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## Scientific Bulletin. Physical and Mathematical Research

#### Article Title

LOCAL AND 2-LOCAL DERIVATIONS OF SOLVABLE LEIBNIZ ALGEBRAS WITH NULL-FILIFORM NILRADICAL

#### Abstract

In the works of Ayupov, Khudoyberdiyev and Yusupov proved that the local and 2-local derivation of solvable Leibniz algebras with model nilradical are derivations. Solvable Leibniz algebras with null-filiform nilradical are the partial case of solvable Leibniz algebras with model nilradical. However, the proof in the paper is different from model nilradical case. The derivation is a fundamental notion in mathematics. Derivations play a prominent role in algebra. There are many generalizations of derivations as antiderivation, δ-derivations, ternary derivations and (α,β,γ)-derivations. One of the important generalizations of derivation is local and 2-local derivations. Local derivations defined by Kadison, Larson and Sourour in 1990. In 1997, P. Šemrl introduced and studied 2-local derivations on the algebra of bounded linear operators on a Hilbert space. Many works were devoted to the study of the local and 2-local derivations.

Leibniz algebras were defined and investigated at the beginning of the 90s of the past century by J.-L.Loday. Leibniz algebras generalize Lie algebras in a natural way. Leibniz algebras have been actively investigated and numerous papers have been devoted to the study of these algebras. In this paper constructed all derivations of considered solvable Leibniz algebras with null-filiform nilradical. In the present paper proved that any local and 2-local derivation of solvable Leibniz algebras with null-filiform nilradical are derivations. In the works of Ayupov, Khudoyberdiyev and Yusupov proved that the local and 2-local derivation of solvable Leibniz algebras with model nilradical are derivations. Solvable Leibniz algebras with null-filiform nilradical are the partial case of solvable Leibniz algebras with model nilradical. However, the proof in the paper is different from model nilradical case.

#### First Page

27

#### Last Page

36

#### References

1. Ayupov Sh., Arzikulov F. (2014). 2-Local derivations on semifinitevon Neumann algebras. *Glasgow Mathematical Journal*. 56. Pp. 9-12.

2. Ayupov Sh., Arzikulov F. (2017). 2-Local derivations on associative and Jordan matrix rings over commutative rings. *LinearAlgebra and its Applications*. 522 Pp. 28-50.

3. Ayupov Sh., Kudaybergenov K. (2012). 2-Local derivations and automorphisms on B(H). *Journal of Mathematical Analysis and Applications**.* 395. Pp. 15-18.

4. Ayupov Sh., Kudaybergenov K., Omirov B. (2020). Local and 2-local derivations and automorphisms on simple Leibniz algebras. *Bulletin of the Malaysian Mathematical Sciences Society**.* 43(3). Pp. 2199–2234.

5. Ayupov Sh., Khudoyberdiyev A., Yusupov B. (2020). Local and 2-Local Derivations of Solvable Leibniz Algebras. *International Journal of Algebra and Computation*. 30(6). Pp. 1185-1197.

6. Casas J.M., Ladra M., Omirov B.A., Karimjanov I.A. (2013) Classification of solvable Leibniz algebras with null-filiform nilradical. *Linear Multilinear Algebra**.* 61 (6). Pp. 758–774.

7. Kadison R. (1990). Local derivations. *Journal of Algebra*. 130(2). Pp. 494–509.

8. Semrl P. (1997). Local automorphisms and derivations on B(H). *Proceedings of the American Mathematical Society*. 125. Pp. 2677-2680.

#### Recommended Citation

Umrzaqov, Sardorbek
(2020)
"LOCAL AND 2-LOCAL DERIVATIONS OF SOLVABLE LEIBNIZ ALGEBRAS WITH NULL-FILIFORM NILRADICAL,"
*Scientific Bulletin. Physical and Mathematical Research*: Vol. 2
:
Iss.
2
, Article 4.

Available at:
https://uzjournals.edu.uz/adu/vol2/iss2/4