Novikov structures on solvable Lie algebras
Mathematical Physics
2009-11-11 v1 math.MP
Rings and Algebras
Abstract
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.
Cite
@article{arxiv.math-ph/0502008,
title = {Novikov structures on solvable Lie algebras},
author = {Dietrich Burde and Karel Dekimpe},
journal= {arXiv preprint arXiv:math-ph/0502008},
year = {2009}
}