Algebra and Combinatorics Seminar

  • May 2, 2022
  • 1:45 PM - 2:45 CST
  • BVM Hall, Room 1102
  • Carmen Rovi, crovi@luc.edu
  • Not open to the public.
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  • Details

    Speaker: Alexander Stolin (Gothenburg)
    Topic: 40 years of Lie bialgebras: From definition to classification.
    Abstract: The history of Lie bialgebras began with the paper where the Lie bialgebras were defined: V. G. Drinfeld, *Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations*, Dokl. Akad. Nauk SSSR, 268:2 (1983) Presented: L.D. Faddeev. Received: 04.06.1982. The aim of my talk is to celebrate 40 years of Lie bialgebras in mathematics and to explain how these important algebraic structures can be classified. This classification goes *hand in hand* with the classification of the so-called Manin triples and Drinfeld doubles also introduced in Drinfeld's paper cited above. The ingenious idea how to classify Drinfeld doubles associated with Lie algebras possessing a root system is due to F. Montaner and E. Zelmanov. In particular, using their approach the speaker classified Lie bialgebras, Manin triples and Drinfeld doubles associated with a simple finite dimensional Lie algebra g (the paper was based on a private communication by E. Zelmanov and it was published in Comm. Alg. in 1999). Further, in 2010, F. Montaner, E. Zelmanov and the speaker published a paper in Selecta Math., where they classified Drinfeld doubles on the Lie algebra of the formal Taylor power series g[[u]] and all Lie bialgebra structures on the polynomial Lie algebra g[u]. Finally, in March 2022 S. Maximov, E. Zelmanov and the speaker published an Arxiv preprint, where they made a crucial progress towards a complete classification of Manin triples and Lie bialgebra structures on g[[u]]. Of course, it is impossible to compress a 40 years history of the subject in one talk but the speaker will try his best to do this.