Classical double, $R$-operators, and negative flows of integrable hierarchies

Using the classical double $\mathcal G$ of a Lie algebra $\mathfrak g$ equipped with the classical $R$-operator, we define two sets of functions commuting with respect to the initial Lie–Poisson bracket on $\mathfrak g^*$ and its extensions. We consider examples of Lie algebras $\mathfrak g$ with the “Adler–Kostant–Symes” $R$-operators and the two corresponding sets of mutually commuting functions in detail. Using the constructed commutative Hamiltonian flows on different extensions of $\mathfrak g$, we obtain zero-curvature equations with $\mathfrak g$-valued $U$–$V$ pairs. The so-called negative flows of soliton hierarchies are among such equations. We illustrate the proposed approach with examples of two-dimensional Abelian and non-Abelian Toda field equations.