The centralizer construction and Yangian-type algebras

Abstract

In the 1980s, Vladimir Drinfeld introduced and studied the notion of Yangian Y(g) associated with an arbitrary simple complex Lie algebra g. The Yangian Y(g) is a deformation of U(g[x]), the universal enveloping algebra for the Lie algebra of polynomial currents g[x]. The general definition of Yangian is radically simplified for the classical series A, and it is even more convenient to work with the reductive algebra g=gl(n). In the same 1980s, it was discovered that the Yangian Y(gl(n)) can be constructed in an alternative way, starting from some centralizers in the universal enveloping algebra U(gl(n+N)) and then letting N go to infinity. This "centralizer construction" was then extended to the classical series B, C, D, which lead to the so-called twisted Yangians. The theory that arose from this is presented in Alexander Molev's book "Yangians and classical Lie algebras", Amer. Math. Soc., 2007. I will report on the recent work arXiv:2208.04809, where another version of the centralizer construction is proposed. It produces a new family of algebras and reveals new effects and connections.