Novikov structures on solvable Lie algebras

We study Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov structure must be solvable. Conversely we present an example of a nilpotent two-step solvable Lie algebra without any Novikov structure. We construct Novikov structures on certain Lie algebras via classical r-matfices and via extensions. In the latter case we lift Novikov structures on an abelian Lie algebra a and a Lie algebra b to certain extensions of b by a. We apply this to prove the existence of affine and Novikov structures on several classes of two-step solvable Lie algebras. In particular we generalize a well known result of Scheuneman concerning affine structures on three-step nilpotent Lie algebras. (c) 2005 Elsevier B.V. All rights reserved.