Title: 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.