Beijing Institute of Mathematical Sciences and Applications

We consider the relationship between a class of generalized Frobenius manifolds and bihamiltonian integrable hierarchies, and present, for any such generalized semisimple Frobenius manifold, an analogue of the construction of the Principal Hierarchy and its topological deformation which is known for a semisimple Frobenius manifold.

Traditionally KP hierarchy is treated as a classical integrable system. We will treat it instead as a quantum system by introducing a Planck constant into the system. By considering the dispersionless limit when this Planck constant goes to zero, some symplectic structures naturally emerge, and the KP hierarchy can be reexamined from the point of view of the first quantizations and the second quantizations.

It is well-known that differential Painlevé equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique - there are many very different Hamiltonians that result in the same differential Painlevé equation. In this paper we describe a systematic procedure of finding changes of coordinates transforming different Hamiltonian systems into some canonical form. Our approach is based on the Okamoto-Sakais geometric approach to Painlevé equations. We explain this approach mainly using the differential P-IV equation as an example, but the procedure is general and can be easily adapted to other Painlevé equations as well. This is a joint work with Galina Filipuk, Adam Ligeza and Alex Stokes.

The isomonodromy deformation equation of a linear meromorphic linear systems of ODEs with Poincaré rank 1 has appeared in many subjects. Since the isomonodromy equation is a higher rank analog of Painlevé VI, it is natural to ask if various results for Painlevé VI can be generalized. This talk generalizes several fundamental results of Painlevé VI to the higher rank case. In particular, it gives the asymptotic expansion of solutions of the isomonodromy equation, and the explicit expression of the Stokes/monodromy data, as well as solves a nonlinear connection problem. It also discusses the initial value space and the WKB approximation of the isomonodromy equation. Part of the talk is based on a joint work with Qian Tang.

The quantum integrable long-range spin chains of Haldane-Shastry and Inozemtsev types describe the pairwise interaction of spins attached to the equidistant points on the circle with the trigonometric and elliptic interactions respectfully. The integrable properties of these spin chains is not based on the standard quantum inverse scattering method with commuting transfer matrices, and a lot of questions related to related to commuting operators and their eigenvectors are still open. In the talk I will describe a method which allows to construct the integrals of motion of these long-range spin chains and their spin anisotopic generalizations based on the quantum R-matrix-valued Lax pairs of the Calogero-Moser classical integrable systems of particles. The construction is based on the quadratic identities on quantum R-matrices, known as the associative Yang-Baxter equation. I will discuss the possible generalizations of the anisotropic long-range spin chains on the other classical Lie algebras.

I will talk about how the Gauss decomposition describes the isomorphism between the RTT realization and the current Drinfeld realization of quantum affine algebras and Yangians. Using this relation one can describe Bethe vectors. It turns out that this construction is quite universal.

Solutions of the Yang-Baxter equation give rise to quantum integrable models such as Heisenberg spin chains. Trigonometric solutions (e.g. the XXZ/6-vertex R-matrix) appear in tensor products of finite-dimensional modules of quantum affine algebras (Drinfeld-Jimbo quantum groups of affine type), roughly as the action of the universal R-matrix. Further, as pointed out by Bazhanov-Lukyanov-Zamolodchikov in the 1990s, certain infinite-dimensional representations of the standard Borel subalgebra of the quantum affine algebra play an important role in the theory of Baxter's Q-operator, an additional tool in the diagonalization of the transfer matrices for these models. By specifying also a solution of the reflection equation (RE) one can define quantum integrable models with a boundary. This has been studied since the 1980s by Cherednik, Sklyanin, Kulish-Sklyanin and many others. Analogous to the above construction, A. Appel and I have shown that universal K-matrices for quantum affine algebras provide many trigonometric solutions of the RE (and hence integrable models with boundaries, such as open spin chains). The theory is very general but we will illustrate it with concrete XXZ-related examples (quantum affine $sl_2$). In this case, also the Q-operator has a boundary counterpart, due to work by Baseilhac-Tsuboi and, more recently, my joint work with A. Cooper and R. Weston in which the universal K-matrix comes to our aid.

We study the representations of the Iwahori algebras. We prove that the category of their representations has highest weight structure. We deduce from this property the analogue of Peter-Weyl theorem. We also prove the analogue of Howe duality for this algebra of type A.