izvuzmathИзвестия высших учебных заведений. МатематикаIzvestiya Vysshikh Uchebnykh Zavedenii. Matematika0021-34462076-4626Казанский (Приволжский) федеральный университет10.26907/0021-3446-2026-4-60-83izvuzmath-291Research ArticleСтатьиМономиальные операторы Роты–Бакстера нулевого веса и операторы усреднения на алгебре многочленовMonomial Rota–Baxter operators of weight zero and averaging operators on the polynomial algebraХодзицкийА. Ф.KhodzitskiiA. F.

Артем Федорович Ходзицкий

Пирогова, д. 1, г. Новосибирск, 630090

Artem Fedorovich Khodzitskii

1 Pirogova str., Novosibirsk, 630090 

a.khodzitskii@g.nsu.ruНовосибирский государственный университетРоссияNovosibirsk State UniversityRussian Federation202608052026046083Copyright © Ходзицкий А.Ф., 20262026Ходзицкий А.Ф.Khodzitskii A.F.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://izvuzmath.elpub.ru/jour/article/view/291

Операторы Роты–Бакстера на алгебре многочленов начали изучать С.Х. Женг, Л. Гуо и М. Розенкранц в 2015 году.  Инъективные операторы Роты–Бакстера нулевого веса на $F[x]$ были описаны в 2021 году.  Мы описываем следующие классы мономиальных операторов Роты–Бакстера нулевого веса  на алгебре многочленов $F[x, y]$ и ее идеале $F_0[x, y]$:  1) не повышающие степень с ядром, не содержащим мономов,  2) отображающие каждый моном в себя с некоторым коэффициентом.  Для таких классов операторов мы показываем, как можно задать  мономиальный оператор усреднения по данному оператору Роты–Бакстера и наоборот.

The study of Rota–Baxter operators on polynomial algebras was initiated by S.H. Zheng, L. Guo, and M. Rosenkranz in 2015. Injective Rota–Baxter operators of weight zero on $F[x]$ were described in 2021. We classify the following classes of monomial Rota–Baxter operators of weight zero on the polynomial algebra $F[x,y]$ and its augmentation ideal $F_0[x,y]$:  1) non-increasing in degree and whose kernels contain no monomials, 2) mapping each monomial to a scalar multiple of itself. In the context of these classes of operators, we show how one may  define a monomial averaging operator by a given Rota–Baxter operator and vice versa.

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The authors declare that there are no conflicts of interest present.