2024/11/22 by Dietrich Burde, Burde, Dietrich, Friedrich Wagemann +1 · 1 citation
Mathematics · #17B10 #17B56 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2411.14952
openalex publication_date 2024/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the adjoint cohomology of perfect Lie algebras over the complex numbers. For the family of perfect Lie algebras \mathfrakg=\mathfraksl2(\Bbb C)\ltimes Vm we obtain some explicit results for Hk(\mathfrakg,\mathfrakg) with k≥ 0. Here Vm is the irreducible representation of \mathfraksl2(\Bbb C) of dimension m+1. For the computation of the cohomology we use the Hochschild-Serre formula, a long exact sequence in the cohomology and explicit formulas for the multiplicities of Vk in the exterior product Λj(Vm) for j≤ 4. In general we cannot determine the total adjoint cohomology for \mathfraksl2(\Bbb C)\ltimes Vm, but for some small m this is possible. We also give a classification of complex perfect Lie algebras \mathfrakg of dimension n≤ 9 and explicitly compute the cohomology spaces Hk(\mathfrakg,\mathfrakg) with k=0,1,2 for all Lie algebras from the classification list.