On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications
In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary...
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| Format: | Article |
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MDPI AG
2022-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/10/1680 |
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| author | Sherali S. Ibraev Larissa S. Kainbaeva Angisin Z. Seitmuratov |
| author_facet | Sherali S. Ibraev Larissa S. Kainbaeva Angisin Z. Seitmuratov |
| author_sort | Sherali S. Ibraev |
| collection | DOAJ |
| description | In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and cohomology of algebraic groups. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> be a semisimple and simply connected algebraic group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> over an algebraically closed field of characteristic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>></mo><mi>h</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>h</mi></semantics></math></inline-formula> is a Coxeter number. Denote the first Frobenius kernel and Lie algebra of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">g</mi></semantics></math></inline-formula>, respectively. First, we calculate the restricted cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">g</mi></semantics></math></inline-formula> with coefficients in simple modules for two families of restricted simple modules. Since in the restricted region the restricted cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">g</mi></semantics></math></inline-formula> is equivalent to the corresponding cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub><mo>,</mo></mrow></semantics></math></inline-formula> we describe them as the cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> in terms of the cohomology for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> with coefficients in dual Weyl modules. Then, we give a necessary and sufficient condition for the isomorphisms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><msub><mi>G</mi><mn>1</mn></msub><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow><mo>≅</mo><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><mi>G</mi><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><mi mathvariant="fraktur">g</mi><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow><mo>≅</mo><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><mi>G</mi><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> and a necessary condition for the isomorphism <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><mi mathvariant="fraktur">g</mi><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow><mo>≅</mo><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><msub><mi>G</mi><mn>1</mn></msub><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>V</mi></semantics></math></inline-formula> is a simple module with highest restricted weight. Using these results, we obtain all non-trivial isomorphisms between the cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">g</mi></semantics></math></inline-formula> with coefficients in the considered simple modules. |
| first_indexed | 2024-03-10T03:29:14Z |
| format | Article |
| id | doaj.art-b684fbba60474fe7b044b2db3122f23e |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:29:14Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b684fbba60474fe7b044b2db3122f23e2023-11-23T12:00:51ZengMDPI AGMathematics2227-73902022-05-011010168010.3390/math10101680On Restricted Cohomology of Modular Classical Lie Algebras and Their ApplicationsSherali S. Ibraev0Larissa S. Kainbaeva1Angisin Z. Seitmuratov2Department of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda 120014, KazakhstanDepartment of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda 120014, KazakhstanDepartment of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda 120014, KazakhstanIn this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and cohomology of algebraic groups. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> be a semisimple and simply connected algebraic group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> over an algebraically closed field of characteristic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>></mo><mi>h</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>h</mi></semantics></math></inline-formula> is a Coxeter number. Denote the first Frobenius kernel and Lie algebra of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula> by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">g</mi></semantics></math></inline-formula>, respectively. First, we calculate the restricted cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">g</mi></semantics></math></inline-formula> with coefficients in simple modules for two families of restricted simple modules. Since in the restricted region the restricted cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">g</mi></semantics></math></inline-formula> is equivalent to the corresponding cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub><mo>,</mo></mrow></semantics></math></inline-formula> we describe them as the cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> in terms of the cohomology for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> with coefficients in dual Weyl modules. Then, we give a necessary and sufficient condition for the isomorphisms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><msub><mi>G</mi><mn>1</mn></msub><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow><mo>≅</mo><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><mi>G</mi><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><mi mathvariant="fraktur">g</mi><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow><mo>≅</mo><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><mi>G</mi><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> and a necessary condition for the isomorphism <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><mi mathvariant="fraktur">g</mi><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow><mo>≅</mo><msup><mi>H</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><msub><mi>G</mi><mn>1</mn></msub><mo>,</mo><mi>V</mi></mrow><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>V</mi></semantics></math></inline-formula> is a simple module with highest restricted weight. Using these results, we obtain all non-trivial isomorphisms between the cohomology of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>G</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">g</mi></semantics></math></inline-formula> with coefficients in the considered simple modules.https://www.mdpi.com/2227-7390/10/10/1680Lie algebraalgebraic groupcohomologyrestricted cohomologysimple modules |
| spellingShingle | Sherali S. Ibraev Larissa S. Kainbaeva Angisin Z. Seitmuratov On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications Mathematics Lie algebra algebraic group cohomology restricted cohomology simple modules |
| title | On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications |
| title_full | On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications |
| title_fullStr | On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications |
| title_full_unstemmed | On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications |
| title_short | On Restricted Cohomology of Modular Classical Lie Algebras and Their Applications |
| title_sort | on restricted cohomology of modular classical lie algebras and their applications |
| topic | Lie algebra algebraic group cohomology restricted cohomology simple modules |
| url | https://www.mdpi.com/2227-7390/10/10/1680 |
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