Non-abelian Extensions of Lie algebras with derivations
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913071480438784 |
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| author | Jiang, Jun Xu, Kanghe |
| author_facet | Jiang, Jun Xu, Kanghe |
| contents | In this paper, we investigate non-abelian extensions of Lie algebras with derivations using several different approaches. We show that the theory of non-abelian extensions of a Lie algebra with a derivation can be characterized by means of the second non-abelian cohomology, the Deligne groupoid, the homotopy category of strict Lie $2$-algebras with strict derivations, and the notion of a $(\g, D)$-kernel, respectively. Moreover, within this unified framework, we address the following existence problem: given a non-abelian extension of Lie algebras \[\begin{CD} 0@>>>\h@>i>>\hat{\g}@>p>>\g @>>>0, \end{CD}\] let $(K,D)\in\Der(\h)\times\Der(\g)$ be a pair of derivations of $\h$ and $\g$ respectively. When does there exist a derivation $\hat{D}$ of $\hat{\g}$ such that $\hat{D}|_\h=K$ and $D\circ p=p\circ\hat{D}.$ We provide an obstruction class for the existence of such a lift. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26276 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-abelian Extensions of Lie algebras with derivations Jiang, Jun Xu, Kanghe Rings and Algebras 17B40, 17B55, 17B56 In this paper, we investigate non-abelian extensions of Lie algebras with derivations using several different approaches. We show that the theory of non-abelian extensions of a Lie algebra with a derivation can be characterized by means of the second non-abelian cohomology, the Deligne groupoid, the homotopy category of strict Lie $2$-algebras with strict derivations, and the notion of a $(\g, D)$-kernel, respectively. Moreover, within this unified framework, we address the following existence problem: given a non-abelian extension of Lie algebras \[\begin{CD} 0@>>>\h@>i>>\hat{\g}@>p>>\g @>>>0, \end{CD}\] let $(K,D)\in\Der(\h)\times\Der(\g)$ be a pair of derivations of $\h$ and $\g$ respectively. When does there exist a derivation $\hat{D}$ of $\hat{\g}$ such that $\hat{D}|_\h=K$ and $D\circ p=p\circ\hat{D}.$ We provide an obstruction class for the existence of such a lift. |
| title | Non-abelian Extensions of Lie algebras with derivations |
| topic | Rings and Algebras 17B40, 17B55, 17B56 |
| url | https://arxiv.org/abs/2604.26276 |