Andrea Appel (Università di Parma, Italy)

Martina Lanini (Università di Roma Tor Vergata, Italy)

Francesco Sala (Università di Pisa, Italy)


The course will be divided in three parts. The first part will describe Yangians and Quantum Loop Algebras as algebraic objects, focusing on their definitions (motivated by mathematical physics), their several presentations, and their category of finite-dimensional representations. The second part will focus on developing the necessary tools to study the geometry of Nakajima quiver varieties and the structure of their equivariant cohomology. Finally, in the last part, these two aspects will come together, showing the natural appearance of Yangians in the context of algebraic geometry, following the approach of Maulik-Okounkov which led to discovery of new kind of Yangians.


Part one: Drinfeld’s approach to Yangians and quantum loop algebras

RTT, Kac-Moody, and loop presentations

finite-dimensional representations

meromorphic structures and R-matrices

geometric actions on the K-theory and cohomology of flag varieties

Part two: Nakajima quiver varieties

brief introduction to GIT

King’s moduli spaces of semistable representations

Nakajima quiver varieties

Examples: cotangent of the Grassmannian and Hilbert schemes of points

Part three: Maulik-Okounkov approach to geometric actions

Cohomology and K-theory of Nakajima quiver varieties

Stable envelopes in cohomology

Definition of Maulik-Okounkov Yangian

Additional topics

Stable envelopes in K-theory

Definition of Maulik-Okounkov quantum loop algebra

Brief introduction to Casimir and KZ connections

Maulik-Okounkov’s approach to Casimir and KZ connections

Essential Bibliography

Ginzburg, Lectures on Nakajima’s quiver varieties (

Maulik and Okounkov, Quantum groups and quantum cohomology (