Introduction to W-Algebras and Their Representation Theory

Arakawa, Tomoyuki

doi:10.1007/978-3-319-58971-8_4

Cited by 57 publications

(65 citation statements)

References 59 publications

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“…Our assumption (Π\Π l ⊂ Π 1 ) is used here. Since the Miura map is injective by [Fre,A3,Ge], Theorem A therefore follows if we show that the map Ind g l satisfies the formula µ = µ l • Ind g l , which in fact follows from (1.3) and Theorem D. Theorem D (Theorem 5.6). The specialization…”

Section: Introductionmentioning

confidence: 84%

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Screening operators and parabolic inductions for affine W-algebras

Genra

2020

Advances in Mathematics

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Affine) W-algebras are a family of vertex algebras defined by the generalized Drinfeld-Sokolov reductions associated with a finite-dimensional reductive Lie algebra g over C, a nilpotent element f in [g, g], a good grading Γ and a symmetric invariant bilinear form κ on g. We introduce free field realizations of W-algebras by using Wakimoto representations of affine Lie algebras, where W-algebras are described as the intersections of kernels of screening operators. We call these Wakimoto free fields realizations of Walgebras. As applications, under certain conditions that are valid in all cases of type A, we construct parabolic inductions for W-algebras, which we expect to induce the parabolic inductions of finite W-algebras defined by Premet and Losev. In type A, we show that our parabolic inductions are a chiralization of the coproducts for finite W-algebras defined by Brundan-Kleshchev. In type BCD, we are able to obtain some generalizations of the coproducts in some special cases.

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Section: Introductionmentioning

confidence: 84%

“…, which is called the Miura map for W k (g, f ; Γ) and injective for all k ∈ C by [Fre,A3], see also [Ge]. Following [Ge], the Miura map µ coincides with the specialization µ T ⊗ C k of an injective vertex algebra homomorphism…”

Section: 1mentioning

confidence: 99%

Screening operators and parabolic inductions for affine W-algebras

Genra

2020

Advances in Mathematics

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“…called the Miura map, which is injective for all κ (see e.g. [Ara17]). In particular, W κ (g) may be identified with the image of the Miura map inside the Heisenberg vertex algebra π κ .…”

Section: Proof Of Theorem 22mentioning

confidence: 99%

Quantum Langlands duality of representations of -algebras

Arakawa¹,

Frenkel²

2019

Compositio Math.

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“…These algebras are called W-algebras. The standard construction of W-algebras is via Hamiltonian reduction associated with a Lie algebra g and an sl(2)-embedding, see, e.g., [1,2] for reviews for mathematicians respectively physicists. For the case of g = sl(N), sl(2)-embeddings are labeled by partitions of the integer N, and each partition leads to a different algebra.…”

Section: Introductionmentioning

confidence: 99%

Correspondences between WZNW models and CFTs with W-algebra symmetry

Creutzig

Hikida

Rønne

2016

J. High Energ. Phys.

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Abstract:We study theories with W -algebra symmetries and their relation to WZNWtype models on (super-)groups generalizing the H + 3 WZNW to Liouville correspondence. Correlation functions of the WZNW models are expressed in terms of correlators of CFTs with W -algebra symmetry. The symmetries of the theories involved in these correspondences are related by the Drinfeld-Sokolov reduction of Lie algebras to W -algebras. The W -algebras considered in this paper are the Bershadsky-Polyakov algebra for sl(3) and the quasi-superconformal algebra for generic sl(N |M ). The quantum W -algebras obtained from affine sl(N ) are constructed using embeddings of sl(2) into sl(N ), and these can in turn be characterized by partitions of N . The above cases correspond to N + 2 = 2 + N 1 and its supergroup extension. Finally, sl(2N ) and the correspondence corresponding to 2N = N 2 is also analyzed. These are all W -algebras that are generated by fields of at most dimension two.

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Introduction to W-Algebras and Their Representation Theory

Cited by 57 publications

References 59 publications

Screening operators and parabolic inductions for affine W-algebras

Screening operators and parabolic inductions for affine W-algebras

Quantum Langlands duality of representations of -algebras

Correspondences between WZNW models and CFTs with W-algebra symmetry

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