TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMS
In the present paper, we investigate 2-local linear operators on vector spaces. Sufficient conditions are obtained for the linearity of a 2-local linear operator on a finite-dimensional vector space. To do this, families of matrices of a certain type are selected and it is proved that every 2-local...
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Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2025-07-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/892 |
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| author | Farhodjon Arzikulov Feruza Nabijonova Furkat Urinboyev |
| author_facet | Farhodjon Arzikulov Feruza Nabijonova Furkat Urinboyev |
| author_sort | Farhodjon Arzikulov |
| collection | DOAJ |
| description | In the present paper, we investigate 2-local linear operators on vector spaces. Sufficient conditions are obtained for the linearity of a 2-local linear operator on a finite-dimensional vector space. To do this, families of matrices of a certain type are selected and it is proved that every 2-local linear operator generated by these families is a linear operator. Based on these results we prove that each 2-local derivation of a finite-dimensional null-filiform Zinbiel algebra is a derivation. Also, we develop a method of construction of 2-local linear operators which are not linear operators. To this end, we select matrices of a certain type and using these matrices we construct a 2-local linear operator. If these matrices are distinct, then the 2-local linear operator constructed using these matrices is not a linear operator. Applying this method we prove that each finite-dimensional filiform Zinbiel algebra has a 2-local derivation that is not a derivation. We also prove that each finite-dimensional naturally graded quasi-filiform Leibniz algebras of type I has a 2-local automorphism that is not an automorphism. |
| format | Article |
| id | doaj-art-bc4c87d10bfa49208ff3c95cb2863f6a |
| institution | Kabale University |
| issn | 2414-3952 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-bc4c87d10bfa49208ff3c95cb2863f6a2025-08-20T03:56:09ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522025-07-0111110.15826/umj.2025.1.001233TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMSFarhodjon Arzikulov0Feruza Nabijonova1Furkat Urinboyev2V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Univesity Str., 9, Olmazor district, Tashkent, 100174; Andijan State University, Universitet Str., 129, Andijan, 170100Fergana State University, Murabbiylar Str., 19, Fergana, 150100Namangan State University, Boburshoh Str., 161 Namangan, 160107In the present paper, we investigate 2-local linear operators on vector spaces. Sufficient conditions are obtained for the linearity of a 2-local linear operator on a finite-dimensional vector space. To do this, families of matrices of a certain type are selected and it is proved that every 2-local linear operator generated by these families is a linear operator. Based on these results we prove that each 2-local derivation of a finite-dimensional null-filiform Zinbiel algebra is a derivation. Also, we develop a method of construction of 2-local linear operators which are not linear operators. To this end, we select matrices of a certain type and using these matrices we construct a 2-local linear operator. If these matrices are distinct, then the 2-local linear operator constructed using these matrices is not a linear operator. Applying this method we prove that each finite-dimensional filiform Zinbiel algebra has a 2-local derivation that is not a derivation. We also prove that each finite-dimensional naturally graded quasi-filiform Leibniz algebras of type I has a 2-local automorphism that is not an automorphism.https://umjuran.ru/index.php/umj/article/view/892linear operator, 2-local linear operator, leibniz algebra, zinbiel algebra, derivation, 2-local derivations, automorphism, 2-local automorphism |
| spellingShingle | Farhodjon Arzikulov Feruza Nabijonova Furkat Urinboyev TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMS Ural Mathematical Journal linear operator, 2-local linear operator, leibniz algebra, zinbiel algebra, derivation, 2-local derivations, automorphism, 2-local automorphism |
| title | TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMS |
| title_full | TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMS |
| title_fullStr | TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMS |
| title_full_unstemmed | TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMS |
| title_short | TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMS |
| title_sort | two methods of describing 2 local derivations and automorphisms |
| topic | linear operator, 2-local linear operator, leibniz algebra, zinbiel algebra, derivation, 2-local derivations, automorphism, 2-local automorphism |
| url | https://umjuran.ru/index.php/umj/article/view/892 |
| work_keys_str_mv | AT farhodjonarzikulov twomethodsofdescribing2localderivationsandautomorphisms AT feruzanabijonova twomethodsofdescribing2localderivationsandautomorphisms AT furkaturinboyev twomethodsofdescribing2localderivationsandautomorphisms |