Quantum integrable systems in cohomology theories

The convolution algebra associated to d parts partitions of a given number N is the algebra built out of the operators in the torus equivariant cohomology of d-step varieties induced by correspondences defined by natural maps between d and d+1-step flag varieties. We prove that the convolution algebra associated with two parts partitions of a given number is a sum of its two subalgebras each of which is a Yang Baxter algebra, a quantum group type object, and 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 a 3- step flag variety via the convolution construction. 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.