| Summary: | In this paper, we investigate the module algebra structures of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>q</mi></msub><mrow><mo>(</mo><msub><mi>A</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> on the quantum polynomial algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">C</mi><mi>q</mi></msub><mrow><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula>. The automorphism group of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">C</mi><mi>q</mi></msub><mrow><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula> is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Aut</mi><mo>(</mo><msub><mi mathvariant="double-struck">C</mi><mi>q</mi></msub><mrow><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>]</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>, and we give a complete classification of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>q</mi></msub><mrow><mo>(</mo><msub><mi>A</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-module algebra structures on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">C</mi><mi>q</mi></msub><mrow><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mn>1</mn></msub><mo>,</mo><msub><mi>K</mi><mn>2</mn></msub><mo>∈</mo><mi>Aut</mi><mrow><mo>(</mo><msub><mi mathvariant="double-struck">C</mi><mi>q</mi></msub><mrow><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>]</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>.
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