Dual R-matrix integrability

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.