Let X be a simply connected closed oriented manifold, and let LX = Map(S1,X) denote the free loop space over X. Some twenty years ago, M. Chas and D. Sullivan discovered a remarkable Lie algebra structure on the (reduced) S1-equivariant homology ˜HS1 ∗ (LX,Q) of LX called the string topology bracket. This bracket is intrinsically related to the geometry of LX and has many interesting properties. In this talk, we will prove that the string topology bracket preserves a natural direct sum decomposition of ˜HS1 ∗ (LX,Q) induced by the finite coverings of the circle S1 → S1, provided X is of rationally elliptic homotopy type. This gives a (partial) answer to an old question about compatibility of Hodge decomposition with string topology operations. We will deduce the above geometric result from a general algebraic theorem on derived Poisson structures on universal enveloping algebras of homologically finite-dimensional nilpotent L∞-algebras. The proof uses ideas of formal differential geometry, in particular relies on Kontsevich-Duflo Isomorphism Theorem for finite-dimensional formal DG manifolds. The talk is based on joint work with A. C. Ramadoss and Y. Zhang