The Ribbon Elements of the Quantum Double of Generalized Taft–Hopf Algebra

Let <i>s</i>, <i>t</i> be two positive integers and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">k</mi></semantics></math></inline-formula> be an algebraically closed field with char (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">k</mi><mo>)</mo><mo>∤</mo><mi>s</mi><mi>t</mi></mrow></semantics></math></inline-formula>. We show that the Drinfeld double <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><msubsup><mo>⋀</mo><mrow><mi>s</mi><mi>t</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo><mi>c</mi><mi>o</mi><mi>p</mi></mrow></msubsup><mo>)</mo></mrow></semantics></math></inline-formula> of generalized Taft–Hopf algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mo>⋀</mo><mrow><mi>s</mi><mi>t</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo><mi>c</mi><mi>o</mi><mi>p</mi></mrow></msubsup></semantics></math></inline-formula> has ribbon elements if and only if <i>t</i> is odd. Moreover, if <i>s</i> is even and <i>t</i> is odd, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><msubsup><mo>⋀</mo><mrow><mi>s</mi><mi>t</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo><mi>c</mi><mi>o</mi><mi>p</mi></mrow></msubsup><mo>)</mo></mrow></semantics></math></inline-formula> has two ribbon elements, and if both <i>s</i> and <i>t</i> are odd, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><msubsup><mo>⋀</mo><mrow><mi>s</mi><mi>t</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo><mi>c</mi><mi>o</mi><mi>p</mi></mrow></msubsup><mo>)</mo></mrow></semantics></math></inline-formula> has only one ribbon element. Finally, we compute explicitly all ribbon elements of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><msubsup><mo>⋀</mo><mrow><mi>s</mi><mi>t</mi><mo>,</mo><mi>t</mi></mrow><mrow><mo>*</mo><mi>c</mi><mi>o</mi><mi>p</mi></mrow></msubsup><mo>)</mo></mrow></semantics></math></inline-formula>.