Abstract
This short chapter is a brief guide to the background and the topics treated in the book. We begin by recalling the key equations for integrability in two dimensions, motivate a generalization to three dimensions, digest how a class of quantum groups known as quantized coordinate rings play an important role, and mention some fruitful applications.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
In later sections, \(\mathcal {F}\) is taken slightly differently for L, G, J.
- 3.
The quantized Yang–Baxter equation is well known as a version of the tetrahedron equation. See Sect. 2.7 for a historical note. The quantized reflection equation and the quantized \(G_2\) reflection equation were introduced in [85, 105].
- 4.
An intrinsic reason why (1.5) admits such a “physical” presentation in terms of scattering diagrams (Figs. 2.18, 4.6 and 8.1) is yet to be revealed.
- 5.
It is parallel with 2D, where the quantum group symmetry of the form \(\mathcal {R}\mathcal {L}\mathcal {L}=\mathcal {L}\mathcal {L}\mathcal {R}\) automatically implies the Yang–Baxter equation \(\mathcal {R}\mathcal {R}\mathcal {R} = \mathcal {R}\mathcal {R}\mathcal {R}\) [43, 63].
- 6.
- 7.
Such an approach to the tetrahedron equation was first undertaken in [77].
- 8.
- 9.
For the Yang–Baxter equation, one may say that almost any trigonometric solution should be just the image of the universal \(\mathcal {R}\) in principle (top down). True. However, to describe or construct one in a tractable manner is another problem of individual interest (bottom up). A typical recipe of the latter is the fusion construction. The 3D approach briefed in this section is another having its own intriguing scope.
- 10.
To characterize \(\mathcal {X}(z)\) for \(G_2\) in such a quantum group theoretical framework is an open problem.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Kuniba, A. (2022). Introduction. In: Quantum Groups in Three-Dimensional Integrability. Theoretical and Mathematical Physics. Springer, Singapore. https://doi.org/10.1007/978-981-19-3262-5_1
Download citation
DOI: https://doi.org/10.1007/978-981-19-3262-5_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-19-3261-8
Online ISBN: 978-981-19-3262-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)