Questions tagged [w-algebras]
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14 questions
13
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1
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$p$-adic counterpart of W-algebra
Representation theory and geometry over $k((t))$ and $\mathbb{Q}_p$ have many similarities, and there are many similar constructions, usually motivated from the other side (say the study of affine ...
- 1,391
5
votes
0
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147
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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 ...
- 1,391
5
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212
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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 $\mathbf{P}^1$...
- 5,701
2
votes
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121
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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.
...
- 5,701
3
votes
0
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143
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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 ...
user35360
3
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0
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83
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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, ...
user35360
4
votes
1
answer
389
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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 ...
- 723
4
votes
0
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324
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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)...
- 3,357
9
votes
0
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627
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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 ...
- 390
5
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3
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495
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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 ...
- 7,037
3
votes
3
answers
514
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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?
- 13.2k
3
votes
1
answer
409
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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
...
- 13.2k
12
votes
1
answer
709
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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? ...
- 7,037
10
votes
2
answers
791
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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 ...
- 33.3k