2008/12/19 by Jiří Hrivnák, Jiri Hrivnak, Petr Novotný +1 · 9 citations
Mathematics · Physics and Astronomy · #Adjoint representation #Adjoint representation of a Lie algebra #Advanced Topics in Algebra #Affine Lie algebra #Algebra over a field #Algebraic structures and combinatorial models #Cohomology #Current algebra #Fundamental representation #Homotopy and Cohomology in Algebraic Topology #Invariant (physics) #Lie algebra #Lie conformal algebra #Lie group #Mathematical physics #Mathematics #Nilpotent #Pure mathematics #Representation of a Lie group #Representation theory #Weight #math-ph #math.MP
paper · pdf · doi:10.1016/j.laa.2008.11.003
published in Linear Algebra and its Applications 430(4), 1384-1403 (Elsevier BV) · 19 pages
openalex publication_date 2008/12/19 · arxiv created 2009/05/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider finite-dimensional complex Lie algebras. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie algebras. A special case is generalization of 1-cocycles with respect to the adjoint representation - so called (α,β,γ)--derivations. Parametric sets of spaces of cocycles allow us to define complex functions which are invariant under Lie isomorphisms. Such complex functions thus represent useful invariants - we show how they classify three and four-dimensional Lie algebras as well as how they apply to some eight-dimensional one-parametric nilpotent continua of Lie algebras. These functions also provide necessary criteria for existence of 1-parametric continuous contraction.