On Drinfeld realization of quantum affine algebras

Jing, Naihuan

doi:10.1515/9783110801897.195

Cited by 77 publications

(100 citation statements)

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“…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%

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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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Section: Jhep10(2016)047mentioning

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%

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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“…We will also need another presentation of U q . Namely, after [2,19], the algebra U q is isomorphic to an associative C(q)-algebra generated by…”

Section: Preliminariesmentioning

confidence: 99%

Quantum loop modules

Chari

Greenstein

2003

Represent. Theory

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Abstract. We classify the simple infinite dimensional integrable modules with finite dimensional weight spaces over the quantized enveloping algebra of an untwisted affine algebra. We prove that these are either highest (lowest) weight integrable modules or simple submodules of a loop module of a finitedimensional simple integrable module and describe the latter class. Their characters and crystal bases theory are discussed in a special case. IntroductionThe aim of the present paper is to study irreducible integrable modules for quantum affine algebras, which have finite dimensional weight spaces. The best known examples of such representations are the highest weight representations V (λ) (cf. [16, 23]) on which the center of the quantum affine algebra acts via a positive integer power of q. These representations have many pleasant properties, for instance it is known that they admit a canonical global or crystal bases. Another family of integrable modules for the quantum affine algebra are the finite-dimensional modules which have been studied, amongst the others, in [1, 8,11,12,13, 18,24, 25,26]. However, unlike the highest weight representations, these finite-dimensional representations do not respect the natural Z-grading on the quantum affine algebra which arises from the adjoint action of the element of the torus corresponding to the Euler operator. Thus it is natural to look for the graded analogue of the finite-dimensional modules. Besides, in certain cases one has to consider these infinite dimensional modules instead of finite-dimensional ones. For example, a finite-dimensional module cannot appear as a submodule of the ring of linear endomorphisms of V (λ) whilst a simple integrable module with non-trivial zero weight space can be embedded in such a ring for λ sufficiently large (cf. for example [20]).Examples of infinite dimensional integrable modules which are not highest weight modules are easy to construct. Namely, given a finite-dimensional module V , one can define on the space L(V ) = V ⊗ C(q)[t, t −1 ] in an obvious way the structure of a graded module for the quantum affine algebra. However, even if V is irreducible, the resulting representation L(V ) need not remain so.The irreducible finite-dimensional representations of the quantum affine algebra are known to be parametrized by families of polynomials in an indeterminate u

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“…In section 2 we will review some preliminary results about quantum affine algebras and the Drinfeld realization written in a modified form [11]. Based on [11] we calculate the special level zero finite dimensional representations for twisted quantum affine algebras.…”

Section: Naihuan Jing and Kailash C Misramentioning

confidence: 99%