Jacobi–Jordan Conformal Algebras: Basics, Constructions and Related Structures
The main purpose of this paper is to introduce and investigate the notion of Jacobi–Jordan conformal algebras. They are a generalization of Jacobi–Jordan algebras which correspond to the case in which the formal parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&...
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2025-03-01
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| author | Taoufik Chtioui Sami Mabrouk Abdenacer Makhlouf |
| author_facet | Taoufik Chtioui Sami Mabrouk Abdenacer Makhlouf |
| author_sort | Taoufik Chtioui |
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| description | The main purpose of this paper is to introduce and investigate the notion of Jacobi–Jordan conformal algebras. They are a generalization of Jacobi–Jordan algebras which correspond to the case in which the formal parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> equals 0. We consider some related structures such as conformal modules, corresponding representations and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">O</mi></semantics></math></inline-formula>-operators. Therefore, conformal derivations from Jacobi–Jordan conformal algebras to their conformal modules are used to describe conformal derivations of Jacobi–Jordan conformal algebras of the semidirect product type. Moreover, we study a class of Jacobi–Jordan conformal algebras called quadratic Jacobi–Jordan conformal algebras, which are characterized by mock-Gel’fand–Dorfman bialgebras. Finally, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><mo>[</mo><mo>∂</mo><mo>]</mo></mrow></semantics></math></inline-formula>-split extending structures problem for Jacobi–Jordan conformal algebras is studied. Furthermore, we introduce an unified product of a given Jacobi–Jordan conformal algebra <i>J</i> and a given <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><mo>[</mo><mo>∂</mo><mo>]</mo></mrow></semantics></math></inline-formula>-module <i>K</i>. This product includes some other interesting products of Jacobi–Jordan conformal algebras such as the twisted product and crossed product. Using this product, a cohomological type object is constructed to provide a theoretical answer to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><mo>[</mo><mo>∂</mo><mo>]</mo></mrow></semantics></math></inline-formula>-split extending structures problem. |
| format | Article |
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| institution | OA Journals |
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| language | English |
| publishDate | 2025-03-01 |
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| spelling | doaj-art-b6d41f6ef6e24a26953f83e365ef445f2025-08-20T02:05:24ZengMDPI AGMathematics2227-73902025-03-0113584310.3390/math13050843Jacobi–Jordan Conformal Algebras: Basics, Constructions and Related StructuresTaoufik Chtioui0Sami Mabrouk1Abdenacer Makhlouf2Mathematics and Applications Laboratory LR17ES11, Faculty of Sciences, Gabes University, Gabes 6072, TunisiaFaculty of Sciences, University of Gafsa, Gafsa 2112, TunisiaDépartement de Mathématiques, IRIMAS, Université de Haute Alsace, 18, Rue des Frères Lumière, F-68093 Mulhouse, FranceThe main purpose of this paper is to introduce and investigate the notion of Jacobi–Jordan conformal algebras. They are a generalization of Jacobi–Jordan algebras which correspond to the case in which the formal parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> equals 0. We consider some related structures such as conformal modules, corresponding representations and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">O</mi></semantics></math></inline-formula>-operators. Therefore, conformal derivations from Jacobi–Jordan conformal algebras to their conformal modules are used to describe conformal derivations of Jacobi–Jordan conformal algebras of the semidirect product type. Moreover, we study a class of Jacobi–Jordan conformal algebras called quadratic Jacobi–Jordan conformal algebras, which are characterized by mock-Gel’fand–Dorfman bialgebras. Finally, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><mo>[</mo><mo>∂</mo><mo>]</mo></mrow></semantics></math></inline-formula>-split extending structures problem for Jacobi–Jordan conformal algebras is studied. Furthermore, we introduce an unified product of a given Jacobi–Jordan conformal algebra <i>J</i> and a given <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><mo>[</mo><mo>∂</mo><mo>]</mo></mrow></semantics></math></inline-formula>-module <i>K</i>. This product includes some other interesting products of Jacobi–Jordan conformal algebras such as the twisted product and crossed product. Using this product, a cohomological type object is constructed to provide a theoretical answer to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">C</mi><mo>[</mo><mo>∂</mo><mo>]</mo></mrow></semantics></math></inline-formula>-split extending structures problem.https://www.mdpi.com/2227-7390/13/5/843Jacobi–Jordan conformal algebraconformal modulemock-Gel’fand–Dorfman bialgebraunified productextending structure |
| spellingShingle | Taoufik Chtioui Sami Mabrouk Abdenacer Makhlouf Jacobi–Jordan Conformal Algebras: Basics, Constructions and Related Structures Mathematics Jacobi–Jordan conformal algebra conformal module mock-Gel’fand–Dorfman bialgebra unified product extending structure |
| title | Jacobi–Jordan Conformal Algebras: Basics, Constructions and Related Structures |
| title_full | Jacobi–Jordan Conformal Algebras: Basics, Constructions and Related Structures |
| title_fullStr | Jacobi–Jordan Conformal Algebras: Basics, Constructions and Related Structures |
| title_full_unstemmed | Jacobi–Jordan Conformal Algebras: Basics, Constructions and Related Structures |
| title_short | Jacobi–Jordan Conformal Algebras: Basics, Constructions and Related Structures |
| title_sort | jacobi jordan conformal algebras basics constructions and related structures |
| topic | Jacobi–Jordan conformal algebra conformal module mock-Gel’fand–Dorfman bialgebra unified product extending structure |
| url | https://www.mdpi.com/2227-7390/13/5/843 |
| work_keys_str_mv | AT taoufikchtioui jacobijordanconformalalgebrasbasicsconstructionsandrelatedstructures AT samimabrouk jacobijordanconformalalgebrasbasicsconstructionsandrelatedstructures AT abdenacermakhlouf jacobijordanconformalalgebrasbasicsconstructionsandrelatedstructures |