Algebr Represent Theor (2018) 21:589–604
Quantum Twist Maps and Dual Canonical Bases
· Hironori Oya
Received: 29 November 2016 / Accepted: 17 August 2017 / Published online: 26 August 2017
© Springer Science+Business Media B.V. 2017
Abstract In this paper, we show that quantum twist maps, introduced by Lenagan-
Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras.
As a corollary, we show the unitriangular property between dual canonical bases and
e-Birkhoff-Witt type bases under the “reverse” lexicographic order. We also show
that quantum twist maps induce bijections between certain unipotent quantum minors.
Keywords Quantized universal enveloping algebras · Quantum twist maps · Dual
canonical bases · Poincar
e-Birkhoff-Witt type bases · Unipotent quantum minors ·
Quantum T -systems
Mathematics Subject Classification (2010) 17B37 · 13F60
Let g be a symmetrizable Kac-Moody Lie algebra and U
(g) the corresponding
quantized universal enveloping algebra. We mainly focus on the subalgebra U
Presented by Kenneth Goodearl.
The work of the first author was supported by JSPS Grant-in-Aid for Scientific Research (S) 24224001.
The work of the second author was supported by Grant-in-Aid for JSPS Fellows (No. 15J09231) and
the Program for Leading Graduate Schools, MEXT, Japan.
Faculty of Liberal Arts and Sciences, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku,
Sakai, Osaka 599-8531, Japan
Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan