Instructors
Andrea Appel (Università di Parma, Italy)
Martina Lanini (Università di Roma Tor Vergata, Italy)
Francesco Sala (Università di Pisa, Italy)
Timetable
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 francesco.sala@unipi.it.
Description
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.
Prerequisites
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)
Syllabus
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
Part two (Francesco Sala): Stable Envelopes and Maulik-Okounkov Yangians
Borel-Moore homology and correspondences
Equivariant cohomology, localization theorem, Bialynicki-Birula decompositions
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 (https://arxiv.org/pdf/1211.1287.pdf)
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 (https://arxiv.org/abs/1907.03525)
Su, Stable Basis and Quantum Cohomology of Cotangent Bundles of Flag Varieties (https://www.math.toronto.edu/csu/thesis.pdf)