Novikov structures on solvable Lie 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.

References in corpus (2)

• Field Algebras
• Chiral Deformations of Conformal Field Theories

Cited by in corpus (13)

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• Gröbner-Shirshov bases method for Gelfand-Dorfman-Novikov algebras
• Polynomial identities in Novikov algebras
• One-generated nilpotent Novikov algebras
• Classical r-matrices and Novikov algebras
• Pseudo-Kähler Lie algebras with abelian complex structures
• Darboux coordinates for Hamiltonian structures defined by Novikov algebras
• On free Gelfand--Dorfman--Novikov superalgebras and a PBW type theorem
• The algebraic and geometric classification of $\delta$-Novikov algebras
• Left-symmetric algebras, or pre-Lie algebras in geometry and physics
• Classification of orbit closures in the variety of 3-dimensional Novikov algebras
• Filiform Lie algebras with low derived length
• Classification of Novikov algebras