). In some cases he can use a previous result in the finite case: this property of sending irreducibles to irreducibles or zero was proved by Brundan-Kleschev in type A (they proved more: that every simple module over W-finite arises in this way).

4) In the cases when you can prove that a) the affine W-algebra is rational, and b) that every module over the affine W-algebra is the Hamiltonian reduction of a module over the affine lie algebra g, then you get fusion for the W-algebra from fusion on g, and I do not see anything wrong with passing to the finite W-algebra by taking Zhu's algebras everywhere, so in this setting I agree with you, and looking at the list in Kac-Wakimoto '07 of possible rational W-algebras, you should get several examples of finite ones. I don't see anything wrong with this but I may be missing something, perhaps some subtlety between the W-algebra and its simple quotient?