Introduction to W-Algebras/Why W-algebras?

Does anyone know of an introduction and motivation for W-algebras?

Edit: Okay, sorry I try to add some more background. W algebras occur, for example when you study nilpotent orbits: Take a nice algebraic/Lie group. It acts on its Lie-algebra by the adjoint action. Fix a nilpotent element e and make a sl_2 triple out of it. A W algebra is some modification of the universal enveloping, based on this data.

A precise definition is for example given here: https://arxiv.org/pdf/0707.3108.

But this definition looks quite complicated and not very natural to me. I don't see what's going on. Therefor I wonder if there exists an easier introduction.

• This question could use a lot more detail. Especially since there is more than one mathematical object called a "W-algebra." Commented Jan 1, 2010 at 22:11
• Yes. At the very least, a context! Commented Jan 1, 2010 at 22:13
• Asking the author of the paper my email would not be a bad idea! Commented Jan 1, 2010 at 23:05
• Alexander Kleshchev gave a nice series of introductory talks on W-algebras at the University of Virginia a few years ago. Unfortunately, I don't have any notes from those talks, but I do remember my favorite line from the first lecture: "I'm not going to tell you what a W-algebra is. I'm going to tell you what a W-algebra is good for." Commented Aug 31, 2011 at 12:52

W-algebras appear in at least three interrelated contexts.

1. Integrable hierarchies, as in the article by Leonid Dickey that mathphysicist mentions in his/her answer. Integrable PDEs like the KdV equation are bihamiltonian, meaning that the equations of motion can be written in hamiltonian form with respect to two different Poisson structures. One of the Poisson structures is constant, whereas the other (the so-called second Gelfand-Dickey bracket) defines a so-called classical W-algebra. For the KdV equation it is the Virasoro Lie algebra, but for Boussinesq and higher-order reductions of the KP hierarchy one gets more complicated Poisson algebras.

2. Drinfeld-Sokolov reduction, for which you might wish to take a look at the work of Edward Frenkel in the early 1990s. This gives a homological construction of the classical W-algebras starting from an affine Lie algebra and a nilpotent element. You can also construct so-called finite W-algebras in this way, by starting with a finite-dimensional simple Lie algebra and a nilpotent element. The original paper is this one by de Boer and Tjin. A lot of work is going on right on on finite W-algebras. You might wish to check out the work of Premet.

3. Conformal field theory. This is perhaps the original context and certainly the one that gave them their name. This stems from this paper of Zamolodchikov. In this context, a W-algebra is a kind of vertex operator algebra: the vertex operator algebra generated by the Virasoro vector together with a finite number of primary fields. A review about this aspect of W-algebras can be found in this report by Bouwknegt and Schoutens.

There is a lot of literature on W-algebras, of which I know the mathematical physics literature the best. They had their hey-day in Physics around the late 1980s and early 1990s, when they offered a hope to classify rational conformal field theories with arbitrary values of the central charge. The motivation there came from string theory where you would like to have a good understanding of conformal field theories of $$c=15$$. The rational conformal field theories without extended symmetry only exist for $$c<1$$, whence to overcome this bound one had to introduce extra fields (à la Zamolodchikov). Lots of work on W-algebras (in the sense of 3) happened during this time.

The emergence of matrix models for string theory around 1989-90 (i.e., applications of random matrix theory to string theory) focussed attention on the integrable hierarchies, whose $$\tau$$-functions are intimately related to the partition functions of the matrix model. This gave rise to lots of work on classical W-algebras (in the sense of 1 above) and also to the realisation that they could be constructed à la Drinfeld-Sokolov.

The main questions which remained concerned the geometry of W-algebras, by which one means a geometric realisation of W-algebras analogous to the way the Virasoro algebra is (the universal central extension of) the Lie algebra of vector fields on the circle, and the representation theory. I suppose it's this latter question which motivates much of the present-day W-algebraic research in Algebra.

In case you are wondering, the etymology is pretty prosaic. Zamolodchikov's first example was an operator vertex algebra generated by the Virasoro vector and a primary weight field $$W$$ of weight 3. People started referring to this as Zamolodchikov's $$W_3$$ algebra and the rest, as they say, is history.

Ben's answer motivates the study of finite W-algebras from geometric representation theory and points out that a finite W-algebra can be viewed as the quantisation of a particular Poisson reduction of the dual of the Lie algebra with the standard Kirillov Poisson structure. The construction I mentioned above is in some sense doing this in the opposite order: you first quantise the Kirillov Poisson structure and then you take BRST cohomology, which is the quantum analogue of Poisson reduction.

• The text by Dickey I referred to is an article rather than a book :) Also, here is a link to the full text of the English translation of the Drinfeld--Sokolov paper you refer to: springerlink.com/content/rx3531wu1r621353/fulltext.pdf Commented Jan 2, 2010 at 4:03
• Thanks -- I've corrected it now. I was confusing it with his book on integrable solitonic hierarchies. Commented Jan 2, 2010 at 4:15
• Thank you very much for your explanations where W-algebras occur and the motivation. While some parts of your answer are over my head at the moment, it gives me a rough overview. I'll have a look at the various papers you mentioned. Commented Jan 2, 2010 at 8:09

My motivation for studying finite W-algebras comes from geometric representation theory; just as the universal enveloping algebra is a quantization of $$\mathfrak{g}^*$$ with its Kostant-Kirillov Poisson bracket, the W-algebra is a quantization of a Poisson reduction of $$\mathfrak{g}^*$$, the Slodowy slice (see Gan and Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. 2002, no. 5, 243-255, doi:10.1155/S107379280210609X, arXiv:math/0105225 ).

There are very interesting ties between the geometry of this Slodowy slice and the representation theory of the W-algebra. In particular, since Springer representations of Weyl groups arise from this geometry, W-algebras give us hope to categorify these. Also, finite dimensional modules over W-algebras have interesting connections to primitive ideals of the original enveloping algebra.

• Ben: I took the liberty of editing your answer to add backquotes to the LaTeX, since otherwise I could not read what was written. Commented Jan 3, 2010 at 2:27
• Thanks Ben, are there papers about categorification of Springer representations via W-algebras, yet? Commented Oct 6, 2010 at 10:04
• Could you revitalize the Gan&Ginzburg link please? Commented Apr 21, 2021 at 4:20
• @მამუკაჯიბლაძე arxiv.org/pdf/math/0105225.pdf ??? Commented Apr 21, 2021 at 6:35
• @EvgenyKuznetsov Thanks! Also the published version for those who have access: IMRN 5(2002) 243-255 Commented Apr 21, 2021 at 9:10

AGT conjecture predicts that the direct sum of intersection cohomology groups of moduli spaces of $$G$$-instantons on $$\mathbf R^4$$ is a representation of the $$W$$-algebra attached to $$G$$. See https://arxiv.org/abs/1108.5632.

• Langlands dual of $G$, a little more precisely. Commented Aug 31, 2011 at 12:36

You may wish to look into the Lectures on Classical W-Algebras by L.A. Dickey. Also, there are older lecture notes by C.N. Pope (intended for the physicists, I guess) and another introductory text by G.M.T. Watts.

• Both of your links are dead by now, do you know any working ones? Commented Apr 21, 2021 at 4:54
• @მამუკაჯიბლაძე Maybe "introductory text by G.M.T. Watts" is W-algebras and their representations, DOI: 10.1007/BFb0105278, kclpure.kcl.ac.uk/portal/files/5035855/kcl_mth_97_50.pdf (Since the original links is gone, we can only guess. But this seems line a reasonable guess it is also among the reference in the Wikipedia article on W-algebras - current revision). Commented Jan 11, 2022 at 8:33
• Thanks a lot @Martin, your link is very helpful as usual! Regardless of whether it is the same or not, this seems to be an excellent text. Commented Jan 11, 2022 at 8:40

There are some nice notes by Wang about the finite W-algebras. They might be of some help.

• Thanks, I just had a look at the notes, they look very promising from what I've seen. Commented Jan 2, 2010 at 8:10

The motivation did not change. The theory of finite W-algebras advanced quite a bit, imho, mostly thanks to the works of Losev. There is a good introductory text by Arakawa now.