Algebraic properties of quantum quasigroups |
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Authors: | Bokhee Im Alex W Nowak Jonathan DH Smith |
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Institution: | 1. Department of Mathematics, Chonnam National University, Gwangju 61186, Republic of Korea;2. Department of Mathematics, Iowa State University, Ames, IA 50011, USA;1. Department of Mathematics, Northeastern University, Boston, MA 02115, USA;2. Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de Investigação em Matemática e Aplicações, Instituto de Investigação e Formação Avançada, Universidade de Évora, Rua Romão Ramalho, 59, P-7000-671 Évora, Portugal;1. HLM, Institute of Mathematics, Academy of Mathematics & System Sciences, Chinese Academy of Sciences, Beijing 100190, PR China;2. School of Mathematics, University of Chinese Academy of Sciences, Beijing 100049, PR China |
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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. |
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Keywords: | Hopf algebra Quantum group Semisymmetric Quasigroup Loop Mendelsohn triple system |
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