| Summary: | Let <i>H</i> be a finite dimensional <inline-formula><math display="inline"><semantics><msup><mi>C</mi><mo>∗</mo></msup></semantics></math></inline-formula>-Hopf algebra and <inline-formula><math display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> the observable algebra of Hopf spin models. For some coaction of the Drinfeld double <inline-formula><math display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></semantics></math></inline-formula> on <inline-formula><math display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>, the crossed product <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">A</mi><mo>⋊</mo><mover accent="true"><mrow><mi>D</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>^</mo></mover></mrow></semantics></math></inline-formula> can define the field algebra <inline-formula><math display="inline"><semantics><mi mathvariant="script">F</mi></semantics></math></inline-formula> of Hopf spin models. In the paper, we study <inline-formula><math display="inline"><semantics><msup><mi>C</mi><mo>∗</mo></msup></semantics></math></inline-formula>-basic construction for the inclusion <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">A</mi><mo>⊆</mo><mi mathvariant="script">F</mi></mrow></semantics></math></inline-formula> on Hopf spin models. To achieve this, we define the action <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo lspace="0pt">:</mo><mrow><mi>D</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>×</mo><mi mathvariant="script">F</mi><mo>→</mo><mi mathvariant="script">F</mi></mrow></semantics></math></inline-formula>, and then construct the resulting crossed product <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mo>⋊</mo><mrow><mi>D</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which is isomorphic <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">A</mi><mo>⊗</mo><mi>End</mi><mo>(</mo><mover accent="true"><mrow><mi>D</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>^</mo></mover><mo>)</mo></mrow></semantics></math></inline-formula>. Furthermore, we prove that the <inline-formula><math display="inline"><semantics><msup><mi>C</mi><mo>∗</mo></msup></semantics></math></inline-formula>-basic construction for <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">A</mi><mo>⊆</mo><mi mathvariant="script">F</mi></mrow></semantics></math></inline-formula> is consistent to <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">F</mi><mo>⋊</mo><mrow><mi>D</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which yields that the <inline-formula><math display="inline"><semantics><msup><mi>C</mi><mo>∗</mo></msup></semantics></math></inline-formula>-basic constructions for the inclusion <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">A</mi><mo>⊆</mo><mi mathvariant="script">F</mi></mrow></semantics></math></inline-formula> is independent of the choice of the coaction of <inline-formula><math display="inline"><semantics><mrow><mi>D</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></semantics></math></inline-formula> on <inline-formula><math display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>.
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