A (q,γ) analog of the W1+∞ algebra

Miki, Kei

doi:10.1063/1.2823979

Cited by 214 publications

(293 citation statements)

References 25 publications

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“…1 This isomorphism arises as the rational limit of the equivalence between the quantum-deformed W 1+∞ algebra and the quantum toroidal algebra of gl 1 [10], generalizing the construction of [13] to the toroidal case. The toroidal isomorphism was first pointed out in [14], and the definition of the quantum toroidal algebra (or quantum affinization of the affine Lie algebra) of gl 1 was inspired by [15]; the toroidal isomorphism was also independently re-derived in a series of papers [16][17][18]. More recently, the construction of the quantum toroidal algebras was generalized further to arbitrary quiver diagrams in [19].…”

Section: Jhep04(2017)152mentioning

confidence: 99%

Higher spins and Yangian symmetries

Gaberdiel

Gopakumar

et al. 2017

J. High Energ. Phys.

122

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Abstract:The relation between the bosonic higher spin W ∞ [λ] algebra, the affine Yangian of gl 1 , and the SH c algebra is established in detail. For generic λ we find explicit expressions for the low-lying W ∞ [λ] modes in terms of the affine Yangian generators, and deduce from this the precise identification between λ and the parameters of the affine Yangian. Furthermore, for the free field cases corresponding to λ = 0 and λ = 1 we give closed-form expressions for the affine Yangian generators in terms of the free fields. Interestingly, the relation between the W ∞ modes and those of the affine Yangian is a non-local one, in general. We also establish the explicit dictionary between the affine Yangian and the SH c generators. Given that Yangian algebras are the hallmark of integrability, these identifications should pave the way towards uncovering the relation between the integrable and the higher spin symmetries.

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Section: Jhep04(2017)152mentioning

confidence: 99%

Higher spins and Yangian symmetries

Gaberdiel

Gopakumar

et al. 2017

J. High Energ. Phys.

122

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“…It will be important for us that the DIM algebra has a remarkable group of automorphisms SL(2, Z), which are precisely the automorphisms of the integer lattice of generators [27]. Let us also note that the central charges (C 1 , C 2 ) transform as a doublet under this SL(2, Z) symmetry.…”

Section: Jhep10(2016)047mentioning

confidence: 99%

“…According to [27], H k are spectral dual to ψ + k and consequently mutually commuting: [H k , H l ] = 0. Thus, the generalized Macdonald polynomials M λ are automatically eigenfunctions of all H k , i.e.…”

Section: B2 Higher Hamiltoniansmentioning

confidence: 99%

“…We are going to compute the R-matrix of the DIM algebra, also known as quantum toroidal algebra or U q,t ( gl 1 ) [26,27]. It is a double quantum deformation of the double loop algebra of gl 1 .…”

Section: Dim Algebra Generalized Macdonald Polynomials and The R-matrixmentioning

confidence: 99%

“…Let us now briefly summarize how this kind of integrability appears. The central object, on which we will mostly focus in our approach is the R-matrix of the Ding-Iohara-Miki (DIM) algebra [26,27]. R-matrices, which can be considered as emerging in the description of coproducts of group elementsĝ ∈ G ⊗ A(G) [28][29][30][31] for quantum groups [32][33][34][35][36],…”

Section: Jhep10(2016)047mentioning

confidence: 99%

See 2 more Smart Citations

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

Awata

Kanno

Миронов

et al. 2016

J. High Energ. Phys.

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R-matrix is explicitly constructed for simplest representations of the DingIohara-Miki algebra. Calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the Yangian case by A. Smirnov but less general. We investigate the interplay between the R-matrix structure and the structure of DIM algebra intertwiners, i.e. of refined topological vertices and show that the R-matrix is diagonalized by the action of the spectral duality belonging to the SL(2, Z) group of DIM algebra automorphisms. We also construct the T -operators satisfying the RT T relations with the R-matrix from refined amplitudes on resolved conifold. We thus show that topological string theories on the toric Calabi-Yau threefolds can be naturally interpreted as lattice integrable models. Integrals of motion for these systems are related to q-deformation of the reflection matrices of the Liouville/Toda theories.

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Moduli Spaces of Sheaves on Surfaces: Hecke Correspondences and Representation Theory

Neguţ

2019

Lecture Notes in Mathematics

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We construct explicit elements W k ij in (a completion of) the shifted quantum toroidal algebra of type A, and show that these elements act by 0 on the K-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements W k ij will be related to q-deformed W -algebras of type A for arbitrary nilpotent, which would imply a q-deformed version of the AGT correspondence between gauge theory with surface operators and conformal field theory.Proposition 4.5. Under the action of Theorem 4.3, the elements (2.21) act on K r as the operators (4.7) with the same name, hence the abuse of notation.

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A (q,γ) analog of the W1+∞ algebra

Abstract: A (q,γ) analog of the W1+∞ algebra is introduced. Irreducible quasifinite highest weight modules of this algebra and R matrices acting on a tensor product of these modules are investigated. A connection with the q deformed Virasoro and WN algebras is also discussed.

Cited by 214 publications

References 25 publications

Higher spins and Yangian symmetries

Higher spins and Yangian symmetries

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

Moduli Spaces of Sheaves on Surfaces: Hecke Correspondences and Representation Theory

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