TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMS

Farhodjon Arzikulov     (V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Univesity Str., 9, Olmazor district, Tashkent, 100174; Andijan State University, Universitet Str., 129, Andijan, 170100, Uzbekistan)
Feruza Nabijonova     (Fergana State University, Murabbiylar Str., 19, Fergana, 150100, Uzbekistan)
Furkat Urinboyev     (Namangan State University, Boburshoh Str., 161 Namangan, 160107, Uzbekistan)

Abstract


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.


Keywords


Linear operator, 2-Local linear operator, Leibniz algebra, Zinbiel algebra, Derivation, 2-Local derivations, Automorphism, 2-Local automorphism

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References


  1. Abdurasulov K., Adashev J., Kaygorodov I. Maximal solvable Leibniz algebras with a quasi-filiform nilradical. Mathematics, 2023. Vol. 11, No. 5. Art. no. 1120. DOI: 10.3390/math11051120
  2. Adashev J.Q., Khudoyberdiyev A.Kh., Omirov B.A. Classifications of some classes of Zinbiel algebras. J. Generalized Lie Theory Appl., 2010. Vol. 4. Art. no. S090601.
  3. Adashev J., Yusupov B. Local automorphisms of n-dimensional naturally graded quasi-filiform Leibniz algebra of type I. Algebr. Struct. Their Appl., 2024. Vol. 11. P. 11–24.
  4. Ayupov Sh., Arzikulov F. 2-Local derivations on semi-finite von Neumann algebras. Glasg. Math. J., 2014. Vol. 56, No. 1. P. 9–12. DOI: 10.1017/S0017089512000870
  5. Ayupov Sh., Arzikulov F. 2-Local derivations on associative and Jordan matrix rings over commutative rings. Linear Algebra Appl., 2017. Vol. 522. P. 28–50. DOI: 10.1016/j.laa.2017.02.012
  6. Ayupov Sh.A., Arzikulov F.N. Description of 2-local and local derivations on some Lie rings of skew-adjoint matrices. Linear Multilinear Algebra, 2020. Vol. 68, No. 4. P. 764–780. DOI: 10.1080/03081087.2018.1517719
  7. Ayupov Sh.A., Arzikulov F.N., Umrzaqov N.M., Nuriddinov O.O. Description of 2-local derivations and automorphisms on finite-dimensional Jordan algebras. Linear Multilinear Algebra, 2022. Vol. 70, No. 18. P. 3525–3542. DOI: 10.1080/03081087.2020.1845595
  8. Ayupov Sh.A., Arzikulov F.N., Umrzaqov S.M. Local and 2-local derivations on Lie matrix rings over commutative involutive rings. J. Lie Theory, 2022. Vol. 32, No. 4. P. 1053–1071. URL: https://www.heldermann.de/JLT/JLT32/JLT324/jlt32049.htm
  9. Ayupov Sh., Kudaybergenov K. 2-Local derivations and automorphisms on \(B(H)\). J. Math. Anal. Appl., 2012. Vol. 395, No. 1. P. 15–18. DOI: 10.1016/j.jmaa.2012.04.064
  10. Ayupov Sh., Kudaybergenov K. 2-Local derivations on von Neumann algebras. Positivity, 2015. Vol. 19. P. 445–455. DOI: 10.1007/s11117-014-0307-3
  11. Ayupov Sh., Kudaybergenov K., Rakhimov I. 2-Local derivations on finite-dimensional Lie algebras. Linear Algebra Appl., 2015. Vol. 474. P. 1–11. DOI: 10.1016/j.laa.2015.01.016
  12. Ayupov Sh., Kudaybergenov K. 2-Local automorphisms on finite-dimensional Lie algebras. Linear Algebra Appl., 2016. Vol. 507. P. 121–131. DOI: 10.1016/j.laa.2016.05.042
  13. Ayupov Sh., Kudaybergenov K., Omirov B. Local and 2-local derivations and automorphisms on simple Leibniz algebras. Bull. Malays. Math. Sci. Soc., 2020. Vol. 43. P. 2199–2234. DOI: 10.1007/s40840-019-00799-5
  14. Ayupov Sh., Kudaybergenov K., Kalandarov T. 2-Local automorphisms on AW ∗-algebras. In: Positivity and Noncommutative Analysis. Trends Math. G. Buskes et al. (eds.). Cham: Birkhäuser, 2019. P. 1–13. DOI: 10.1007/978-3-030-10850-2_1
  15. Burgos M., Fernáandez-Polo F.J., Garcés J., Peralta A.M. A Kowalski-Słodkowski theorem for 2-local ∗-homomorphisms on von Neumann algebras. Rev. Ser. A Mat. RACSAM, 2015. Vol. 109. P. 551–568. DOI: 10.1007/s13398-014-0200-8
  16. Kashuba I., Martin M.E. Deformations of Jordan algebras of dimension four. J. Algebra, 2014. Vol. 399. P. 277–289. DOI: 10.1016/j.jalgebra.2013.09.040
  17. Kim S.O., Kim J.S. Local automorphisms and derivations on \(M_n\). Proc. Amer. Math. Soc., 2004. Vol. 132, No. 5. P. 1389–1392.
  18. Lai X., Chen Z.X. 2-local derivations of finite-dimensional simple Lie algebras (Chinese). Acta Math. Sinica (Chin. Ser.), 2015. Vol. 58, No. 5. P. 847–852.
  19. Lin Y.-F., Wong T.-L. A note on 2-local maps. Proc. Edinb. Math. Soc., 2006. Vol. 49, No. 3. P. 701–708. DOI: 10.1017/S0013091504001142
  20. Šemrl P. Local automorphisms and derivations on \(B(H)\). Proc. Amer. Math. Soc., 1997. Vol. 125. P. 2677–2680. DOI: 10.1090/S0002-9939-97-04073-2
  21. Umrzaqov S. Local derivations of null-filiform and filiform Zinbiel algebras. Uzbek Math. J., 2023. Vol. 67, No. 2. P. 162–169.




DOI: http://dx.doi.org/10.15826/umj.2025.1.001

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