Andrea Appel (Università di Parma, Italy)

Martina Lanini (Università di Roma Tor Vergata, Italy)

Francesco Sala (Università di Pisa, Italy)

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The lectures will take place on Thursdays from 11 am to 1 pm, starting from March 4 until May 27. We shall use the platform Microsoft Teams. A short presentation of the course and the instructors will take place on Monday February 22 at 5 pm. If you would like to be added to the Teams Channel, please email


The course will be divided in three parts. The first part will describe the algebraic theory of Yangians from their definitions (motivated by mathematical physics), their several presentations, and their category of finite-dimensional representations. The second part will focus on Maulik-Okounkov’s approach to Yangians based on the theory of stable envelopes. Finally, in the last part we will study in detail Maulik-Okounkov’s construction in the case of the cotangent of flag varieties.


Basic notions on representation theory of Lie algebras (e.g. finite-dimensional representations of sl(N), Hopf algebras)

Basic knowledge of algebraic topology (e.g. singular cohomology) and of algebraic geometry (e.g. theory of varieties)


Part one (Andrea Appel): Drinfeld’s approach to Yangians

Motivations from Mathematical Physics: lattice models

The Yangians of gl(N) and sl(N)

The Yangian of a Kac-Moody algebra and Drinfeld’s “new presentation”

Finite-dimensional representations

The meromorphic R-matrix

An explicit example: Varagnolo’s action of Yangians on cohomology of flag varieties

Part two (Francesco Sala): Stable Envelopes and Maulik-Okounkov Yangians

Torus actions, Bialynicki-Birula decompositions

Equivariant cohomology, localization theorem

Stable envelopes in equivariant cohomology

Maulik-Okounkov R-matrix

Part three (Martina Lanini): Stable envelopes and cohomology of cotangent bundles of flag varieties

Flag varieties and their cotangent bundles

Torus actions and Weyl group combinatorics

Explicit characterisation of the stable basis in the flag variety case

Restriction formulae and wall crossing

Essential Bibliography

Carrell, Torus Actions and Cohomology

Maulik and Okounkov, Quantum groups and quantum cohomology (

Molev, Yangians and classical Lie algebras

Chari and Pressley, A guide to quantum groups

Gautam, Toledano Laredo, and Wendlandt, The meromorphic R-matrix of the Yangian (

Su, Stable Basis and Quantum Cohomology of Cotangent Bundles of Flag Varieties (