Quantization of Lie Groups and Lie Algebras

Ram, Arun

doi:10.1142/9789814340960_0034

Cited by 41 publications

(57 citation statements)

References 5 publications

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“…R(u) |0 1 ⊗|0 2 = |0 1 ⊗ |0 2 . We conjecture that by setting such a normalization condition, the solution of Rmatrix obtained in this article satisfies (53), as there is only one unique solution to (27) with this specific normalization. We emphasize again that it is not straightforward to check this conjecture even at the leading order in u of the R-matrix.…”

Section: Conclusion and Discussionmentioning

confidence: 91%

The Maulik–Okounkov R-matrix from the Ding–Iohara–Miki algebra

Fukuda

Harada

Matsuo

et al. 2017

Progress of Theoretical and Experimental Physics

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The integrability of 4d N = 2 gauge theories has been explored in various contexts, for example the Seiberg-Witten curve and its quantization. Recently, Maulik and Okounkov proposed that an integrable lattice model is associated with the gauge theory, through an Rmatrix, to which we refer as MO's R-matrix in this paper, constructed in the instanton moduli space. In this paper, we study the R-matrix using the Ding-Iohara-Miki (DIM) algebra. We provide a concrete boson realization of the universal R-matrix in DIM and show that the defining conditions for MO's R-matrix can be derived from this free boson oscillator expression. Several consistency checks for the oscillator expression are also performed.

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Section: Conclusion and Discussionmentioning

confidence: 91%

The Maulik–Okounkov R-matrix from the Ding–Iohara–Miki algebra

Fukuda

Harada

Matsuo

et al. 2017

Progress of Theoretical and Experimental Physics

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“…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%

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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“…This is a Lie algebra g equipped with a 'Lie cobracket' map δ that forms a Lie 1-cocycle and makes g * into a Lie algebra. The associated connected and simply connected Lie group G is a Poisson-Lie group, the semiclassical analogue of a quantum group; one can think of standard q-deformed quantum groups C q [G] as in [33], as the algebraic version of noncommutative deformations of C ∞ (G) (which is essentially recovered in some algebraic form) if we let λ → 0, where q = e λ 2 . These are dual to the Drinfeld-Jimbo U q (g) enveloping algebra deformations.…”

Section: Introductionmentioning

confidence: 99%

Poisson Principal Bundles

Majid¹,

Williams²

2021

SIGMA

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We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

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Quantization of Lie Groups and Lie Algebras

Cited by 41 publications

References 5 publications

The Maulik–Okounkov R-matrix from the Ding–Iohara–Miki algebra

The Maulik–Okounkov R-matrix from the Ding–Iohara–Miki algebra

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

Poisson Principal Bundles

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