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.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

( slides | video )

---

BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

---