Novikov structures on solvable Lie algebras

Details Novikov structures on solvable Lie algebras Novikov structures on solvable Lie algebras

2005-02-02

arxiv.org/abs/math-ph/0502008v1

Journal of Geometry and Physics 56, Nr. 9, 1837-1855 (2006).

Physics

Mathematical Physics

Scientific paper

10.1016/j.geomphys.2005.10.010

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 2-step solvable Lie algebra without any Novikov structure. We construct Novikov structures on certain Lie algebras via classical r-matrices 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 2-step solvable Lie algebras. In particular we generalize a well known result of Scheuneman concerning affine structures on 3-step nilpotent Lie algebras.

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