Abstract Quantum quasigroups provide a self-dual framework for the unification of quasigroups and Hopf algebras. This paper furthers the transfer program, investigating extensions to quantum quasigroups of various algebraic features of quasigroups and Hopf algebras. Part of the difficulty of the transfer program is the fact that there is no standard model-theoretic procedure for accommodating the coalgebraic aspects of quantum quasigroups. The linear quantum quasigroups, which live in categories of modules under the direct sum, are a notable exception. They form one of the central themes of the paper. From the theory of Hopf algebras, we transfer the study of grouplike and setlike elements, which form separate concepts in quantum quasigroups. From quasigroups, we transfer the study of conjugate quasigroups, which reflect the triality symmetry of the language of quasigroups. In particular, we construct conjugates of cocommutative Hopf algebras. Semisymmetry, Mendelsohn, and distributivity properties are formulated for quantum quasigroups. We classify distributive linear quantum quasigroups that furnish solutions to the quantum Yang-Baxter equation. The transfer of semisymmetry is designed to prepare for a quantization of web geometry.

中文翻译：

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量子拟群的代数性质

摘要 量子拟群为拟群与Hopf 代数的统一提供了一个自对偶框架。本文进一步推进了转移程序，研究了对拟群和 Hopf 代数的各种代数特征的量子拟群的扩展。转移程序的部分困难在于没有标准的模型理论程序来适应量子拟群的代数方面。存在于直和下模类别中的线性量子拟群是一个显着的例外。它们构成了本文的中心主题之一。从 Hopf 代数理论，我们转移了类群元素和类集合元素的研究，它们在量子拟群中形成了独立的概念。从拟群，我们转移对共轭拟群的研究，反映了拟群语言的试验对称性。特别是，我们构建了互易 Hopf 代数的共轭。为量子拟群制定了半对称性、门德尔松和分布特性。我们对提供量子杨-巴克斯特方程解的分布线性量子拟群进行分类。半对称的传递旨在为网络几何的量化做准备。