Quantum integrable systems in cohomology theories I

In these lectures we describe a geometric construction of two Yang Baxter algebras, quantum group type objects, each is a degeneration of the so-called six vertex model from statistical physics. These algebras are defined by the Chern classes of the natural vector bundles over partial flag varieties via the convolution construction common in the geometric representation theory. The major ingredient of the construction of a Yang Baxter algebra is the R matrix, which is a part of a solution of the Yang Baxter equation. It turns out that in our case the R matrix encodes the natural relations between the above Chern classes. As an output we obtain a new description of the Schubert calculus on the cohomology of Grassmannians.