On classification up to isotopy of simple 3-dimensional commutative algebras of nil-rank 1

Authors

  • Vita I. Glizburg Moscow City University, 4 Vtoroj Selskohozjajstvennyj proezd, 1 building, Moscow 129226, Russia
  • Sergey V. Pchelintsev Financial University under the Government of the Russian Federation, 49/2 Leningradskij Avenue, Moscow 125167, Russia

Keywords:

simple 3-dimensional algebra, commutative algebra, nil-element, nil-rank, isotope
Supporting Agencies
This work was carried out with the financial support of the São Paulo Research Foundation (FAPESP) (grant No. 2023/01159-5). The authors express their gratitude to I. B. Kaigorodov, A. P. Pozhidaev, and I. P. Shestakov for moral support, as well as to O. V. Shashkov for his comprehensive assistance and help in preparing the article.

Abstract

It is proved that any two simple 3-dimensional unital commutative algebras of nil-rank 1 are isotopic if the ground field is algebraically closed and has characteristic different from 2 and 3. Thus, the classification up to isotopy of simple 3-dimensional unital commutative algebras containing nil-elements of index 2 is completed.

Author Biographies

  • Vita I. Glizburg, Moscow City University, 4 Vtoroj Selskohozjajstvennyj proezd, 1 building, Moscow 129226, Russia

    PhD (physics and mathematics), doctor of science (pedagogy), docent; professor at the department of teaching methods, Institute of Pedagogy and Psychology of Education

  • Sergey V. Pchelintsev, Financial University under the Government of the Russian Federation, 49/2 Leningradskij Avenue, Moscow 125167, Russia

    doctor of science (physics and mathematics), full professor; professor at the department of mathematics and data analysis, faculty of information technology and big data analysis

References

  1. Bruck RH. Some results in the theory of linear non-associative algebras. Transactions of the American Mathematical Society. 1944;56(2):141–199. DOI: 10.2307/1990248.
  2. Krylov AA, Pchelintsev SV. The isotopically simple algebras with a nil-basis. Communications in Algebra. 2020;48(4):1697–1712. DOI: 10.1080/00927872.2019.1705467.
  3. Tkachev VG. Inner isotopes associated with automorphisms of commutative associative algebras. Communications in Mathematics. 2024;32(2):153 –184. DOI: 10.46298/cm.12223.
  4. Pchelintsev SV. Isotopes of alternative algebras of characteristic different from 3. Izvestiya: Mathematics. 2020;84(5):1002–1015. DOI: 10.1070/IM8930.
  5. Pchelintsev SV. Central isotopes of (-1, 1)-algebras. Siberian Mathematical Journal. 2021;62(4):678 – 690. DOI: 10.1134/S003744662104011X.
  6. Borisova LR, Glizburg VI, Pchelintsev SV. One-sided isotopes of finite-dimensional algebras. Journal of Mathematical Sciences. 2023;269(4):443 – 452. DOI: 10.1007/s10958-023-06291-7.
  7. Glizburg VI, Pchelintsev SV. Isotopes of simple algebras of arbitrary dimension. Asian-European Journal of Mathematics. 2020;13(6):2050108. DOI: 10.1142/S1793557120501089.
  8. Glizburg VI, Pchelintsev SV. On commutative isotopes of Jordan algebra of a symmetric bilinear non-degenerate form on a two-dimensional vector space. arXiv:2504.00868v1 [Preprint]. 2025 [cited 2025 April 4]: [14 p.]. Available from: https://arxiv.org/abs/2504.00868v1.
  9. Albert AA. Non-associative algebras. I. Fundamental concepts and isotopy. Annals of Mathematics. 1942;43(4):685 –707. DOI: 10.2307/1968960.

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Published

2026-09-25

Issue

No. 2 (2026): Journal of the Belarusian State University. Mathematics and Informatics

Section

Mathematical Logic, Algebra and Number Theory

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Copyright (c) 2026 Journal of the Belarusian State University. Mathematics and Informatics

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How to Cite

[1]
Glizburg, V. and Pchelintsev, S.V. 2026. On classification up to isotopy of simple 3-dimensional commutative algebras of nil-rank 1. Journal of the Belarusian State University. Mathematics and Informatics. 2 (Sep. 2026), 14–27.