Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0607228

Affine quantum groups

G. W. Delius·N. J. MacKay
View original source ↗Read source document ↗Export metadata ↓

Abstract

Affine quantum groups are certain pseudo-quasitriangular Hopf algebras that arise in mathematical physics in the context of integrable quantum field theory, integrable quantum spin chains, and solvable lattice models. They provide the algebraic framework behind the spectral parameter dependent Yang-Baxter equation. One can distinguish three classes of affine quantum groups, each leading to a different dependence of the R-matrices on the spectral parameter: Yangians lead to rational R-matrices, quantum affine algebras lead to trigonometric R-matrices and elliptic quantum groups lead to elliptic R-matrices. We will mostly concentrate on the quantum affine algebras but many results hold similarly for the other classes. After giving mathematical details about quantum affine algebras and Yangians in the first two section, we describe how these algebras arise in different areas of mathematical physics in the three following sections. We end with a description of boundary quantum groups which extend the formalism to the boundary Yang-Baxter (reflection) equation.

Explore related subjects

math.QAmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

G. W. Delius, N. J. MacKay. 2006-07-09. Affine quantum groups. https://arxiv.org/abs/math/0607228

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence

We describe infinitesimal deformations of post-Lie algebras and post-Hopf algebras and prove that the universal enveloping algebra and primitive elements adjunction is compatible with the infinitesimal structure. In characteristic zero, this restricts to an equivalence for connected cocommutative infinitesimal post-Hopf algebras, extending the Cartier--Milnor--Moore theorem. We parametrize all infinitesimal post-Lie structures on $\mathfrak{sl}(2)$ and exhibit a second-order obstruction. From a geometric point of view, infinitesimal post-Lie algebras emerge from flat connections with covariantly constant torsion and an additional compatible symmetric $(1,2)$-tensor. This leads to explicit examples of infinitesimal post-Lie structures based on Heisenberg Lie algebras. On the Hopf side, we classify infinitesimal structures above the trivial action, prove rigidity for group algebras, and describe the structures on Sweedler's Hopf algebra. The subadjacent construction induces a Hopf algebra deformation with fixed coproduct. Finally, we prove that the quadratic operad of infinitesimal post-Lie algebras is Koszul, using a filtered distributive law between the operads of Lie algebras and bi-magmas.

math.QA↗
arXiv2026

Galois-modular obstructions to the categorification of near-group fusion rings

In this article we establish new obstructions to the categorification of near-group fusion rings by combining the values of irreducible characters of the Grothendieck ring with the Galois symmetry of the induced unit object $I(1)$ in the Drinfeld center. As a result, we show that a near-group ring of type $G+m$, where $G$ is an abelian group and both $|G|$ and $m$ are odd, can only categorify if $m=|G|$.

math.QA↗
arXiv2026

Reflective algebras over factorizable Hopf algebras

The reflective algebra $(R_H(A),K_{\mathrm{ref}}(A))$, introduced by Laugwitz, Walton and Yakimov, is a quasitriangular left $H$-comodule algebra constructed from a quasitriangular Hopf algebra $(H,R)$ and a left $H$-comodule algebra $A$. We study the structure and factorizability of finite-dimensional reflective algebras. When $(H,R)$ is factorizable, we show that $(R_H(A),K_{\mathrm{ref}}(A))$ is isomorphic, as a quasitriangular left $H$-comodule algebra, to $A\otimes H$ equipped with the trivial left $H$-coaction on $A$ and the universal $K$-matrix induced by $R_{21}R$. In this case, we also classify all universal $K$-matrices on $R_H(A)$. Finally, we prove that $(R_H(A),K_{\mathrm{ref}}(A))$ is factorizable if and only if $(H,R)$ is factorizable and $A$ is simple as an algebra.

math.QA↗