Correspondence, Wells and Hochschild-Serre sequences for nonabelian extensions of multiplicative Lie algebras

2025/09/03 by Alexander Wires, Wires, Alexander, Dev Karan Singh +3
Mathematics · #08A35 #17A30 #17B36 #17B56 #18G50 #19C09 #20D45 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2509.03019

openalex publication_date 2025/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For nonabelian 2^nd-cohomology of multiplicative Lie algebras, we properly generalize from the group case three classic results. We prove a Correspondence theorem which compares 2^nd-cohomology associated to a realized abstract kernel to the abelian 2^nd-cohomology group over the algebraic center. For arbitrary extensions, we prove a Wells's Theorem characterizing ideal-preserving automorphisms and establish the 1-dimensional Lyndon-Hochschild-Serre exact sequence. Several previously established results are recovered when restricted to extensions with group-abelian or Lie-trivial ideals.

Citations

• Automorphisms of Multiplicative Lie algebra Extensions
• Theory of Extensions of Multiplicative Lie Algebras
• A seven-term exact sequence for the cohomology of a group extension

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