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On Virasoro-type reductions and inverse Hamiltonian reductions for $W$-algebras and $W_\infty$-algebras

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Pith's one-line read The authors prove that a Virasoro-type reduction sends the height-two W-algebras of classical types to the W-algebras of the smallest nilpotent orbit containing them, and that the passage is reversible after tensoring with a free-field…

desk verdict Strong extension of reduction-by-stages for height-two classical W-algebras, but the BCD main theorem rests on omitted screening computations that a referee should require. read the letter →

arxiv 2411.10694 v2 pith:DXDQ5OJV submitted 2024-11-16 math.QA math.RT

Justine Fasquel , Vladimir Kovalchuk , Shigenori Nakatsuka This is my paper · ORCID
classification math.QAmath.RT MSC 17B6917B6717B08
keywords Virasoro-typereductionW-algebrasquantumHamiltonianinverseWakimotorealizationnilpotentorbitsW-infinityalgebrasscreeningoperators
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that, for a family of W-algebras attached to height-two nilpotent orbits in classical Lie algebras, the Virasoro-type quantum Hamiltonian reduction $H^0$ sends $W_k(g,O)$ isomorphically to $W_k(g,\hat{O})$, where $\hat{O}$ is the smallest nilpotent orbit whose closure contains $O$. It also constructs the inverse Hamiltonian reduction: $W_k(g,O)$ embeds into $\Pi \otimes W_k(g,\hat{O})$, with $\Pi$ the half-lattice vertex algebra. The same statements are promoted to a module level and to the universal two-parameter $W_\infty$-algebra $W_\infty^{\mathrm{sp}}(c,k)$, whose Virasoro-type reduction is shown to be a simple freely generated vertex algebra of type $W(2^3,3,4^3,5,6^3,\ldots)$. If correct, this gives a systematic reduction and inverse reduction connecting neighboring W-algebras in the closure order of nilpotent orbits, and lifts the pattern to universal objects.

What carries the argument

The carrying object is the Wakimoto free-field realization of W-algebras at generic levels, which places $W_k(g,O)$ inside $\beta\gamma^{\star} \otimes \Phi(\mathfrak{g}_{1/2}) \otimes \pi_{\mathfrak{h}}^{k+h^\vee}$ as the common kernel of screening operators $S_i^O = \int Y(P_i^O e^{-\alpha_i/(k+h^\vee)}, z) dz$. The reduction under study is the Virasoro-type BRST complex with differential $d = \int Y((G^+ + 1)\phi^*, z) dz$, where $G^+$ is the strong generator playing the role of the positive root of $\mathfrak{sl}_2$. Gauging the free fields transforms $d$ into a derivative of a single $\beta$-gamma pair, reducing the cohomology to an intersection of transformed screening operators that is precisely the Wakimoto realization of $W_k(g,\hat{O})$. The inverse reduction is obtained by localizing that $\beta$-gamma pair into the half-lattice vertex algebra $\Pi$, which matches the screening operators on both sides.

What would settle it

Work out the rank N=3 case of W_k(sp_6, O_[$3^{2}$]) using the stated Wakimoto realization: if the kernel intersection does not reproduce W_k(g,O) or if the reduced screening operators fail to match the Wakimoto realization of W_k(sp_6, O_[4,2]), the chain of proof breaks.

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Extended reading notes

Core claim

At generic level, each W-algebra in Table 1 is realized inside a free field algebra as the common kernel of screening operators coming from its Wakimoto realization. The paper shows that the strong generator $G^+$, an analogue of the upper nilpotent element of $\mathfrak{sl}_2$, is a sum of $\beta$-fields, and that the Virasoro-type BRST differential can be gauged into a differential acting on a single $\beta$-gamma pair. Computing the cohomology of the Wakimoto modules leaves an intersection of transformed screening operators that exactly matches the Wakimoto realization of $W_k(g,\hat{O})$. Hence $H^0(W_k(g,O))$ is isomorphic to $W_k(g,\hat{O})$ with cohomology vanishing, and localizing the relevant $\beta$-gamma pair into $\Pi$ identifies the screening operators on both sides, producing the inverse embedding $W_k(g,O) \hookrightarrow \Pi \otimes W_k(g,\hat{O})$.

Load-bearing premise

The paper relies on the explicit Wakimoto realizations of Propositions 4.3–4.5, whose proofs are omitted, and on the freeness, simplicity, and structure of W_∞^sp(c,k) imported from the authors' preprint [21]; if either input fails, the screening-operator identification and the universal reduction no longer follow.

Editorial extensions

If this is right

  • For each pair in Table 1, $H^0(W_k(g,O))$ is isomorphic to $W_k(g,\hat{O})$ at generic level, with cohomology vanishing in all but the unstarred rows of Table 1; type A and the starred rows extend to all levels.
  • There is an embedding $W_k(g,O) \hookrightarrow \Pi \otimes W_k(g,\hat{O})$ for all levels, realizing the inverse Hamiltonian reduction.
  • For $M$ in the Kazhdan–Lusztig category $KL_k(g)$, $H^0(H_O(M))$ is isomorphic to $H_{\hat{O}}(M)$, so the functors $H^0 \circ H_O$ and $H_{\hat{O}}$ are naturally isomorphic.
  • The Virasoro-type reduction of the universal $W_\infty^{\mathrm{sp}}(c,k)$ is a simple freely generated vertex algebra of type $W(2^3,3,4^3,5,6^3,\ldots)$, free over its base ring, and the W-algebras $W_k(\mathfrak{sp}_{4n+2}, O_{[2n,2n+2]})$ and $W_k(\mathfrak{so}_{4n}, O_{[2n-1,2n+1]})$ arise as 1-parameter quotients of it over $\mathbb{C}(k)$.
  • The type A statement extends to $W_k(\mathfrak{sl}_{N+M}, O_{[N,M]})$: $H^0$ gives $W_k(\mathfrak{sl}_{N+M}, O_{[N+1,M-1]})$ at generic level, with an inverse embedding for all levels.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Iterating the Virasoro-type reduction along chains in the Hesse diagram would produce W-algebras of larger orbits from smaller ones by elementary steps, so the reduction may serve as a building block for the conjectural reduction-by-stages descriptions of W-algebras.
  • The module-level isomorphism suggests that, at generic level, the Virasoro-type reduction and the inverse localization define an equivalence between weight-module categories of neighboring W-algebras; that equivalence is not constructed in the paper.
  • The universal theorem motivates searching for analogous two-parameter algebras of type $W(1,2^3,3,4^3,5,\ldots)$ whose Virasoro-type reduction would be an extension of two even-spin $W_\infty$-algebras; the explicit two commuting Virasoro vectors and weight-4 primaries in (9.31)-(9.36) give a concrete OPE check.
  • For the unstarred rows of Table 1, the continuity argument does not settle all levels; a direct attack on the finite-dimensional Slodowy-slice group-action problem the authors identify would decide the remaining cases.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

6 major / 5 minor

Summary. The paper studies Virasoro-type quantum Hamiltonian reductions for W-algebras associated with nilpotent orbits of height two in classical types, and their inverse Hamiltonian reductions. The main results are: Theorem A, giving isomorphisms H^0(W_k(g,O)) ≅ W_k(g,\hat O) for the pairs in Table 1 at generic level and embeddings W_k(g,O) ↪ Π ⊗ W_k(g,\hat O); Theorem B, a module-level analogue for modules in the Kazhdan–Lusztig category; and Theorem C, computing the Virasoro-type reduction of the universal two-parameter algebra W^{sp}_∞(c,k) as a simple freely generated algebra of type W(2^3,3,4^3,5,6^3,...), with applications to W-algebras of types C and D. The proofs use Wakimoto realizations and screening operators, following and extending previous work on partial reductions in type A.

Significance. If the main theorems are correct, the paper provides a systematic mechanism relating adjacent W-algebras via small, explicit reductions and inverse reductions, including new universal statements for W^{sp}_∞. This would be a valuable contribution to the structure theory of W-algebras, with potential consequences for representation theory and for the program of building W-algebras from fundamental reductions. The manuscript has genuine strengths: the type A proof is explicit, the screening-operator framework is concrete and falsifiable, no fitted parameters are introduced, and the module-level functor isomorphism in Theorem B is a useful extension. The BCD and universal parts, however, currently rest on several omitted computations and on external results, as detailed below; the verdict is therefore conditional.

major comments (6)
  1. [§4.2.2–4.2.4, Props. 4.3–4.5] The Wakimoto realizations for types B, C, and D, including the zero-graded sets ⋆ and all screening coefficients P_i^O, are central inputs for the proof of Theorem 3.6. Proposition 4.3 states that 'the computations of ⋆ and the P_i's are parallel to the proof of Proposition 4.2, which we omit', and Propositions 4.4 and 4.5 have no proof at all. These formulas are used without further verification in Section 5: the change of variables in (5.19), (5.30), (5.38), (5.49), (5.57), and (5.63), and the identification of the reduced screening operators with those of W_k(g,\hat O), depend directly on every coefficient. I therefore regard the BCD rows of Theorem A(1) and Corollary 9.2 as conditional on unverified computations. Please provide the omitted computations or, failing that, independent checks such as characters or strong-generating-type tests for small N.
  2. [Theorem 3.6, Remark 3.7] Theorem 3.6 is stated for generic k, and Remark 3.7 explains that the all-level isomorphism and cohomology vanishing are obtained only when (g,O) is (so_{2N},O_{[N^2]}) or (so_{2N+1},O_{[N^2,1]}) with N even, or (sp_{2N},O_{[N^2]}) with N odd; the other parity cases remain open. The abstract, however, says the reductions are 'established for classical Lie type and nilpotent orbits of height two' without this generic-level qualification. The paper should either soften the abstract and the introductory framing to state the generic-level result, or prove the missing parity cases; as written, the claims exceed the results.
  3. [§6, Theorem 6.1(2)] Theorem 6.1(2), the inverse Hamiltonian embedding for all BCD pairs in Table 4, is a main conclusion of the paper, but its proof is one sentence: 'The proof of (2) is similar, we omit it.' Since the embedding is built from the localization trick and an automorphism analogous to (6.12), and since it is claimed for all levels, the omission is load-bearing. Please provide the construction at least for one representative parity case in each type, or restrict the statement to the cases that are actually proved.
  4. [§7, after Eq. (7.10)] The proof of Theorem 7.1 for general [N,M] is only sketched. After deriving the induced screenings (7.10), the text says 'We show with the same argument as for Theorem 3.5 that this set of screenings can be obtained by [the Wakimoto realization] ... We omit the details.' This is precisely the point where H^0(W_k(sl_{N+M},O_{[N,M]})) is identified with W_k(sl_{N+M},O_{[N+1,M-1]}). Please provide the omitted pyramid/good-pair computation, or explicitly restrict the theorem to the case M=1 proved in Section 5.
  5. [§5.2–5.4, Eqs. (5.24), (5.34), (5.44), (5.54), (5.61), (5.66)] The conjugacy claims that identify the formal pairs (f_c, Γ_c) with standard good pairs are not demonstrated. For example, in type B even, f_c in (5.24) contains the term 2E^o_{-N,N-1}, while the pyramid in Figure 12 is said to give f_{\hat O} containing E^o_{-N+1,N}; the displayed conjugation matrix (5.26) is not accompanied by any computation showing that it preserves the relevant bilinear form and sends f_c to f_{\hat O}. The same issue occurs in (5.34), (5.44), (5.54), (5.61), and (5.66). Since the good-pair property is required to invoke Theorem 4.1, these checks cannot be omitted.
  6. [§9, proof of Theorem 9.1] The proof of Theorem 9.1 uses, without re-proof, the freeness, complete reducibility over Q(R), and simplicity of W^{sp}_∞(c,k) from [21], as well as the Kazhdan–Lusztig categorical facts in (9.18) from [20,35]. Since [21] is a preprint by one of the authors, the universality part of the paper is conditional on an external result whose status should be made explicit. I recommend either including a short proof of the needed facts or clearly stating this dependency in Theorem 9.1 and Corollary 9.2.
minor comments (5)
  1. [§4.2.4, §5.3.1] There are typos in 'vertex algberas' in Proposition 4.4 and 'Propositionn' in §5.3.1; please correct them.
  2. [§5.4.1] In the paragraph following Eq. (5.58), the cohomology is written as H^p(βγ_{N+1}) but the reduced pair is βγ_N; please fix the index.
  3. [§7] The text refers to 'Theorem 4.7' after Eq. (7.10), but the only Wakimoto realization theorem in the paper is Theorem 4.1; please correct the cross-reference.
  4. [Tables 1 and 4] The tables rely on 'upper row' and 'lower row' to distinguish N even and odd, but this convention is not stated in the captions; please add explicit labels such as 'N even' and 'N odd'.
  5. [§5.1 and §7] The isomorphisms (5.6) and (7.6) are described as isomorphisms of vertex algebras between βγ-systems, but the domain and codomain indexing is not fully explained; a sentence indicating that these are the standard change-of-basis isomorphisms for βγ vertex algebras would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main isomorphisms are verified by explicit Wakimoto screening-operator computations rather than by fitting or renaming the target; the only same-author reliance is on prior structure results for the input W∞-algebra that do not contain the reduction conclusions.

full rationale

The derivation chain is not circular. In Section 5, Theorems 3.5 and 3.6 are proved by computing the Virasoro-type reduction on the Wakimoto realization of Wk(g,O), obtaining explicit reduced screening operators, and then recognizing them as the screening operators characterizing Wk(g,Ohat) via Theorem 4.1. The identification is a genuine computation: for instance, the type-B even case changes coordinates by (5.19), reduces the differential to d=∫Y(γ0+1)ϕ*,z)dz, and then matches the resulting coefficients (5.23) with the Wakimoto data of a conjugate good pair for O[N+1,N−1,1], via the explicit grading-preserving conjugation (5.24)–(5.27). Similar matching occurs in types C and D, and in the type-A case. The target algebra is not assumed or built into the input; both sides are independently characterized as kernels of screening operators and compared. The same holds for Theorem B, whose proof (Section 8) reduces the statement to the same identification of screening operators through Corollary 8.2 and Proposition 8.1. For the universal part, Theorem 9.1 uses the Kac–Wakimoto reduction method on Wsp∞(c,k), whose construction and freeness over R come from [21], a preprint sharing an author. That citation supplies the input algebra and its basic structure, not the claimed cohomology or generating type; the simplicity and generating type of H0(Wsp∞(c,k)) are derived in this paper from the decomposition (9.5)–(9.15) and the explicit generators (9.10). Corollary 9.2 uses [21] for the quotient statements about Wsp∞ and then applies Theorem 3.6 over C(k), so the target W-algebras are not encoded in the assumptions. The omitted proofs of the explicit Wakimoto coefficients in Propositions 4.3–4.5 are a completeness and correctness risk, but not a circularity: the coefficients are computational inputs, not consequences of the target isomorphisms. Overall, the paper contains some self-citation but no load-bearing circular reduction; the score is therefore low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims depend on established background theorems (Wakimoto realizations, BRST cohomology vanishing) and on the prior construction of W^{sp}_∞ in [21]. No numerical parameters are fitted to data; the level k and central charge c are variables in the universal algebra.

assumptions (5)
  • standard math Wakimoto realization of W-algebras at generic levels (Theorem 4.1, citing [37]): Wk(g,O) is isomorphic to the intersection of kernels of screening operators inside a free field algebra.
    Used as the starting point for all reduction computations in Sections 5-8. The paper does not prove this theorem.
  • standard math Cohomology vanishing H^{≠0}_O(Vk(g)) = 0 for quantum Hamiltonian reductions of affine vertex algebras (citing [9]).
    Needed to ensure exactness of the sequence (4.6) and to identify H^0 with the kernel of screenings.
  • standard math Exactness of the BRST reduction functor H^• on the category O_k of sl2 (citing [6]), used to extend type A results to all levels.
    Invoked in Section 5.1 for the all-level statement in Theorem 3.5.
  • domain assumption Continuity argument for deforming the level k from generic to all values, as developed in [26,27,30].
    Used repeatedly to pass from generic-level isomorphisms to all-level embeddings/isomorphisms; the authors note in Remark 3.7 that the argument does not cover all cases, so the all-level isomorphism remains conditional for some BCD families.
  • domain assumption Structure, freeness, and simplicity of the universal W∞ algebra W^{sp}_∞(c,k) from [21].
    Theorem 9.1 and Corollary 9.2 use the generating type, the Kazhdan-Lusztig category properties, and simplicity of W^{sp}_∞ from [21], a preprint by a co-author, as input rather than proving them here.

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Cite this review

Pith. "Pith review of On Virasoro-type reductions and inverse Hamiltonian reductions for $W$-algebras and $W_\infty$-algebras." pith.science (2026). https://pith.science/paper/DXDQ5OJV

@misc{pith2026241110694,
  author       = {Pith},
  title        = {Pith review of: On Virasoro-type reductions and inverse Hamiltonian reductions for $W$-algebras and $W_\infty$-algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DXDQ5OJV}},
  note         = {Machine review of arXiv:2411.10694}
}
abstract

In this article, the Virasoro-type reduction and the corresponding inverse reductions are established for W-algebras associated with classical Lie type and nilpotent orbits of height two. Moreover, these results are lifted to the universal objects by analyzing the Virasoro-type reduction of the vertex algebra $\mathcal{W}^{\mathfrak{sp}}_{\infty}$.

Figures

Figures reproduced from arXiv: 2411.10694 by the authors.

Figure 1
Figure 1. Generator τ for Γ(Dn) α1 α2 α3 α4 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 3
Figure 3. Pyramid for [N 2 ] in type A fN2 = 2 X N−2 i=1 Ei+2,i (3.14) as representative of the orbit and the following good even grading1 α1 α2 α3 α4 α2N−2 α2N−1 0 1 0 1 0 1 0 ΓN2 : . (3.15) Then the W-algebra associated to O[N2] has strong generating type Wk (sl2N , O[N2]) = W(13 , 2 4 , . . . , N4 ). (3.16) The three weight-1 generators, which we denote by G+, J, G−, correspond to the vectors e = X 1≤i≤N i odd Ei,i+1, h = … view at source ↗
Figure 4
Figure 4. Pyramid for [N 2 , 1], N = 2n, in type B 0 N-1 . . . -N 4 2 1 . . . × -3 -5 N . . . -1 -2 -4 . . . 1-N 5 3 × [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figures from the paper (7 more)
Figure 6
Figure 6. Figure 6: Pyramid for [N 2 ], N = 2n, in type C 1 2 3 4 . . . . . . N-2 N-1 N -N 1-N 2-N . . . . . . -4 -3 -2 -1 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 8
Figure 8. Figure 8: Pyramid for [N 2 ], N = 2n, in type D 5 3 N . . . -1 -2 -4 . . . 1-N 4 2 1 N-1 . . . -3 -5 . . . -N [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 11
Figure 11. Figure 11: Pyramid for [N + 1, N − 1] in type A and the same grading Γc. The nilpotent elements fc and fOb are conjugate with each other Adg(fc) = fOb by the grading-preserving adjoint action Adg ∈ SL2N with g =   1 A . . . A 1   , A =  1 0 1 1 . (5.12) Therefore…
Figure 13
Figure 13. Figure 13: Pyramid for [N + 2, N − 2, 1], N = 2n + 1, in type B Indeed the pyramid in [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 15
Figure 15. Figure 15: Pyramid for [N + 1, N − 1], N = 2n + 1, in type C standard good pair associated with the pyramid in [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]
Figure 16
Figure 16. Figure 16: Pyramid for [N + 1, N − 1], N = 2n, in type D 1 × 2 × 4 3 6 5 . . . . . . N-1 N-2 -N N 2-N 1-N . . . . . . -5 -6 -3 -4 × -2 × -1 [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 18
Figure 18. Figure 18: Pyramid for [N, M] in type A where Λ = P(−1)a̟a is the alternating sum of all the fundamental weights ̟a, it follows that Wk (g, O) ⊂ 2N \−1 i=1 Ker[S O i ] ⊂ βγ⋆c ⊗ Π ⊗ π k+h ∨ h (6.13) where [S O i ] are the screening operators in (5.9) acting on the component βγ⋆c …

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Works this paper leans on

54 extracted references · 46 canonical work pages · cited by 1 Pith paper

  1. [21]

    Creutzig, V

    T. Creutzig, V. Kovalchuk, and A. R. Linshaw. Building blocks for W-algebras of classical types. 2024. arXiv: 2409.03465 [math.RT]

    work page arXiv 2024

  2. [1]

    Realizations of Simple Affine Vertex Algebras and T heir Modules: The Cases ˆsl(2) and ˆosp(1, 2)

    D. Adamovi´ c. “Realizations of Simple Affine Vertex Algebras and T heir Modules: The Cases ˆsl(2) and ˆosp(1, 2)”. In: Commun. Math. Phys. 366 (2019), pp. 1025–1067

    work page 2019

  3. [2]

    Relaxed and logarith mic modules of ˆsl3

    D. Adamovi´ c, T. Creutzig, and N. Genra. “Relaxed and logarith mic modules of ˆsl3”. In: Math. Ann. 389 (2023), pp. 281–324

    work page 2023

  4. [3]

    A Realisation of th e Bershadsky–Polyakov Algebras and Their Relaxed Modules

    D. Adamovi´ c, K. Kawasetsu, and D. Ridout. “A Realisation of th e Bershadsky–Polyakov Algebras and Their Relaxed Modules”. In: Lett. Math. Phys. 111.38 (2021)

    work page 2021

  5. [4]

    Weight module clas sifications for Bershadsky– Polyakov algebras

    D. Adamovi´ c, K. Kawasetsu, and D. Ridout. “Weight module clas sifications for Bershadsky– Polyakov algebras”. In: Commun. Contemp. Math. 26.10 (2024), Paper No. 2350063

    work page 2024

  6. [5]

    Rationality and fusion rules of exce ptional W-algebras

    T Arakawa and J van Ekeren. “Rationality and fusion rules of exce ptional W-algebras”. In: J. Eur. Math. Soc. (JEMS) 25.7 (2023), pp. 2763–2813

    work page 2023

  7. [6]

    Representation Theory of Superconformal Alge bras and the Kac-Roan-Wakimoto Conjecture

    T. Arakawa. “Representation Theory of Superconformal Alge bras and the Kac-Roan-Wakimoto Conjecture”. In: Duke Math. J. 130 (2005), pp. 435–478

    work page 2005

  8. [7]

    Representation Theory of W-Algebras

    T. Arakawa. “Representation Theory of W-Algebras”. In: Invent. math. 169 (2007), pp. 219– 320

    work page 2007

Show all 54 references
  1. [8]

    Two-sided BGG resolutions of admissible represent ations

    T. Arakawa. “Two-sided BGG resolutions of admissible represent ations”. In: Represent. Theory 18 (2014), pp. 183–222

    2014

  2. [9]

    Associated Varieties of Modules over Kac-Moody A lgebras and C2-Cofiniteness of W-Algebras

    T. Arakawa. “Associated Varieties of Modules over Kac-Moody A lgebras and C2-Cofiniteness of W-Algebras”. In: Int. Math. Res. Not. 22 (2015), pp. 11605–11666

    2015

  3. [10]

    Arakawa, T

    T. Arakawa, T. Creutzig, and K. Kawasetsu. Weight representations of affine Kac-Moody algebras and small quantum groups . 2023. arXiv: 2311.10233 [math.RT]

    2023 arXiv

  4. [11]

    W -algebras as coset vertex algebras

    T. Arakawa, T. Creutzig, and A. R. Linshaw. “ W -algebras as coset vertex algebras”. In: Invent. Math. 218.1 (2019), pp. 145–195

    2019

  5. [12]

    Quantum Langlands Duality of Repr esentations of W-Algebras

    T. Arakawa and E. Frenkel. “Quantum Langlands Duality of Repr esentations of W-Algebras”. In: Compos. Math. 155 (2019), pp. 2235–2262. REFERENCES 39

    2019

  6. [13]

    Localization of Affine W-Algebras

    T. Arakawa, T. Kuwabara, and F. Malikov. “Localization of Affine W-Algebras”. In: Com- mun. Math. Phys. 335 (2015), pp. 143–182

    2015

  7. [14]

    Beilinson and V

    A. Beilinson and V. Drinfeld. Chiral Algebras . Vol. 51. Colloquium Publications. Amer. Math. Soc., 2004

    2004

  8. [15]

    Good Grading Polytopes

    J. Brundan and S. Goodwin. “Good Grading Polytopes”. In: Proc. Lond. Math. Soc. 94 (2005), pp. 155–180

    2005

  9. [16]

    D. Butson. Vertex algebras from divisors on Calabi-Yau threefolds. 2023. arXiv: 2312.03648 [math.RT]

    2023 arXiv

  10. [17]

    R. W. Carter. Lie algebras of finite and affine type . Vol. 96. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2005, pp. x viii+632

    2005

  11. [18]

    Collingwood and W

    D. Collingwood and W. McGovern. Nilpotent Orbits in Semisimple Lie Algebra: An Intro- duction. Vol. 65. New York: Van Nostrand Reinhold Co., 1993

    1993

  12. [19]

    Creutzig, J

    T. Creutzig, J. Fasquel, A. R. Linshaw, and S. Nakatsuka. On the structure of W-algebras in type A. 2024. arXiv: 2403.08212 [math.RT]

    2024 arXiv

  13. [20]

    Tensor categories arising from the Virasoro algebra

    T. Creutzig, C. Jiang, F. Orosz Hunziker, D. Ridout, and J. Yan g. “Tensor categories arising from the Virasoro algebra”. In: Adv. Math. 380 (2021), Paper No. 107601, 35

    2021

  14. [22]

    Trialities of W-Algebras

    T. Creutzig and A. R. Linshaw. “Trialities of W-Algebras”. In: Cambridge J. Math. 10.1 (2022), pp. 69–194

    2022

  15. [23]

    Trialities of Orthosymplectic W-Algebras

    T. Creutzig and A. R. Linshaw. “Trialities of Orthosymplectic W-Algebras”. In: Adv. Math. 409.108678 (2022)

    2022

  16. [24]

    Finite vs Affine W-Algebras

    A. De Sole and V. Kac. “Finite vs Affine W-Algebras”. In: Japan. J. Math. 1 (2006), pp. 137– 261

    2006

  17. [25]

    Classification of Good Gradings of Simple L ie Algebras

    A. Elashvili and V. Kac. “Classification of Good Gradings of Simple L ie Algebras”. In: Lie Groups and Invariant Theory . Ernest Vinberg Editor. Vol. 213. Amer. Math. Soc. Transl. Ser. 2. American Mathematical Society, 2005, pp. 85–104

    2005

  18. [26]

    Fasquel, Z

    J. Fasquel, Z. Fehily, E. Fursman, and S. Nakatsuka. Connecting affine W-algebras: A case study on sl4. 2024. arXiv: 2408.13785 [math.QA]

    2024

  19. [27]

    Fasquel and S

    J. Fasquel and S. Nakatsuka. Orthosymplectic Feigin-Semikhatov duality. 2023. arXiv: 2307.14574 [math.RT]

    2023

  20. [28]

    Fasquel, C

    J. Fasquel, C. Raymond, and D. Ridout. Modularity of admissible-level sl3 minimal models with denominator 2. 2024. arXiv: 2406.10646 [math.QA]

    2024 arXiv

  21. [29]

    Subregular W-algebras of type A

    Z. Fehily. “Subregular W-algebras of type A”. In: Commun. Contemp. Math. 25.9 (2023), no. 2250049.44

    2023

  22. [30]

    Inverse reduction for hook-type W-algebras

    Z. Fehily. “Inverse reduction for hook-type W-algebras”. In : Comm. Math. Phys. 405.9 (2024), Paper No. 214, 38

    2024

  23. [31]

    Modularity of Bershadsky–Polyakov min imal models

    Z. Fehily and D. Ridout. “Modularity of Bershadsky–Polyakov min imal models”. In: Lett. Math. Phys. 122(46) (2022)

    2022

  24. [32]

    Quantization of the Drinfeld-Sokolov R eduction

    B. Feigin and E. Frenkel. “Quantization of the Drinfeld-Sokolov R eduction”. In: Phys. Lett. B 246 (1990), pp. 75–81

    1990

  25. [33]

    E. Frenkel. Langlands Correspondence for Loop Groups . Vol. 103. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2 007

  26. [34]

    Frenkel and D

    E. Frenkel and D. Ben-Zvi. Vertex Algebras and Algebraic Curves . Second. Vol. 88. Mathe- matical Surveys and Monographs. Amer. Math. Soc., 2004

    2004

  27. [35]

    Vertex operator algebras associate d to representations of affine and Virasoro algebras

    I. Frenkel and Y. Zhu. “Vertex operator algebras associate d to representations of affine and Virasoro algebras”. In: Duke Math. J. 66.1 (1992), pp. 123–168

    1992

  28. [36]

    Conformal invarian ce, supersymmetry and string theory

    D. Friedan, E. Martinec, and S. Shenker. “Conformal invarian ce, supersymmetry and string theory”. In: Nucl. Phys. B. 271.1 (1986), pp. 93–165

    1986

  29. [37]

    Screening operators and parabolic inductions for a ffine W-algebras

    N. Genra. “Screening operators and parabolic inductions for a ffine W-algebras”. In: Adv. Math. 369 (2020), no. 107179

    2020

  30. [38]

    Reduction by stages for finite W-algeb ras

    N. Genra and T. Juillard. “Reduction by stages for finite W-algeb ras”. In: Math. Z. 308.15 (2024)

    2024

  31. [39]

    Genra and T

    N. Genra and T. Juillard. Reduction by stages for affine W-algebras . 2025. arXiv: 2501.04501 [math.RT]

    2025 arXiv

  32. [40]

    Generalized and degene rate Whittaker models

    R. Gomez, D. Gourevitch, and S. Sahi. “Generalized and degene rate Whittaker models”. In: Compos. Math. 153(2) (2017), pp. 223–256

    2017

  33. [41]

    Humphreys

    J. Humphreys. Introduction to Lie Algebras and Representation Theory . Graduate texts in mathematics. Springer, 1972. 40 REFERENCES

    1972

  34. [42]

    Quantum Reduction for Affin e Superalgebras

    V. Kac, S. Roan, and M. Wakimoto. “Quantum Reduction for Affin e Superalgebras”. In: Comm. Math. Phys. 241 (2003), pp. 307–342

    2003

  35. [43]

    Quantum Reduction and Representat ion Theory of Supercon- formal Algebras

    V. Kac and M. Wakimoto. “Quantum Reduction and Representat ion Theory of Supercon- formal Algebras”. In: Adv. Math. 185 (2004), pp. 400–458

    2004

  36. [44]

    Universal two-parameter even spin W∞-Algebra

    S. Kanade and A. R. Linshaw. “Universal two-parameter even spin W∞-Algebra”. In: Adv. Math. 355.106774 (2019)

    2019

  37. [45]

    On Whittaker vectors and representatio n theory

    Bertram Kostant. “On Whittaker vectors and representatio n theory”. In: Invent. Math. 48.2 (1978), pp. 101–184

    1978

  38. [46]

    Quantized symplectic actions and W -algebras

    I. Losev. “Quantized symplectic actions and W -algebras”. In: J. Amer. Math. Soc. 23.1 (2010), pp. 35–59

    2010

  39. [47]

    Secondary quantum Hamiltonian re ductions

    J. Madsen and E. Ragoucy. “Secondary quantum Hamiltonian re ductions”. In: Comm. Math. Phys. 185.3 (1997), pp. 509–541

    1997

  40. [48]

    Chiral de Rham co mplex

    F. Malikov, V. Schechtman, and A. Vaintrob. “Chiral de Rham co mplex”. In: Comm. Math. Phys. 204.2 (1999), pp. 439–473

    1999

  41. [49]

    R. McRae. On Rationality for C2-Cofinite Vertex Operator Algebras. 2021. arXiv: 2108.01898 [math.QA]

    2021

  42. [50]

    S. Morgan. Quantum Hamiltonian reduction of W-algebras and category O . Thesis (Ph.D.)– University of Toronto (Canada). ProQuest LLC, Ann Arbor, MI, 2 014. 66 pp. isbn: 978- 1339-37342-3

  43. [51]

    Special Transverse Slices and Their Enveloping Alge bras

    A. Premet. “Special Transverse Slices and Their Enveloping Alge bras”. In: Adv. Math. 170 (2002), pp. 1–55

    2002

  44. [52]

    W-algebras and Whittaker categories

    S. Raskin. “ W-algebras and Whittaker categories”. In: Selecta Math. (N.S.) 27.3 (2021), Paper No. 46, 114

    2021

  45. [53]

    Inverting the Hamiltonian reduction in string th eory

    A. Semikhatov. “Inverting the Hamiltonian reduction in string th eory”. In: 28th Interna- tional Symposium on Particle Theory . Wendisch-Rietz, 1994, pp. 156–167

    1994

  46. [54]

    Semi-infinite induction and Wakimoto modules

    A. Voronov. “Semi-infinite induction and Wakimoto modules”. In: Amer. J. Math. 121 (1999), pp. 1079–1094. (J.F.) School of Mathematics and Statistics, University of Melbour ne, Parkville, Australia, 3010 Email address : justine.fasquel@unimelb.edu.au (V.K.) Department of Mathem...

    1999

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