Home > ajphysmat.uz > Vol. 2 > Iss. 1 (2020)

## Scientific Bulletin. Physical and Mathematical Research

#### Abstract

Associative algebras are introduced into mathematics of the 19th century and are still intensively studied. The classification of associative algebras of small dimensions first appeared in the works of Pierce in 1881. In 2018, the German scientist William de Graf gave a classification of nilpotent associative algebras of small sizes. The article describes all the Rota-Baxter operators on 3-dimensional nilpotent associative algebras.

Rota-Baxter operators were defined by Baxter to solve an analytic formula in probability. It has been related to other areas in mathematical physics and mathematics.

Throughout this paper algebras are considered over the field of complex numbers.

A Rota-Baxter operator on an associative algebra A over a field F is defined to be a linear map P : A → A satisfying

P(x)P(y) = P(xP(y) + P(x)y + λxy), ∀x,y ∈ A, λ ∈ F.

Note that, if P is a Rota-Baxter operator of weight λ≠0, then λ^{−1}P is a Rota-Baxter operator of weight 1. Therefore, it is sufficient to consider Rota-Baxter operators of weight 0 and 1.

Any three-dimensional complex nilpotent associative algebra A is isomorphic to one of the five pairwise non-isomorphic algebras named by A_{1}, A_{2}, A_{3}, A4 and A5 which presented above.

We proof that there are five types for 3-dimensional associative algebra A_{1}, three types for A_{2} and A_{3}, two types for A_{5} of Rota-Baxter operators of weight 0. Moreover, we have showed there are eight types for 3-dimensional associative algebra A_{1}, one type for A_{2}, four types for A_{3}, one type for A_{4} and two types for A_{5} of Rota-Baxter operators of weight 1.

#### First Page

70

#### Last Page

76

#### References

1. Bai R., Guo L., Li J., Wu Y. (2013) Rota-Baxter 3-Lie algebras. Journal of Mathematical Physics, 54(6).

2. Baxter G. (1960) An analytic problem whose solution follows from a simple algebraic identity. Pacific Journal of Mathematics. 10. Pp. 731-742.

3. Belavinand A.A., Drinfel’d V.G. (1982) Solutions of the classical Yang-Baxter equation for simple Lie algebras. Functional Analysis and its Applications. 16(3). Pp. 159-180.

4. Benito P., Gubarev V., Pozhidaev A. (2018) Rota-Baxter operators on quadratic algebras. Mediterranean Journal of Mathematics. 15. Pp. 1-23.

5. De Graaf W.A. (2018) Classification of nilpotent associative algebras of small dimension. Internatioal Journal of Algebra and Computtation. 28(1). Pp. 133-161.

6. Karimjanov I., Kaygorodov I., Ladra M. (2020) Rota-type operators on null-filiform associative algebras. Linear and Multilinear algebra. 68(1). Pp. 205-219.

7. Pan Y., Liu Q., Bai C., et. al. (2012) Post Lie algebra structures on the Lie algebra sl(2;C). Electron Journal of Linear Algebra. 23. Pp. 180-197.

8. Tang X., Zhang Y., Sun Q. (2014) Rota-Baxter operators on 4-dimensional complex simple associative algebras. Applied Mathematics and Computation. 229. Pp. 173-186.

9. Yu H. (2015) Classification of monomial Rota-Baxter operators on k[x]. Journal of Algebra and its Applications. 15(5).

10. Zheng S., Guo L., Rosenkranz M. (2015) Rota-Baxter operators on the polynomial algebra, integration, and averaging operators. Pacific Journal of Mathematics. 275(2). Pp. 481-507.

#### Recommended Citation

Aliyeva, Jamila R.; Karimjanova, Hushruyahon M.; and Holmirzayeva, Ziyodahon B.
(2020)
"ROTA-BAXTER OPERATORS ON 3-DIMENSIONAL NILPOTENT
ASSOCIATIVE ALGEBRAS,"
*Scientific Bulletin. Physical and Mathematical Research*: Vol. 2
:
Iss.
1
, Article 9.

Available at:
https://uzjournals.edu.uz/adu/vol2/iss1/9