Coboundary Lie bialgebras and commutative subalgebras of universal enveloping algebras. math.QA/0503608

Abstract. We solve a functional version of the problem of twist quantization of a cobound-ary Lie bialgebra (g, r, Z). We derive from this the following results: (a) the formal Pois-son manifolds g ∗ and G ∗ are isomorphic; (b) we construct an injective algebra morphism S(g∗)g → ֒ U(g∗). When (g, r, Z) can be quantized, we construct a deformation of this mor-phism. In the particular case when g is quasitriangular and nondegenerate, we compare our construction with Semenov-Tian-Shansky’s construction of a commutative subalgebra of U(g∗). We also show that the canonical derivation of the function ring of G ∗ is Hamiltonian. Let (g, r, Z) be a coboundary Lie bialgebra over a field K of characteristic 0. This means that g is a Lie bialgebra, the Lie cobracket δ of which is the coboundary of r ∈ ∧2(g): δ(x) = [x ⊗ 1 + 1 ⊗ x, r] for any x ∈ g. This condition means that Z: = CYB(r) belongs to ∧3(g)g (here CYB is the l.h.s. of the classical Yang-Baxter equation). Quasi-triangular and triangula