Quantum Groups in Lattice Models

1992 ◽  
pp. 355-380
Author(s):  
V. Pasquier
Keyword(s):  
Quantum Groups ◽  
Lattice Models

YANG-BAXTER ALGEBRAS, INTEGRABLE THEORIES AND QUANTUM GROUPS

1989 ◽  
Vol 04 (10) ◽  
pp. 2371-2463 ◽  
Author(s):  
H.J. De VEGA
Keyword(s):  
Quantum Groups ◽  
Scaling Limit ◽  
Finite Size ◽  
Lattice Models ◽  
Physical Models ◽  
Quantum Field Theories ◽  
Integrable Lattice ◽  
Ba Construction ◽  
Integrable Lattice Models ◽  
Conformal Models

The Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and their many applications are the subject of this reivew. I start by the solvable lattice statistical models constructed from YBZF algebras. All two-dimensional integrable vertex models follow in this way and are solvable via Bethe Ansatz (BA) and their generalizations. The six-vertex model solution and its q(2q−1) vertex generalization including its nested BA construction are exposed. YBZF algebras and their associated physical models are classified in terms of simple Lie algebras. It is shown how these lattice models yield both solvable massive quantum field theories (QFT) and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is described as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YBZF algebras. A discussion on braid and quantum groups concludes this review.

YANG-BAXTER ALGEBRAS, INTEGRABLE THEORIES AND BETHE ANSATZ

1990 ◽  
Vol 04 (05) ◽  
pp. 735-801 ◽  
Author(s):  
H.J. DE VEGA
Keyword(s):  
Lie Algebras ◽  
Bethe Ansatz ◽  
Quantum Groups ◽  
General Framework ◽  
Continuum Limit ◽  
Lattice Models ◽  
Model Solution ◽  
Sos Model ◽  
Recent Developments ◽  
Conformal Models

The Yang-Baxter algebras (YBA) are introduced in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Bethe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. This approach permit to define and solve rigorously massive QFT as an appropiate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underly the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter. To conclude, some recent developments in the domain of integrable theories are summarized.

Temperley-Lieb lattice models arising from quantum groups

1992 ◽  
Vol 25 (4) ◽  
pp. 1019-1019 ◽  
Author(s):  
M T Batchelor ◽  
A Kuniba
Keyword(s):  
Quantum Groups ◽  
Lattice Models ◽  
Lieb Lattice

Temperley-Lieb lattice models arising from quantum groups

1991 ◽  
Vol 24 (11) ◽  
pp. 2599-2614 ◽  
Author(s):  
M T Batchelor ◽  
A Kuniba
Keyword(s):  
Quantum Groups ◽  
Lattice Models ◽  
Lieb Lattice

Continuum limit of lattice models built on quantum groups

1988 ◽  
Vol 295 (4) ◽  
pp. 491-510 ◽  
Author(s):  
V. Pasquier
Keyword(s):  
Quantum Groups ◽  
Continuum Limit ◽  
Lattice Models

Series Expansion Methods for Strongly Interacting Lattice Models

2006 ◽  
Author(s):  
Jaan Oitmaa ◽  
Chris Hamer ◽  
Weihong Zheng
Keyword(s):  
Series Expansion ◽  
Lattice Models ◽  
Strongly Interacting

scholarly journals On cocycle twisting of compact quantum groups

2010 ◽  
Vol 258 (10) ◽  
pp. 3362-3375 ◽  
Author(s):  
Kenny De Commer
Keyword(s):  
Quantum Groups ◽  
Compact Quantum Groups

scholarly journals Curved Rickard complexes and link homologies

2020 ◽  
Vol 2020 (769) ◽  
pp. 87-119
Author(s):  
Sabin Cautis ◽  
Aaron D. Lauda ◽  
Joshua Sussan
Keyword(s):  
Spectral Sequence ◽  
Quantum Groups ◽  
Braid Group ◽  
Derived Categories ◽  
Duke Math ◽  
Khovanov Homology ◽  
Hilbert Schemes ◽  
Coherent Sheaves ◽  
Knot Homology ◽  
Original Motivation

AbstractRickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [3] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

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