BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Tim Van der Linden (Université catholique de Louvain) DTSTART:20210928T133000Z DTEND:20210928T143000Z DTSTAMP:20260423T021221Z UID:itaca/8 DESCRIPTION:Title: Al gebras with representable representations\nby Tim Van der Linden (Univ ersité catholique de Louvain) as part of ItaCa Fest 2021\n\n\nAbstract\n( Joint work with Xabier García-Martínez\, Matsvei Tsishyn and Corentin Vi enne)\n\nJust 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$\\to\\mathbf{Der}$(X) from B to the Lie algebra $\\mat hbf{Der}$(X) of derivations on X. The aim of this talk is to elaborate on the question\, whether the concept of a derivation can be extended to othe r types of non-associative algebras over a field $\\mathbf{K}$\, in such a way that these generalised derivations characterise the $\\mathbf{K}$-alg ebra actions. We prove that the answer is no\, as soon as the field $\\mat hbf{K}$ is infinite. In fact\, we prove a stronger result: already the rep resentability of all abelian actions – which are usually called represen tations or Beck modules – suffices for this to be true. Thus we characte rise 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 $\\math bf{gl}$(V) as a representing object for the representations on a vector sp ace V.\n LOCATION:https://researchseminars.org/talk/itaca/8/ END:VEVENT END:VCALENDAR