Elliptic Quantum Group U_{q,p}(\hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series

Konno, Hitoshi

doi:10.48550/arxiv.0803.2292

Cited by 3 publications

(4 citation statements)

References 42 publications

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“…While different universal structures related to the Yang-Baxter equations are well studied for arbitrary simple Lie group in trigonometric and rational cases [30][31][32][33][34][35][36][37][38], the elliptic solution of the QDYB equation with spectral parameter (2.13) is known only in the SL(N, C) case [40][41][42][43][44].…”

Section: Introductionmentioning

confidence: 99%

Characteristic classes of ${\rm SL}(N, {\mathbb {C}})$-bundles and quantum dynamical elliptic R-matrices

Levin¹,

Olshanetsky²,

Smirnov³

et al. 2012

J. Phys. A: Math. Theor.

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We discuss quantum dynamical elliptic R-matrices related to arbitrary complex simple Lie group G. They generalize the known vertex and dynamical R-matrices and play an intermediate role between these two types. The R-matrices are defined by the corresponding characteristic classes describing the underlying vector bundles. The latter are related to elements of the center Z(G) of G. While the known dynamical R-matrices are related to the bundles with trivial characteristic classes, the Baxter-Belavin-Drinfeld-Sklyanin vertex Rmatrix corresponds to the generator of the center Z N of SL(N). We construct the R-matrices related to SL(N)-bundles with an arbitrary characteristic class explicitly and discuss the corresponding IRF models.

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Section: Introductionmentioning

confidence: 99%

Characteristic classes of ${\rm SL}(N, {\mathbb {C}})$-bundles and quantum dynamical elliptic R-matrices

Levin¹,

Olshanetsky²,

Smirnov³

et al. 2012

J. Phys. A: Math. Theor.

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“…[168,169]). Recently, Konno [132] reported a systematic construction of both finite and infinite-dimensional dynamical representations of a H-Hopf algebroid(introduced in [87]), and their parallel structures to the quantum affine algebra U q ( ŝl 2 ). Such generally non-Abelian structures are constructed in terms of the Drinfel'd generators of the quantum affine algebra U q ( ŝl 2 ) and a Heisenberg algebra.…”

Section: Graded Lie Algebroids and Bialgebroidsmentioning

confidence: 99%

“…Such generally non-Abelian structures are constructed in terms of the Drinfel'd generators of the quantum affine algebra U q ( ŝl 2 ) and a Heisenberg algebra. The structure of the tensor product of two evaluation representations was also provided by Konno [132], and an elliptic analogue of the Clebsch-Gordan coefficients was expressed by using certain balanced elliptic hypergeometric series 12 V 11 .…”

Section: Graded Lie Algebroids and Bialgebroidsmentioning

confidence: 99%

Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

Bâianu¹

2009

SIGMA

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Abstract. A novel algebraic topology approach to supersymmetry (SUSY) and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of nonAbelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized FourierStieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, 'string-net condensed' (ground) states. I.C. Baianu, J.F. Glazebrook and R. BrownKey words: extended quantum symmetries; groupoids and algebroids; quantum algebraic topology (QAT); algebraic topology of quantum systems; symmetry breaking, paracrystals, superfluids, spin networks and spin glasses; convolution algebras and quantum algebroids; nuclear Fréchet spaces and GNS representations of quantum state spaces (QSS); groupoid and functor representations in relation to extended quantum symmetries in QAT; quantization procedures; quantum algebras: von Neumann algebra factors; paragroups and Kac algebras; quantum groups and ring structures; Lie algebras; Lie algebroids; GrassmannHopf, weak C * -Hopf and graded Lie algebras; weak C * -Hopf algebroids; compact quantum groupoids; quantum groupoid C * -algebras; relativistic quantum gravity (RQG); supergravity and supersymmetry theories; fluctuating quantum spacetimes; intense gravitational fields; Hamiltonian algebroids in quantum gravity; Poisson-Lie manifolds and quantum gravity theories; quantum fundamental groupoids; tensor products of algebroids and categories; quantum double groupoids and algebroids; higher dimensional quantum symmetries; applications of generalized van Kampen theorem (GvKT) to quantum spacetime invariants

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“…To construct the deformed vertex operators, one can take an approach that utilizes algebras having the coproduct and that is closely connected with the q-Viraroso/W algebras. There are at least two such algebras: the Ding-Iohara-Miki (DIM) algebras [38,39] and the elliptic algebra U q,p ( g) [40,41,42,43,44,45,46,47]. Here g is an untwisted affine Lie algebra.…”

Section: Introductionmentioning

confidence: 99%

Elliptic algebra, Frenkel–Kac construction and root of unity limit

Itoyama¹,

Oota²,

Yoshioka³

2017

J. Phys. A: Math. Theor.

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We argue that the level-1 elliptic algebra U q,p ( g) is a dynamical symmetry realized as a part of 2d/5d correspondence where the Drinfeld currents are the screening currents to the q-Virasoro/W block in the 2d side. For the case of U q,p ( sl(2)), the level-1 module has a realization by an elliptic version of the Frenkel-Kac construction. The module admits the action of the deformed Virasoro algebra. In a r-th root of unity limit of p with q 2 → 1, the Z r -parafermions and a free boson appear and the value of the central charge that we obtain agrees with that of the 2d coset CFT with para-Virasoro symmetry, which corresponds to the 4d N = 2 SU(2) gauge theory on R 4 /Z

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Elliptic Quantum Group U_{q,p}(\hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series

Cited by 3 publications

References 42 publications

Characteristic classes of ${\rm SL}(N, {\mathbb {C}})$-bundles and quantum dynamical elliptic R-matrices

Characteristic classes of ${\rm SL}(N, {\mathbb {C}})$-bundles and quantum dynamical elliptic R-matrices

Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

Elliptic algebra, Frenkel–Kac construction and root of unity limit

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