A CHARACTERIZATION OF DERIVATIONS AND AUTOMORPHISMS ON SOME SIMPLE ALGEBRAS

Farhodjon Arzikulov     (V.I. Romanovskiy Institute of Mathematics, Universitet street 9, Tashkent, 100174, Uzbekistan)
Furqatjon Urinboyev     (Namangan State University, Uychi street 316, Namangan, 716019, Uzbekistan)
Shahlo Ergasheva     (Kokand State Pedagogical Institute, Turon street 23, Kokand, 150700, Uzbekistan)

Abstract


In the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple finite-dimensional algebras over a field of characteristic 0 without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticommutative algebra  \(\mathcal{D}\) and the seven-dimensional central simple commutative algebra \(\mathcal{C}\). We prove that every local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is a derivation, and every 2-local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also a derivation. We also prove that every local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is an automorphism, and every 2-local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also an automorphism.


Keywords


Simple algebra, Derivation, Local derivation, 2-Local derivation, Automorphism, Local automorphism, 2-Local automorphism, Basis of identities

Full Text:

PDF

References


  1. 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
  2. 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
  3. 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
  4. 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
  5. 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
  6. Ayupov Sh., Kudaybergenov K. Local derivations on finite-dimensional Lie algebras. Linear Algebra Appl., 2016. Vol. 493. P. 381—398. DOI: 10.1016/j.laa.2015.11.034
  7. 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
  8. 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
  9. Chen Z., Wang D. 2-Local automorphisms of finite-dimensional simple Lie algebras. Linear Algebra Appl., 2015. Vol. 486. P. 335–344. DOI: 10.1016/j.laa.2015.08.025
  10. Costantini M. Local automorphisms of finite dimensional simple Lie algebras. Linear Algebra Appl., 2019. Vol. 562. P. 123–134. DOI: 10.1016/j.laa.2018.10.009
  11. Filippov V.T. \(\delta\)-derivations of prime alternative and Mal’tsev algebras. Algebra Logic, 2000. Vol. 39. P. 354–358. DOI: 10.1007/BF02681620
  12. Kadison R.V. Local derivations. J. Algebra, 1990. Vol. 130, No. 2. P. 494–509. DOI: 10.1016/0021-8693(90)90095-6
  13. Kaigorodov I. On \((n + 1)\)-ary derivations of simple \(n\)-ary Mal’tsev algebras. St. Petersburg Math. J., 2014. Vol. 25. P. 575–585. DOI: 10.1090/S1061-0022-2014-01307-6
  14. Kaygorodov I., Popov Yu. A characterization of nilpotent nonassociative algebras by invertible Leibniz-derivations. J. Algebra, 2016. Vol. 456. P. 323–347. DOI: 10.1016/j.jalgebra.2016.02.016
  15. Khrypchenko M. Local derivations of finitary incidence algebras. Acta Math. Hungar., 2018. Vol. 154. P. 48–55. DOI: 10.1007/s10474-017-0758-7
  16. 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.
  17. Isaev I.M., Kislitsin A.V. An example of a simple finite-dimensional algebra with no finite basis of identities. Dokl. Math., 2012. Vol. 86, No. 3. P. 774–775. DOI: 10.1134/S1064562412060154    
  18. Isaev I.M., Kislitsin A.V. Example of simple finite dimensional algebra with no finite basis of its identities. Comm. Algebra, 2013. Vol. 41, No. 12. P 4593–4601. DOI: 10.1080/00927872.2012.706348
  19. Kislitsin A.V. An example of a central simple commutative finite-dimensional algebra with an infinite basis of identities. Algebra Logic, 2015. Vol 54. P. 204–210. DOI: 10.1007/s10469-015-9341-x
  20. Kislitsin A.V. Simple finite-dimensional algebras without finite basis of identities. Sib. Math. J., 2017. Vol. 58. P. 461–466. DOI: 10.1134/S00374466 17030090
  21. Larson D.R., Sourour A.R. Local derivations and local automorphisms of \(B(X)\). In: Proc. Sympos. Pure Math., Providence, Rhode Island Part 2, 1990. Vol. 51. P. 187–194. URL: http://hdl.handle.net/1828/2373
  22. Lin Y.-F., Wong T.-L. A note on 2-local maps. Proc. Edinb. Math. Soc. (2), 2006. Vol. 49, No. 3. P. 701–708. DOI: 10.1017/S0013091504001142
  23. Mal’tsev A.I. Analytic loops. Mat. Sb. (N.S.), 1955. Vol. 36(78), No. 3. P 569–576. (in Russian)
  24. 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




DOI: http://dx.doi.org/10.15826/umj.2022.2.004

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.