Work

On locally nilpotent derivations of polynomial algebra in three variables

Year: 2025

Type: article

Abstract: In this paper we investigate locally nilpotent derivations on the polynomial algebra in three variables over a field of characteristic zero. We introduce an iterating construction giving all locally nilpotent derivations of rank $2$. This construction allows us to obtain examples of non-triangularizable locally nilpotent derivations of rank $2$. We also show that the well-known example of a locally nilpotent derivation of rank $3$, given by Freudenburg, is a member of a large family of new examples of rank $3$ locally nilpotent derivations. Our approach is based on considering all locally nilpotent derivations commuting with a given derivation. We obtain a characterization of locally nilpotent derivations with a given rank in terms of sets of commuting locally nilpotent derivations. Bibliography: 32 titles.

Source: Sbornik Mathematics

DOI: 10.4213/sm10094e

Authors: Nikhilesh Dasgupta, Sergey Aleksandrovich Gaifullin

Institutions: GITAM University, National Research University Higher School of Economics, Lomonosov Moscow State University, Moscow Center of Fundamental and Applied Mathematics

Language: English

Topic: Advanced Topics in Algebra

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: Department of Science and Technology, Ministry of Science and Technology, India, Russian Science Foundation

Awards: G3183471198, G410114771, G4334174149, G5846989363, G8935180567