
Journal of Lie Theory 33 (2023), No. 2, 663686 Copyright Heldermann Verlag 2023 Kronecker's Method and Complete Systems of Functions in BiInvolution on Classical Lie Algebras Aleksandra Garazha Faculty of Mechanics and Mathematics, Lomonosov State University, Moscow, Russia garazha.alex.andr@gmail.com [Abstractpdf] \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 biinvolution 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 biinvolution 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 biinvolution and the elements $A$ such that we can construct the Kronecker part of a complete system of functions in biinvolution. Also, we prove that the constructed functions freely generate some limits of MishchenkoFomenko 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. Keywords: BiHamiltonian systems, JordanKronecker invariants, argument shift method. MSC: 17B80. [ Fulltextpdf (219 KB)] for subscribers only. 