Abstract:
Using the $R$-operator on a Lie algebra $\mathfrak{g}$ satisfying the modified
classical Yang–Baxter equation, we define two sets of functions that
mutually commute with respect to the initial Lie–Poisson bracket on $\mathfrak{g}^*$.
We consider examples of the Lie algebras $\mathfrak{g}$ with the Kostant–Adler–Symes
and triangular decompositions, their $R$-operators, and the corresponding two
sets of mutually commuting functions in detail. We answer the question for
which $R$-operators the constructed sets of functions also commute with
respect to the $R$-bracket. We briefly discuss the Euler–Arnold-type
integrable equations for which the constructed commutative functions
constitute the algebra of first integrals.
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