Quantization of web geometry: Semisymmetrization of linear quantum quasigroups

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics Pub Date : 2026-05-01 Epub Date: 2026-02-02 DOI:10.1016/j.geomphys.2026.105781

Jonathan D.H. Smith

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引用次数: 0

Abstract

Classical quasigroups coordinatize structures called 3-nets in combinatorics, and 3-webs in geometry. The coordinatization is up to isotopy, a relation coarser than isomorphism. The semisymmetrization of a classical quasigroup is built on the cube of the underlying set of the quasigroup. Isotopic quasigroups have isomorphic semisymmetrizations.

Quantum quasigroups provide a self-dual unification (with both a multiplication and a comultiplication) of quasigroups and Hopf algebras, in the general setting of symmetric monoidal categories. Linear quantum quasigroups are quantum quasigroups in categories of vector spaces or modules over a commutative ring, with the direct sum as the Cartesian monoidal product.

With a view to addressing the quantization of web geometry, the paper determines linear quantum quasigroup structures that provide comultiplications to extend the semisymmetrization multiplication of a linear quasigroup. In particular, if the linear quasigroup structure comes from a real or complex affine plane, a complete classification of the quantum semisymmetric comultiplications is provided, based on the solution of a system of cubic equations.

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网几何的量化：线性量子拟群的半对称化

经典拟群协调的结构在组合学中称为3-网，在几何中称为3-网。这种配位达到了同位素关系，这是一种比同构关系更粗糙的关系。一个经典拟群的半对称是建立在拟群的下集的立方上的。同位素准群具有同构半对称性。在对称一元范畴的一般情况下，量子拟群提供了拟群和Hopf代数的自对偶统一（既有乘法也有乘法）。线性量子拟群是交换环上向量空间或模范畴内的量子拟群，其和为笛卡尔单轴积。为了解决网几何的量子化问题，本文确定了线性量子拟群结构，该结构提供了扩展线性拟群的半对称乘法的乘法。特别地，如果线性拟群结构来自实或复仿射平面，则基于三次方程组的解，给出了量子半对称乘法的完整分类。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Geometry and Physics 物理-物理：数学物理

CiteScore

2.90

自引率

6.70%

发文量

205

审稿时长

64 days

期刊介绍： The Journal of Geometry and Physics is an International Journal in Mathematical Physics. The Journal stimulates the interaction between geometry and physics by publishing primary research, feature and review articles which are of common interest to practitioners in both fields. The Journal of Geometry and Physics now also accepts Letters, allowing for rapid dissemination of outstanding results in the field of geometry and physics. Letters should not exceed a maximum of five printed journal pages (or contain a maximum of 5000 words) and should contain novel, cutting edge results that are of broad interest to the mathematical physics community. Only Letters which are expected to make a significant addition to the literature in the field will be considered. The Journal covers the following areas of research: Methods of: • Algebraic and Differential Topology • Algebraic Geometry • Real and Complex Differential Geometry • Riemannian Manifolds • Symplectic Geometry • Global Analysis, Analysis on Manifolds • Geometric Theory of Differential Equations • Geometric Control Theory • Lie Groups and Lie Algebras • Supermanifolds and Supergroups • Discrete Geometry • Spinors and Twistors Applications to: • Strings and Superstrings • Noncommutative Topology and Geometry • Quantum Groups • Geometric Methods in Statistics and Probability • Geometry Approaches to Thermodynamics • Classical and Quantum Dynamical Systems • Classical and Quantum Integrable Systems • Classical and Quantum Mechanics • Classical and Quantum Field Theory • General Relativity • Quantum Information • Quantum Gravity