Questions tagged [w-algebras]

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Functoriality of Feigin–Frenkel duality

For a simple Lie algebra $\mathfrak{g}$, we have the W-algebra of level $k$, denoted by $\mathcal{W}^k(\mathfrak{g})$. Using Wakimoto free field realization and screening operators, Feigin and Frenkel ...
4 votes
0 answers
187 views

Relations between Whittaker functions/W algebras and Stokes data/resurgence

Skippable background: A Whittaker function is more or less a function on a flag manifold which is twisted-invariant for the action of a unipotent subgroup. E.g. consider functions $f$ on $\mathbf{P}^1$...
5 votes
3 answers
488 views

Good even grading and principal Levi type

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e$ be a nilpotent element in it. In the theory of finite W-algebras one often encounters the following two conditions: 1) $e$ is ...
2 votes
0 answers
108 views

What is the factorization algebra/space of an affine W algebra?

The affine vertex algebra $V_k(\mathfrak{g})$ factorizes, i.e. comes from a factorisation space, the Beilinson Drinfeld Grassmannian. Similarly, lattice vertex algebras have a factorization analogue. ...
3 votes
3 answers
500 views

Irreducible representations of W-algebra in case $\mathfrak sl_3$

Is there a paper in which are all the irreps of the finite W-algebra with trivial action of the center are classified, in the case of $\mathfrak sl_3(\mathbb C)$ and the minimal orbit?
3 votes
0 answers
136 views

Twisted screening operators and twisted free-field realizations of $\mathcal{W}_n$ algebras

Let $\mathfrak{g}=\mathfrak{sl}_{n+1}$ and I am interested in the principal $\mathcal{W}$-algebra of $\mathcal{g}$ at self-dual level i.e. $k=- h ^{\vee} +1$, usually denoted by $\mathcal{W}_n$. Now ...
3 votes
0 answers
80 views

Composition of operators in $w_{1+\infty}$ and $W_{1+\infty}$

The algebra $W_{1+\infty}$ can be defined as a central extension of the lie algebra $w_{1+\infty}$ (defined as being spanned by $\left(-\partial_z \right)^m z^{-k}$ ). See for example: Alexandrov, ...
4 votes
1 answer
361 views

Globalizing Feigin--Frenkel duality

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{g}^L$ be its Langlands dual. Feigin--Frenkel duality says $$ W^k(\mathfrak{g})=W^{k_L}(\mathfrak{g}^L) $$ if $r'(k+h^{'})(k_L+h'_L)=1$, where ...
3 votes
0 answers
306 views

Center of affine W-algebras

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $k$ a complex number. Denote by $\hat{\mathfrak{g}}$ the corresponding affine Lie algebra ($\hat{\mathfrak{g}}=\mathfrak{g}((t)...
9 votes
0 answers
605 views

Quantum Drinfeld-Sokolov reduction for a module

There is a well-established procedure for quantizing the Drinfeld-Sokolov reduction for an affine Lie algebra. In particular, this paper of de Boer and Tjin describes an algorithm to produce the ...
12 votes
1 answer
700 views

Cartan involution for finite W-algebras

Does anybody know if there is an analog of the Cartan (anti)involution for W-algebra associated to a nilpotent element e, which is principal in some Levi subalgebra of semi-simple Lie algebra g? ...
10 votes
2 answers
770 views

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

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 ...
3 votes
1 answer
401 views

A question on the construction of finite W-algebras

In a well known construction of finite W-algebras, one first constructs a certain nilpotent subalgebra $\mathfrak{m}$ along with a character $\chi:\mathfrak{m}\rightarrow \mathbb{C}$. Then one defines ...