Kronecker's Method and Complete Systems of Functions in Bi-Involution on Classical Lie Algebras

Aleksandra Garazha¹

¹Faculty of Mechanics and Mathematics, Lomonosov State University, Moscow, Russia

Journal of Lie Theory, Tome 33 (2023) no. 2, pp. 663-686

Résumé

\newcommand\gh{\mathfrak{g}} \newcommand\ssl{\mathfrak {sl}} \newcommand\sso{\mathfrak {so}} \newcommand\ssp{\mathfrak {sp}} We use Kronecker's method to construct systems of functions in bi-involution with respect to two Poisson brackets: the canonical one and the bracket with frozen argument $A\in \gh$. For the Lie algebras $\ssl_n$ and $\ssp_{2n}$, we construct complete systems of functions in bi-involution for any $A \in \gh$. For the Lie algebras $\sso_{2n+1}$ and $\sso_{2n}$, we describe elements $A$ such that we can construct a complete system of functions in bi-involution and the elements $A$ such that we can construct the Kronecker part of a complete system of functions in bi-involution. Also, we prove that the constructed functions freely generate some limits of Mishchenko-Fomenko subalgebras. Finally, for the Lie algebras $\ssl_n$ and $\ssp_{2n}$, we show that the Kronecker indices are the same for all elements $A$ in any given sheet, while for the Lie algebras $\sso_{2n}$ and $\sso_{2n+1}$, we give examples of sheets such that this is not true.

Détail

Classification : 17B80
Mots-clés : Bi-Hamiltonian systems, Jordan-Kronecker invariants, argument shift method