Module Algebra Structures of Non-Standard Quantum Group Xq(A1) on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>[x,y,z]

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>(&l...

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Main Author:	Dong Su
Format:	Article
Language:	English
Published:	MDPI AG 2025-07-01
Series:	Mathematics
Subjects:	nonstandard quantum group quantum polynomial algebra Hopf action module algebra weight
Online Access:	https://www.mdpi.com/2227-7390/13/14/2227
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author	Dong Su
author_facet	Dong Su
author_sort	Dong Su
collection	DOAJ
description	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>.
format	Article
id	doaj-art-d2a6c82a204e4aa49ea2a4d4c5eef37e
institution	DOAJ
issn	2227-7390
language	English
publishDate	2025-07-01
publisher	MDPI AG
record_format	Article
series	Mathematics
spelling	doaj-art-d2a6c82a204e4aa49ea2a4d4c5eef37e2025-08-20T03:08:12ZengMDPI AGMathematics2227-73902025-07-011314222710.3390/math13142227Module Algebra Structures of Non-Standard Quantum Group <i>X</i><sub>q</sub>(<i>A</i><sub>1</sub>) on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>[<i>x</i>,<i>y</i>,<i>z</i>]Dong Su0School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, ChinaIn 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>.https://www.mdpi.com/2227-7390/13/14/2227nonstandard quantum groupquantum polynomial algebraHopf actionmodule algebraweight
spellingShingle	Dong Su Module Algebra Structures of Non-Standard Quantum Group <i>X</i><sub>q</sub>(<i>A</i><sub>1</sub>) on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>[<i>x</i>,<i>y</i>,<i>z</i>] Mathematics nonstandard quantum group quantum polynomial algebra Hopf action module algebra weight
title	Module Algebra Structures of Non-Standard Quantum Group <i>X</i><sub>q</sub>(<i>A</i><sub>1</sub>) on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>[<i>x</i>,<i>y</i>,<i>z</i>]
title_full	Module Algebra Structures of Non-Standard Quantum Group <i>X</i><sub>q</sub>(<i>A</i><sub>1</sub>) on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>[<i>x</i>,<i>y</i>,<i>z</i>]
title_fullStr	Module Algebra Structures of Non-Standard Quantum Group <i>X</i><sub>q</sub>(<i>A</i><sub>1</sub>) on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>[<i>x</i>,<i>y</i>,<i>z</i>]
title_full_unstemmed	Module Algebra Structures of Non-Standard Quantum Group <i>X</i><sub>q</sub>(<i>A</i><sub>1</sub>) on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>[<i>x</i>,<i>y</i>,<i>z</i>]
title_short	Module Algebra Structures of Non-Standard Quantum Group <i>X</i><sub>q</sub>(<i>A</i><sub>1</sub>) on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>[<i>x</i>,<i>y</i>,<i>z</i>]
title_sort	module algebra structures of non standard quantum group i x i sub q sub i a i sub 1 sub on inline formula math display inline semantics mrow mi mathvariant double struck c mi mrow semantics math inline formula i x i i y i i z i
topic	nonstandard quantum group quantum polynomial algebra Hopf action module algebra weight
url	https://www.mdpi.com/2227-7390/13/14/2227
work_keys_str_mv	AT dongsu modulealgebrastructuresofnonstandardquantumgroupixisubqsubiaisub1suboninlineformulamathdisplayinlinesemanticsmrowmimathvariantdoublestruckcmimrowsemanticsmathinlineformulaixiiyiizi

Module Algebra Structures of Non-Standard Quantum Group <i>X</i><sub>q</sub>(<i>A</i><sub>1</sub>) on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula>[<i>x</i>,<i>y</i>,<i>z</i>]

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