iQuantum groups and iHopf algebras II: dual canonical bases

TL;DR

This paper constructs dual canonical bases for universal iquantum groups of finite type, demonstrating their invariance under braid group actions and connecting them to known bases, thereby advancing the understanding of quantum group structures.

Contribution

It introduces the dual canonical basis for arbitrary finite type iquantum groups and links it to existing bases, confirming conjectures and extending previous geometric results.

Findings

Dual canonical bases are preserved by braid group actions.

The dual canonical basis for the Drinfeld double quantum group matches Berenstein-Greenstein's double canonical basis.

Settles several conjectures regarding the structure of these bases.

Abstract

Building on the iHopf algebra realization of quasi-split universal iquantum groups developed in a prequel, we construct the dual canonical basis for a universal iquantum group of arbitrary finite type, which are further shown to be preserved by the ibraid group action; this recovers the results of Lu-Pan in ADE type obtained earlier in a geometric approach. Moreover, we identify the dual canonical basis for the Drinfeld double quantum group of arbitrary finite type, which is realized via iHopf algebra on the double Borel, with Berenstein-Greenstein's double canonical basis, settling several of their conjectures.