AP14870282 “Defining Identities for Varieties of Non-Associative Algebras”
Project Abstract:
In this project, the main focus is on generalizations of non-associative algebras, such as Leibniz, Novikov, Zinbiel, bicommutative, and assosymmetric algebras with one and two generators.
The project is aimed at describing defining identities for varieties of mono- and binary Novikov algebras, bicommutative and assosymmetric algebras, as well as varieties of algebras related to binary Leibniz and binary Zinbiel algebras.
In addition, differential Novikov algebras, associative and perm algebras, and related classes obtained via new binary operations will be considered.
The project is aimed at describing defining identities for varieties of mono- and binary Novikov algebras, bicommutative and assosymmetric algebras, as well as varieties of algebras related to binary Leibniz and binary Zinbiel algebras.
In addition, differential Novikov algebras, associative and perm algebras, and related classes obtained via new binary operations will be considered.
Project Objective:
The objective of this project is to study the defining identities of binary and mono Novikov, bicommutative, and assosymmetric algebras. We plan to investigate classes of algebras embeddable into differential associative, perm algebras, and Novikov algebras with new operations, and to construct their bases.
The objective of this project is to study the defining identities of binary and mono Novikov, bicommutative, and assosymmetric algebras. We plan to investigate classes of algebras embeddable into differential associative, perm algebras, and Novikov algebras with new operations, and to construct their bases.
Project Tasks:
The main tasks of the project include:
— finding polynomial identities of a subvariety of the variety of binary Leibniz algebras that contains all Leibniz and Malcev algebras;
— studying the nilpotency of binary Zinbiel algebras;
— determining defining identities for varieties of mono and binary Novikov, bicommutative, and assosymmetric algebras;
— describing special identities of classes of algebras embeddable into associative, perm, and Novikov algebras with one derivation with respect to left and right multiplication, and verifying Kohn’s criterion for these classes of algebras;
— constructing bases of free algebras embeddable into associative, perm, and Novikov algebras with one derivation with respect to left and right multiplication.
The main tasks of the project include:
— finding polynomial identities of a subvariety of the variety of binary Leibniz algebras that contains all Leibniz and Malcev algebras;
— studying the nilpotency of binary Zinbiel algebras;
— determining defining identities for varieties of mono and binary Novikov, bicommutative, and assosymmetric algebras;
— describing special identities of classes of algebras embeddable into associative, perm, and Novikov algebras with one derivation with respect to left and right multiplication, and verifying Kohn’s criterion for these classes of algebras;
— constructing bases of free algebras embeddable into associative, perm, and Novikov algebras with one derivation with respect to left and right multiplication.
Project Implementation Stages:
— polynomial identities of a subvariety of the variety of binary Leibniz algebras containing all Leibniz and Malcev algebras will be determined;
— the nilpotency of binary Zinbiel algebras will be studied;
— defining identities of varieties of mono and binary Novikov, bicommutative, and assosymmetric algebras will be established;
— special identities of classes of algebras embeddable into associative, perm, and Novikov algebras with one derivation with respect to left and right multiplication will be described, and Kohn’s criterion will be verified for these classes;
— bases of free algebras embeddable into associative, perm, and Novikov algebras with one derivation (with respect to left and right multiplication) will be constructed.
— polynomial identities of a subvariety of the variety of binary Leibniz algebras containing all Leibniz and Malcev algebras will be determined;
— the nilpotency of binary Zinbiel algebras will be studied;
— defining identities of varieties of mono and binary Novikov, bicommutative, and assosymmetric algebras will be established;
— special identities of classes of algebras embeddable into associative, perm, and Novikov algebras with one derivation with respect to left and right multiplication will be described, and Kohn’s criterion will be verified for these classes;
— bases of free algebras embeddable into associative, perm, and Novikov algebras with one derivation (with respect to left and right multiplication) will be constructed.
Expected Results:
As a result of the project, polynomial identities of the subvariety (if it exists) of the variety of binary Leibniz algebras containing all Leibniz and Malcev algebras will be determined. The nilpotency of binary Zinbiel algebras will be investigated. Defining identities of varieties of mono and binary Novikov, bicommutative, and assosymmetric algebras will be established. Special identities of classes of algebras embeddable into associative, perm, and Novikov algebras with one derivation (with respect to left and right multiplication) will be described, and Kohn’s criterion will be verified for these classes. Bases of free algebras embeddable into associative, perm, and Novikov algebras with one derivation will be constructed.
As a result of the project, polynomial identities of the subvariety (if it exists) of the variety of binary Leibniz algebras containing all Leibniz and Malcev algebras will be determined. The nilpotency of binary Zinbiel algebras will be investigated. Defining identities of varieties of mono and binary Novikov, bicommutative, and assosymmetric algebras will be established. Special identities of classes of algebras embeddable into associative, perm, and Novikov algebras with one derivation (with respect to left and right multiplication) will be described, and Kohn’s criterion will be verified for these classes. Bases of free algebras embeddable into associative, perm, and Novikov algebras with one derivation will be constructed.
Project Team:
Nurlan Amankeldievich Ismailov, PhD, Associate Professor, Project Scientific Supervisor. Role in the project: Project management, execution of all stages according to the project schedule, and ensuring the achievement of required results.
Askar Serkulovich Dzhumadildaev, Doctor of Physical and Mathematical Sciences, Academician of the National Academy of Sciences of the Republic of Kazakhstan, Professor, Chief Researcher. Role in the project: Conducting core research on identities of binary and mono algebras.
Pavel Sergeevich Kolesnikov, Doctor of Physical and Mathematical Sciences, Professor, Chief Researcher. Role in the project: Conducting core research on identities of algebras with derivations.
Bekzat Kopzhasarovich Zhakhaev, PhD, Assistant Professor, Senior Researcher. Role in the project: Conducting core research on identities of algebras using the representation theory of symmetric groups.
Farukh Arkinovich Mashurov, PhD, Junior Researcher. Role in the project:
Conducting research on polynomial identities satisfied by perm algebras with respect to left and right multiplication.
Bauyrzhan Kairbekovich Sartaev, PhD, Junior Researcher. Role in the project: Conducting research on special identities of classes of algebras embeddable into associative algebras with one derivation.
Nurken Utepbergenuly Smadyarov, PhD doctoral student (2nd year), Junior Researcher. Role in the project: Conducting research on the description of a subvariety (if it exists) of the variety of binary Leibniz algebras containing all Leibniz and Malcev algebras.
Tolkynai Elemes, PhD doctoral student, Junior Researcher. Role in the project: Conducting research on the description of a subvariety (if it exists) of the variety of binary Leibniz algebras containing all Leibniz and Malcev algebras.
Nurlan Amankeldievich Ismailov, PhD, Associate Professor, Project Scientific Supervisor. Role in the project: Project management, execution of all stages according to the project schedule, and ensuring the achievement of required results.
Askar Serkulovich Dzhumadildaev, Doctor of Physical and Mathematical Sciences, Academician of the National Academy of Sciences of the Republic of Kazakhstan, Professor, Chief Researcher. Role in the project: Conducting core research on identities of binary and mono algebras.
Pavel Sergeevich Kolesnikov, Doctor of Physical and Mathematical Sciences, Professor, Chief Researcher. Role in the project: Conducting core research on identities of algebras with derivations.
Bekzat Kopzhasarovich Zhakhaev, PhD, Assistant Professor, Senior Researcher. Role in the project: Conducting core research on identities of algebras using the representation theory of symmetric groups.
Farukh Arkinovich Mashurov, PhD, Junior Researcher. Role in the project:
Conducting research on polynomial identities satisfied by perm algebras with respect to left and right multiplication.
Bauyrzhan Kairbekovich Sartaev, PhD, Junior Researcher. Role in the project: Conducting research on special identities of classes of algebras embeddable into associative algebras with one derivation.
Nurken Utepbergenuly Smadyarov, PhD doctoral student (2nd year), Junior Researcher. Role in the project: Conducting research on the description of a subvariety (if it exists) of the variety of binary Leibniz algebras containing all Leibniz and Malcev algebras.
Tolkynai Elemes, PhD doctoral student, Junior Researcher. Role in the project: Conducting research on the description of a subvariety (if it exists) of the variety of binary Leibniz algebras containing all Leibniz and Malcev algebras.
Project Results:
In accordance with the project schedule, the following new scientific results have been obtained:
— a subvariety of the variety of binary Leibniz algebras containing all Leibniz and Malcev algebras has been described. The validity of an analogue of Engel’s theorem for binary Leibniz algebras has also been verified;
— the nilpotency of binary Zinbiel algebras over a field of characteristic zero has been investigated. An example of a non-nilpotent binary Zinbiel algebra has been constructed;
— polynomial identities defining varieties of binary and mono Novikov and bicommutative algebras have been determined. Generalizations of assosymmetric algebras with one and two generators have been constructed.
— a subvariety of the variety of binary Leibniz algebras containing all Leibniz and Malcev algebras has been described. The validity of an analogue of Engel’s theorem for binary Leibniz algebras has also been verified;
— the nilpotency of binary Zinbiel algebras over a field of characteristic zero has been investigated. An example of a non-nilpotent binary Zinbiel algebra has been constructed;
— polynomial identities defining varieties of binary and mono Novikov and bicommutative algebras have been determined. Generalizations of assosymmetric algebras with one and two generators have been constructed.
Based on the research results, 3 articles have been published in ranked journals indexed in Web of Science (Clarivate Analytics, with impact factor) and Scopus:
1. V. Dotsenko, N. Ismailov and U. Umirbaev, Polynomial identities of Novikov algebras. Mathematische Zeitschrift, 303(3), 60 (2023). https://doi.org/10.1007/s00209-023-03231-8 (Quartile: Q2, Percentile: 64%, Impact Factor: 0.82)
2. F. Mashurov, B. Sartayev, Metabelian Lie and perm algebras. Journal of Algebra and Its Applications (2024) 2450065. https://doi.org/10.1142/S0219498824500658 (Quartile: Q3, Percentile: 57%, Impact Factor: 0.762)
3. B. Sartayev and P. Kolesnikov, Noncommutative Novikov algebras. European Journal of Mathematics, Volume 9, Article number: 35 (2023). https://doi.org/10.1007/s40879-023-00632-1 (Quartile: Q3, Percentile: 50%, Impact Factor: 0.624)
1. V. Dotsenko, N. Ismailov and U. Umirbaev, Polynomial identities of Novikov algebras. Mathematische Zeitschrift, 303(3), 60 (2023). https://doi.org/10.1007/s00209-023-03231-8 (Quartile: Q2, Percentile: 64%, Impact Factor: 0.82)
2. F. Mashurov, B. Sartayev, Metabelian Lie and perm algebras. Journal of Algebra and Its Applications (2024) 2450065. https://doi.org/10.1142/S0219498824500658 (Quartile: Q3, Percentile: 57%, Impact Factor: 0.762)
3. B. Sartayev and P. Kolesnikov, Noncommutative Novikov algebras. European Journal of Mathematics, Volume 9, Article number: 35 (2023). https://doi.org/10.1007/s40879-023-00632-1 (Quartile: Q3, Percentile: 50%, Impact Factor: 0.624)
Conference Abstracts:
1. N. Ismailov, Polynomial identities of Novikov algebras, International Conference “Malcev Readings”, November 14–19, 2022, Novosibirsk, Russia, p. 19.
2. B.K. Sartayev, Basis of the free noncommutative Novikov algebra, Abstracts of the Traditional International April Mathematical Conference dedicated to the Day of Science Workers of the Republic of Kazakhstan, p. 51. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan, April 5–7, 2023.
3. N. Ismailov, Polynomial identities of Novikov algebras, Abstracts of the Traditional International April Mathematical Conference dedicated to the Day of Science Workers of the Republic of Kazakhstan, p. 47. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan, April 5–7, 2023.
1. N. Ismailov, Polynomial identities of Novikov algebras, International Conference “Malcev Readings”, November 14–19, 2022, Novosibirsk, Russia, p. 19.
2. B.K. Sartayev, Basis of the free noncommutative Novikov algebra, Abstracts of the Traditional International April Mathematical Conference dedicated to the Day of Science Workers of the Republic of Kazakhstan, p. 51. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan, April 5–7, 2023.
3. N. Ismailov, Polynomial identities of Novikov algebras, Abstracts of the Traditional International April Mathematical Conference dedicated to the Day of Science Workers of the Republic of Kazakhstan, p. 47. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan, April 5–7, 2023.