Advances in R-matrices and their applications (after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,...)

Abstract

R-matrices are the solutions of the Yang-Baxter equation. At the origin of the quantum group theory, they may be interpreted as intertwining operators. Recent advances have been made independently in different directions. Maulik-Okounkov have given a geometric approach to R-matrices with new tools in symplectic geometry, the stable envelopes. Kang-Kashiwara-Kim-Oh proved a conjecture on the categorification of cluster algebras by using R-matrices in a crucial way. Eventually, a better understanding of the action of transfer-matrices obtained from R-matrices led to the proof of several conjectures about the corresponding quantum integrable systems.

This is an English translation of the Bourbaki seminar 1129 (March 2017). The French version will appear in Astérisque

References in corpus (3)

• 2-Kac-Moody algebras
• Baxter Q-operator from quantum K-theory
• Monoidal categorification of cluster algebras II

Cited by in corpus (4)

• Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
• A functor for constructing $R$-matrices in the category $\mathcal{O}$ of Borel quantum loop algebras
• Convolution Products and R-Matrices for KLR Algebras of Type $B$
• Inflations for representations of shifted quantum affine algebras