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Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

Abstract

The present paper is devoted to study 2-local derivations on infinite-dimensional Lie algebras over a field of characteristic zero. We show that every derivation on Virasoro algebra is inner and prove that all 2-local derivations on this algebra is a derivation. We give an example of infinite-dimensional Lie algebra with a 2-local derivation which is not a derivation.

First Page

217

Last Page

230

References

1. Alizadeh R., Bitarafan M.J. Local derivations of full matrix rings. Acta Mathematica Hungarica, Vol. 145, Issue 2, 433–439 (2015).

2. Ayupov Sh.A., Kudaybergenov K.K. 2-Local derivations on von Neumann algebras. Positivity, Vol. 19, Issue 3, 445–455 (2015).

3. Ayupov Sh.A., Kudaybergenov K.K. 2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras. Journal of Physics: Conference Series, Vol. 697, 012001 (2016).

4. Ayupov Sh.A., Kudaybergenov K.K. 2-Local automorphisms on finite dimensional Lie algebras. Linear Algebra and its Applications, Vol. 507, 121–131 (2016).

5. Ayupov Sh.A., Kudaybergenov K.K., Rakhimov I.S. 2-Local derivations on finite-dimensional Lie algebras. Linear Algebra and its Applications, Vol. 474, 1–11 (2015).

6. Ayupov Sh.A., Kudaybergenov K.K., Omirov B.A. Local and 2-local derivations and automorphisms on simple Leibniz algebras. Bull. Malays. Math. Sci. Soc., (2019), https://doi.org/10.1007/s40840-019-00799-5.

7. Ayupov Sh.A., Yusupov B.B. 2-Local derivations of infinite-dimensional Lie algebras. Journal of Algebra and its Applications, (2019), doi: 10.1142/S0219498820501005.

8. Ayupov Sh.A., Kudaybergenov K.K., Alauadinov A.K. 2-Local derivations on matrix algebras over commutative regular algebras. Linear Algebra and its Applications, Vol. 439, 1294–1311 (2013).

9. Ayupov Sh.A., Kudaybergenov K.K., Yusupov B.B. 2-Local derivations on generalized Witt algebras. Linear and Multilinear Algebra, (in press).

10. Chen Z. Biderivations and linear commuting maps on simple generalized Witt algebras over a field. Electronic Journal of Linear Algebra, Vol. 31, 1–12 (2016).

11. Chen Z., Wang D. 2-Local automorphisms of finite-dimensional simple Lie algebras. Linear Algebra and its Applications, Vol. 486, 335–344 (2015).

12. Millionshchikov D.V. Naturally graded lie algebras (carnot algebras) of slow growth. Sbornik: Mathematics, Vol. 210, No. 6, 862–909 (2019). doi: 10.1070/SM9055.

13. Ikeda T., Kawamoto N. On the derivations of generalized Witt algebras over a field of characteristic zero. Hiroshima Math. J., Vol. 20, Issue 1, 47–55 (1990).

14. Kac V., Raina A. Bombay lectures on highest weight representations of infinite-dimensional Lie algebras. World Sci., Singapore, (1987).

15. Kadison R.V. Local derivations. J. Algebra, Vol. 130, Issue 2, 494–509 (1990).

16. Kim S.O., Kim J.S. Local automorphisms and derivations on ��n. Proc. Amer. Math. Soc., Vol. 132, Issue 5, 1389–1392 (2004).

17. Khakimdjanova K., Khakimdjanov Yu. Sur une classe d'algebres de Lie de dimension infinie. Communication in Algebra, Vol. 29, Issue 1, 177–191 (2001).

18. Patera J., Zassenhaus H. The higher rank Virasoro algebras. Commun. Math. Phys., Vol. 136, Issue 1, 1–14 (1991).

19. Šemrl P. Local automorphisms and derivations on B(H). Proc. Amer. Math. Soc., Vol. 125, Issue 9, 2677–2680 (1997).

20. Yongping W., Ying X., Lamei Y. Derivations and automorphism groups of completed Witt Lie algebra. Algebra Colloquium, Vol. 19, Issue 3, 581–590 (2012).

21. Yusupov B.B. 2-local derivations on Witt algebras. Uzbek Mathematical Journal, No. 1, 160–166 (2018).

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