Construction of polynomials in bi-involution for singular elements of the space dual to a Lie algebra

F. I. Lobzin<sup>ab

a</sup> Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

Abstract: A generalization of the well-known problem of the construction of complete bi-involutive sets of polynomials on the conjugate space of a Lie algebra to the case of singular covectors is considered. A generalization of the Mishchenko–Fomenko argument shift method to singular covectors is proposed and sufficient conditions for the completeness of the resulting sets are found. Using this method, it is shown that complete bi-involutive sets of polynomials can be constructed for singular covectors in all reductive Lie algebras.
Bibliography: 19 titles.

Keywords: Lie–Poisson bracket, compatible Poisson brackets, bi-involutive sets of polynomials, Mishchenko–Fomenko argument shift method.

MSC: Primary 17B05; Secondary 15A22

Received: 15.10.2024 and 10.12.2024

DOI: 10.4213/sm10216

 English version: Sbornik: Mathematics, 2025, 216:10, 1393–1405

Bibliographic databases:

• 
• 
• 
•