Representation theoretic research of Painleve systems

2015 Fiscal Year Final Research Report

• Project/Area Number: 23540003
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tohoku University
• Principal Investigator: Kuroki Gen 東北大学, 理学(系)研究科(研究院), 助教 (10234593)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Keywords: 量子パンルヴェ系 / 量子群 / カッツ・ムーディ代数 / ワイル群双有理作用 / ベックルント変換 / τ函数 / 共形場理論

Outline of Final Research Achievements

We quantize the tau-functions generated by the birational action of the Weyl group associated to any symmetrizable generalized Cartan matix (GCM). For example, if the GCM is of the affine A_2 type, then the quantized tau-functions are identified with the ones for the quantum Painleve IV equation. The classical tau-functions are polynomials in independent variables. We establish the quantized version that the quantized tau-functions are non-commutative polynomials in the quantized independent variables. The proof is derived from the certain formulae of the translation functors in the BGG category for the Kac-Moody algebra. Using the theory of quantum groups, we can generalize these results to the cases of q-difference analogues.

Free Research Field

表現論と量子可積分系