2005/02/02 by Dietrich Burde, Karel Dekimpe · 2 citations
Mathematics · Physics and Astronomy · #Adjoint representation of a Lie algebra #Advanced Topics in Algebra #Affine Lie algebra #Algebra over a field #Algebraic structures and combinatorial models #Current algebra #Lie algebra #Lie conformal algebra #Mathematics #Nilpotent #Nonlinear Waves and Solitons #Novikov self-consistency principle #Pure mathematics #Universal enveloping algebra #math-ph #math.MP #math.RA #msc:17B30 #msc:17B69 #msc:17D25
paper · pdf · doi:10.1016/j.geomphys.2005.10.010
published as Journal of Geometry and Physics 56, Nr. 9, 1837-1855 (2006).
arxiv created 2005/02/02 · openalex publication_date 2005/12/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.