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A new approach to braided monoidal categories

Let H be a Hopf quasigroup over k possessing an adjoint quasiaction. Then we first show that if M is any right H-comodule and N is any right H-quasimodule such that τN,M2=idN⊗M, where τN,M : N ⊗ M → M ⊗ N is a favourable map, then we have H = k as a generalization of Pareigis’ theorem [J. Pure Appl. Algebra 155, 91 (2001)]. Then we mainly give a new approach to a braided monoidal category by generalizing one of Schauenburg’s main results [Schauenburg, P., J. Algebra 169, 874–890 (1994)] in the setting of Hopf quasigroups introduced by Klim and Majid [J. Algebra 323, 3067–3110 (2010)]. This yields new sources of braidings which provide solutions to the Yang-Baxter equation playing an important role in various areas of mathematics.Let H be a Hopf quasigroup over k possessing an adjoint quasiaction. Then we first show that if M is any right H-comodule and N is any right H-quasimodule such that τN,M2=idN⊗M, where τN,M : N ⊗ M → M ⊗ N is a favourable map, then we have H = k as a generalization of Pareigis’ theorem [J. Pure Appl. Algebra 155, 91 (2001)]. Then we mainly give a new approach to a braided monoidal category by generalizing one of Schauenburg’s main results [Schauenburg, P., J. Algebra 169, 874–890 (1994)] in the setting of Hopf quasigroups introduced by Klim and Majid [J. Algebra 323, 3067–3110 (2010)]. This yields new sources of braidings which provide solutions to the Yang-Baxter equation playing an important role in various areas of mathematics.