Article Title

ROTA-BAXTER OPERATORS ON 3-DIMENSIONAL NILPOTENT ASSOCIATIVE ALGEBRAS

Authors

Jamila R. Aliyeva, Andijan State University, Andijan, 170100, Str. University, 129 (Uzbekistan). E-mail: agsu_info@edu.uzFollow
Hushruyahon M. Karimjanova, School №31 Andijan city, Andijan, 170100, Str. Tashkent, 24 (Uzbekistan). E-mail: 31_maktab_31@inbox.uzFollow
Ziyodahon B. Holmirzayeva, Andijan State University, Andijan, 170100, Str. University, 129 (Uzbekistan). E-mail: agsu_info@edu.uzFollow

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⁻¹P 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 A₁, A₂, A₃, A4 and A5 which presented above.

We proof that there are five types for 3-dimensional associative algebra A₁, three types for A₂ and A₃, two types for A₅ of Rota-Baxter operators of weight 0. Moreover, we have showed there are eight types for 3-dimensional associative algebra A₁, one type for A₂, four types for A₃, one type for A₄ and two types for A₅ of Rota-Baxter operators of weight 1.

First Page

70

Last Page

76

References

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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