English

Dual canonical bases of quantum groups and $\imath$quantum groups II: geometry

Quantum Algebra 2026-03-03 v1 Representation Theory

Abstract

The ı\imathquantum groups admit two realizations: one via the ı\imathHall algebras and the other via the quantum Grothendieck rings of quiver varieties, as developed by the first author and Wang. Based on these two realizations, we establish the dual canonical bases for ı\imathquantum groups of type ADE in two distinct ways, using perverse sheaves and Hall algebras respectively. In this paper, we prove that these two dual canonical bases coincide, thereby proving their invariance under braid group actions, and that their structural constants are integral and positive. Furthermore, we establish the positivity of the coefficients of the transition matrix from the Hall basis (and PBW basis) to the dual canonical basis.

Keywords

Cite

@article{arxiv.2603.01350,
  title  = {Dual canonical bases of quantum groups and $\imath$quantum groups II: geometry},
  author = {Ming Lu and Xiaolong Pan},
  journal= {arXiv preprint arXiv:2603.01350},
  year   = {2026}
}

Comments

49 pages. This is separated from Section 4, 6, 7.3, 7.4, 9 of arXiv:2504.19073 v1