| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>H</mi><mo>,</mo><msub><mi>μ</mi><mi>H</mi></msub><mo>,</mo><msub><mo>Δ</mo><mi>H</mi></msub><mo>,</mo><msub><mi>α</mi><mi>H</mi></msub><mo>,</mo><msub><mi>β</mi><mi>H</mi></msub><mo>,</mo><msub><mi>ψ</mi><mi>H</mi></msub><mo>,</mo><msub><mi>ω</mi><mi>H</mi></msub><mo>,</mo><msub><mi>S</mi><mi>H</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> be a BiHom–Hopf algebra. First, we provide a non-trivial example of a left–left BiHom–Yetter–Drinfeld module and show that the category <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow></mrow><mi>H</mi><mi>H</mi></msubsup><mi mathvariant="script">BHYD</mi></mrow></semantics></math></inline-formula> is a braided monoidal category. We also study the connection between the category <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow></mrow><mi>H</mi><mi>H</mi></msubsup><mi mathvariant="script">BHYD</mi></mrow></semantics></math></inline-formula> and the category <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mi>H</mi></msup><mi mathvariant="script">M</mi></mrow></semantics></math></inline-formula> of the left co-modules over a coquasitriangular BiHom–bialgebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>H</mi><mo>,</mo><mi>σ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Secondly, we prove that the category of finitely generated projective left–left BiHom–Yetter–Drinfeld modules is closed for left and right duality.
|
|---|