On the Liouville-Arnold integrable flows related with quantum algebras and their Poissonian representations

Based on the structure of Casimir elements associated with general Hopf algebras there are constructed Liouville-Arnold integrable flows related with naturally induced Poisson structures on arbitrary co-algebra and their deformations. Some interesting special cases including the oscillatory Heisenberg-Weil algebra related co-algebra structures and adjoint with them integrable Hamiltonian systems are considered. 1 Hopf algebras and co-algebras: main definitions Consider a Hopf algebra A over C endowed with two special homomorphisms called coproduct ∆ : A → A⊗A and counit ε : A →C, as well an antihomomorphism (antipode) ν : A → A, such that for any a ∈ A (id⊗∆)∆(a) = (∆⊗ id)∆(a), (1.1) (id⊗ ε)∆(a) = (ε⊗ id)∆(a) = a, m((id⊗ ν)∆(a)) = m((ν ⊗ id)∆(a)) = ε(a)I, where m : A⊗A → A is the usual multiplication mapping, that is for any a, b ∈ A m(a ⊗ b) = ab. The conditions (1.1) were introduced by Hopf [1] in a cohomological context. Since most of the Hopf algebras properties depend on the coproduct operation ∆ : A → A⊗A and related with it Casimir elements, below we shall dwell mainly on the objects called co-algebras endowed with this coproduct. The most interesting examples of co-algebras are provided by the universal enveloping algebras U(G) of Lie algebras G. If, for instance, a Lie algebra G possesses generators Xi ∈ G, i = 1, n, n = dimG, the corresponding enveloping