Quantum twist maps and dual canonical bases (Various Issues relating to Representation Theory and Non-commutative Harmonic Analysis)

Abstract

本稿では "量子捻り写像"(quantum twist map)と呼ばれる写像と双対標準基底に関連して得られた結果を報告する. 量子捻り写像はLenagan-Yakimov [LeY]によって導入された量子包絡環の反代数自己同型である. 主結果の一つは量子捻り写像が, 量子幕零部分代数間の反代数同型と見た場合に双対標準基底を双対標準基底に移すということである. また, その系として, 双対標準基底を双対Poincaré-Birkhoff-Witt型基底で展開した際の展開係数のある対称性が得られることを述べる. 本研究は木村嘉之氏との共同研究である.

In this report, we explain our results on the compatibility between quantum twist maps and dual canonical bases. The quantum twist maps that we deal with are introduced by Lenagan-Yakimov [LeY], and they are algebra anti-automorphisms on quantized enveloping algebras. We show that they induce bijections between dual canonical bases if we regard them as anti-isomorphisms between quantum nilpotent subalgebras. As a corollary, we show certain symmetries of the entries of transition matrices between dual Poincaré-Birkhoff-Witt type bases and dual canonical bases. This report is based on joint work with Yoshiyuki Kimura.