{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/algebras-with-representable-representations","title":"Algebras with representable representations","arxiv_id":"2002.05924","date":"2020-02-14","proceeding":null,"authors":["Xabier García-Martínez","Matsvei Tsishyn","Tim Van der Linden","Corentin Vienne"],"abstract":"Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\\to \\mathit{Der}(X)$ from $B$ to the Lie algebra $\\mathit{Der}(X)$ of derivations on $X$. In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field $\\mathbb{K}$, in such a way that these generalised derivations characterise the $\\mathbb{K}$-algebra actions. We prove that the answer is no, as soon as the field $\\mathbb{K}$ is infinite. In fact, we prove a stronger result: already the representability of all abelian actions -- which are usually called representations or Beck modules -- suffices for this to be true. Thus we characterise the variety of Lie algebras over an infinite field of characteristic different from $2$ as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasises the unique role played by the Lie algebra of linear endomorphisms $\\mathfrak{gl}(V)$ as a representing object for the representations on a vector space $V$.","url_abs":"https://arxiv.org/abs/2002.05924v3","url_pdf":"https://arxiv.org/pdf/2002.05924v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"algebras-with-representable-representations","repo_url":"https://github.com/MatveiT/code-for-ARR","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}