On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface Singularities
It is a natural question to ask whether there is any Lie algebra that completely characterize simple singularities? The higher Nash blow-up derivation Lie algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><m...
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MDPI AG
2023-04-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/8/1935 |
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| author | Muhammad Asif Ahmad N. Al-Kenani Naveed Hussain Muhammad Ahsan Binyamin |
| author_facet | Muhammad Asif Ahmad N. Al-Kenani Naveed Hussain Muhammad Ahsan Binyamin |
| author_sort | Muhammad Asif |
| collection | DOAJ |
| description | It is a natural question to ask whether there is any Lie algebra that completely characterize simple singularities? The higher Nash blow-up derivation Lie algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">L</mi><mi>k</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> associated to isolated hypersurface singularities defined to be the Lie algebra of derivations of the local Artinian algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">M</mi><mi>n</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msub><mi mathvariant="script">O</mi><mi>l</mi></msub><mo>/</mo><mrow><mo>⟨</mo><mi>F</mi><mo>,</mo><mi mathvariant="script">J</mi><mi>n</mi><mo>⟩</mo></mrow></mrow></semantics></math></inline-formula>, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">L</mi><mi>k</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>=</mo><mi>D</mi><mi>e</mi><mi>r</mi><mrow><mo>(</mo><msubsup><mi mathvariant="script">M</mi><mi>n</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we construct a new conjecture for the complete characterization of simple hypersurface singularities using the Lie algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">L</mi><mi>k</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> under certain condition and prove it true for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">L</mi><mi>k</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>,</mo><mi>l</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-11T04:46:39Z |
| format | Article |
| id | doaj.art-40489913400d4dcf904f8ea131daf9ff |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:46:39Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-40489913400d4dcf904f8ea131daf9ff2023-11-17T20:18:42ZengMDPI AGMathematics2227-73902023-04-01118193510.3390/math11081935On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface SingularitiesMuhammad Asif0Ahmad N. Al-Kenani1Naveed Hussain2Muhammad Ahsan Binyamin3Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Punjab, PakistanDepartment of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics and Statistics, University of Agriculture, Faisalabad 38000, Punjab, PakistanDepartment of Mathematics, GC University Faisalabad, Faisalabad 38000, Punjab, PakistanIt is a natural question to ask whether there is any Lie algebra that completely characterize simple singularities? The higher Nash blow-up derivation Lie algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">L</mi><mi>k</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> associated to isolated hypersurface singularities defined to be the Lie algebra of derivations of the local Artinian algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">M</mi><mi>n</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msub><mi mathvariant="script">O</mi><mi>l</mi></msub><mo>/</mo><mrow><mo>⟨</mo><mi>F</mi><mo>,</mo><mi mathvariant="script">J</mi><mi>n</mi><mo>⟩</mo></mrow></mrow></semantics></math></inline-formula>, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">L</mi><mi>k</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>=</mo><mi>D</mi><mi>e</mi><mi>r</mi><mrow><mo>(</mo><msubsup><mi mathvariant="script">M</mi><mi>n</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we construct a new conjecture for the complete characterization of simple hypersurface singularities using the Lie algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">L</mi><mi>k</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> under certain condition and prove it true for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">L</mi><mi>k</mi><mi>l</mi></msubsup><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>,</mo><mi>l</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/11/8/1935higher Nash blow-upJacobian matrixCartan matrixisolated singularity |
| spellingShingle | Muhammad Asif Ahmad N. Al-Kenani Naveed Hussain Muhammad Ahsan Binyamin On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface Singularities Mathematics higher Nash blow-up Jacobian matrix Cartan matrix isolated singularity |
| title | On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface Singularities |
| title_full | On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface Singularities |
| title_fullStr | On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface Singularities |
| title_full_unstemmed | On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface Singularities |
| title_short | On the Higher Nash Blow-Up Derivation Lie Algebras of Isolated Hypersurface Singularities |
| title_sort | on the higher nash blow up derivation lie algebras of isolated hypersurface singularities |
| topic | higher Nash blow-up Jacobian matrix Cartan matrix isolated singularity |
| url | https://www.mdpi.com/2227-7390/11/8/1935 |
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