Correspondence, Wells and Hochschild-Serre sequences for nonabelian extensions of multiplicative Lie algebras
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| Format: | Preprint |
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2025
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| _version_ | 1866914018836348928 |
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| author | Wires, Alexander Singh, Dev Karan Kumar, Shiv Datt |
| author_facet | Wires, Alexander Singh, Dev Karan Kumar, Shiv Datt |
| contents | For nonabelian $2^{\mathrm{nd}}$-cohomology of multiplicative Lie algebras, we properly generalize from the group case three classic results. We prove a Correspondence theorem which compares $2^{\mathrm{nd}}$-cohomology associated to a realized abstract kernel to the abelian $2^{\mathrm{nd}}$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03019 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Correspondence, Wells and Hochschild-Serre sequences for nonabelian extensions of multiplicative Lie algebras Wires, Alexander Singh, Dev Karan Kumar, Shiv Datt Rings and Algebras Group Theory 08A35, 17A30, 17B36, 17B56, 18G50, 19C09, 20D45 For nonabelian $2^{\mathrm{nd}}$-cohomology of multiplicative Lie algebras, we properly generalize from the group case three classic results. We prove a Correspondence theorem which compares $2^{\mathrm{nd}}$-cohomology associated to a realized abstract kernel to the abelian $2^{\mathrm{nd}}$-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. |
| title | Correspondence, Wells and Hochschild-Serre sequences for nonabelian extensions of multiplicative Lie algebras |
| topic | Rings and Algebras Group Theory 08A35, 17A30, 17B36, 17B56, 18G50, 19C09, 20D45 |
| url | https://arxiv.org/abs/2509.03019 |