BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Bernardo Leite da Cunha (University of Santiago de Compostela\, Sp ain) DTSTART:20260126T150000Z DTEND:20260126T160000Z DTSTAMP:20260423T021151Z UID:ENAAS/158 DESCRIPTION:Title: Representations of two-dimensional compatible Lie algebras\nby Bernard o Leite da Cunha (University of Santiago de Compostela\, Spain) as part of European Non-Associative Algebra Seminar\n\n\nAbstract\nA compatible Lie algebra is a vector space equipped with two Lie products such that any lin ear combination of them is also a Lie product. These algebras arose from t he related class of compatible Poisson algebras in the context of mathemat ical physics and Hamiltonian mechanics.\nIn this talk\, we start by statin g some basic definitions and results about compatible Lie algebras. We the n present counterexamples to analogues of some of the most important theor ems in Lie algebra theory\, namely the theorems of Weyl and Levi\, highlig hting how compatible Lie algebras can behave very differently from Lie alg ebras.\nWe then move on to studying the representation theory of a family of simple two-dimensional compatible Lie algebras. We construct a family o f irreducible representations for each algebra of this family\, and therea fter\, we focus on one specific simple two-dimensional compatible Lie alge bra in order to make the computations simpler and results easier to state and prove. In this setting\, we prove a Clebsch-Gordan formula for the irr educible representations previously described\, and we also exhibit a seco nd family of representations\, this time "breaking" Weyl's theorem (i.e.\, reducible and indecomposable representations over the field of complex nu mbers).\nTime permitting\, we finish by discussing the failure of further central results from Lie algebra theory in this broader context\, includin g the characterization of semisimple algebras and the Whitehead Lemmas.\n LOCATION:https://researchseminars.org/talk/ENAAS/158/ END:VEVENT END:VCALENDAR