Representations of Quantum groups and Cherednik algebras

24th of June 2016

School of Mathematics and Statistics, Newcastle University

The meeting

This is a one day conference to celebrate the new appointment of Martina Balagovic as a lecturer at the School of Mathematics and Statistics at Newcastle University. The format of the conference is the LMS "Celebrating New Appointments". We acknowledge the financial support of the LMS and the School of Mathematics and Statistics. There is a small amount of funding available to help early career researchers (PhD students and postdocs) with their travel costs.

Schedule, titles and abstracts

• 1:30-2:30, HERB.3.TR3
  • Maxim Nazarov (University of York)
  • Cherednik algebras and Zhelobenko operators
  • This is a joint work with Sergey Khoroshkin. We study canonical intertwining operators between modules of the trigonometric Cherednik algebra, induced from the standard modules of the degenerate affine Hecke algebra. We show that these operators correspond to the Zhelobenko operators for the affine Lie algebra of series A. To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain modules of the affine Lie algebra to modules of the Cherednik algebra.
  
  
• 2:40-3:40, HERB.3.TR3
  • Alexander P. Veselov (Loughborough University)
  • Representations of holonomy Lie algebras and logarithmic Frobenius structures
  • The classification of the logarithmic Frobenius structures is one of the main open problems in this area. I will explain that for given hyperplane arrangement this problem is equivalent to the description of a certain class of representations of the corresponding holonomy Lie algebra. The flat sections of the corresponding V-connection can be interpreted as vector fields, which are both logarithmic and gradient. We show that for a special class of V-systems called harmonic, which includes all Coxeter systems, the corresponding hyperplane arrangement must be free. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations.
    
    The talk is based on a joint work with M.V. Feigin.
  
  
• 3:40-4:20
  • Coffee break, HERB.3.TR4
  
  
• 4:20-5:20, HERB.3.TR3
  • Iain Grant Gordon (University of Edinburgh)
  • Robinson-Schensted algorithm and Bethe Algebras
  • I will discuss a recent conjecture of Bonnafe-Rouquier which proposes a Kazhdan-Lusztig cell theory for all finite complex reflection groups. This conjecture is a puzzle even in the case of the symmetric group. I will explain an approach to the conjecture (still to be confirmed!) by using recent work of Mukhin-Tarasov-Varchenko on the Bethe Ansatz for Gaudin Hamiltonians. This is joint work with Adrien Brochier and features thesis work of Noah White. The talk will not assume any sophisticated background in combinatorics or integrable systems.
  
  
• 5:30-6:30, HERB.4.TR3
  • Martina Balagovic (Newcastle University)
  • Universal K-matrices via quantum symmetric pairs
  • The construction of the universal R-matrix for quantum groups produces solutions of the Yang-Baxter equation on tensor products of representations of that quantum group. This gives an action of the braid group of type A, endowing the category of finite dimensional Uq(g)-representations with a structure of a braided tensor category.
    
    I will explain how the theory of quantum symmetric pairs allows an analogous construction of a universal K- matrix, which produces solutions of the reflection equation on tensor products of representations of that quantum group. This gives a representation of the braid group of type B, endowing the category of finite dimensional Uq(g)-representations with a structure of a braided tensor category with a cylinder twist.
• 7:15
  • Dinner, Sabatini restaurant on Quayside (cca 15 minute walk from the Cental Station).
  
  

Location

All talks and coffee breaks will be on the 4th floor of the Herschel building on the campus of Newcastle University. The building is in the city center, about 20 minutes walk or 2 Metro stops from the Central station. Here are some maps of the campus and the city center. Talks will be in HERB.4.TR3 and the coffee break in HERB.4.TR4.