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154. 26/01/2026 Bernardo Leite da Cunha (University of Santiago de Compostela, Spain)
Title: Representations of two-dimensional compatible Lie algebras
Abstract: A compatible Lie algebra is a vector space equipped with two Lie products such that any linear combination of them is also a Lie product. These algebras arose from the related class of compatible Poisson algebras in the context of mathematical physics and Hamiltonian mechanics. In this talk, we start by stating some basic definitions and results about compatible Lie algebras. We then present counterexamples to analogues of some of the most important theorems in Lie algebra theory, namely the theorems of Weyl and Levi, highlighting how compatible Lie algebras can behave very differently from Lie algebras. We then move on to studying the representation theory of a family of simple two-dimensional compatible Lie algebras. We construct a family of irreducible representations for each algebra of this family, and thereafter, we focus on one specific simple two-dimensional compatible Lie algebra 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 irreducible representations previously described, and we also exhibit a second family of representations, this time "breaking" Weyl's theorem (i.e., reducible and indecomposable representations over the field of complex numbers). Time permitting, we finish by discussing the failure of further central results from Lie algebra theory in this broader context, including the characterization of semisimple algebras and the Whitehead Lemmas.
Future talks:
155. 02/02/2026 Zahra Nazemian (University of Graz, Austria)
156. 09/02/2026 Hans Cuypers (Eindhoven University of Technology, Netherlands)
157. 16/02/2026 Davide Ferri (University of Torino, Italy)
158. 23/02/2026 Saikat Goswami (Institute for Advancing Intelligence, India)
159. 02/03/2026 Ilan Levin (Bar Ilan University, Israel)
160. 09/03/2026 Lucia Bagnoli (Sapienza University of Rome, Italy)
161. 16/03/2026 Carsten Dietzel (University of Caen Normandy, France)
162. 23/03/2026 István Heckenberger (Philipps University Marburg, Germany)
163. 30/03/2026 Willian Franca (Federal University of Juiz de Fora, Brazil)
164. 06/04/2026 Rong Tang (Jilin University, China)
165. 13/04/2026 Maria Ferrara (University of Campania Luigi Vanvitelli, Italy)
166. 20/04/2026 Michael Lau (Laval University, Canada)
167. 27/04/2026 Abdenacer Makhlouf (University of Haute Alsace, France)
168. 04/05/2026 Per Bäck (Mälardalen University, Sweden)
169. 11/05/2026 Krzysztof Radziszewski (University of Białystok, Poland)
170. 18/05/2026 Ágota Figula (University of Debrecen, Hungary)
171. 25/05/2026 Andrés Pérez-Rodríguez (University of Santiago de Compostela, Spain)
172. 01/06/2026 Geoffrey Powell (University of Angers, France)
173. 08/06/2026 Ivan Dimitrov (Queen’s University, Canada)
174. 15/06/2026 Rinat Kashaev (University of Geneva, Switzerland)
175. 22/06/2026 María Cueto-Avellaneda (University of Murcia, Spain)
176. 29/06/2026 Michael Kinyon (University of Denver, USA)
177. 06/07/2026 Atabey Kaygun (Istanbul Technical University, Turkey)
178. 13/07/2026 Marco Castelli (University of Salento, Italy)
179. 20/07/2026 Yannic Vargas (CUNEF University, Spain)
180. 27/07/2026 Marijana Butorac (University of Rijeka, Croatia)
181. 03/08/2026 Elad Paran (The Open University of Israel, Israel)
182. 10/08/2026 Diogo Diniz (Federal University of Campina Grande, Brazil)
183. 17/08/2026 Lực Ta (University of Pittsburgh, USA)
184. 24/08/2026 Jethro van Ekeren (Institute of Pure and Applied Mathematics, Brazil)
185. 31/08/2026 Silvia Properzi (Vrije Universiteit Brussel, Belgium)
186. 07/09/2026 Quentin Ehret (New York University Abu Dhabi, United Arab Emirates)
Ivan Kaygorodov • Salvatore Siciliano • Mykola Khrypсhenko • Jobir Adashev