Classifying Critical Points
1 hour ago
An occasional blog by a "working mathematician" in the UK, focussing on mathematical and science-related items. The title is in homage to the classic text "Categories for the Working Mathematician" by Saunders Mac Lane. This book was where I discovered the language of categories and a perspective that has influenced my work ever since. Non-specialists will be pleased to hear that I don't intend blogging on this subject (well, not much!).
We give several explicit examples of quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky, on quantized coordinate rings of partial flag varieties and their associated unipotent radicals. These structures are shown to be quantizations of the cluster algebra structures found on the corresponding classical objects by Geiß, Leclerc and Schröer, whose work generalizes that of several other authors. We also exhibit quantum cluster algebra structures on the quantized enveloping algebras of the Lie algebras of the unipotent radicals.