Novikov structures on solvable Lie algebras

TL;DR

This paper investigates the existence of Novikov structures on finite-dimensional Lie algebras, showing that such structures imply solvability and providing methods to construct them on certain classes of solvable Lie algebras.

Contribution

It establishes the relationship between Novikov structures and solvability, and introduces new construction techniques using classical r-matrices and extensions.

Findings

Lie algebra admitting a Novikov structure must be solvable

Existence of Novikov structures on certain 2-step solvable Lie algebras

Extension methods for constructing Novikov structures

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.