Twist automorphisms on quantum unipotent cells and dual canonical bases
arXiv:1701.02268 · doi:10.1093/imrn/rnz040
Abstract
In this paper, we construct twist automorphisms on quantum unipotent cells, which are quantum analogues of the Berenstein-Fomin-Zelevinsky twist automorphisms on unipotent cells. We show that those quantum twist automorphisms preserve the dual canonical bases of quantum unipotent cells. Moreover, we prove that quantum twist automorphisms are described by the syzygy functors for representations of preprojective algebras in the symmetric case. This is the quantum analogue of Geiß-Leclerc-Schröer's description, and Geiß-Leclerc-Schröer's results are essential in our proof. As a consequence, we show that quantum twist automorphisms are compatible with quantum cluster monomials. The 6-periodicity of specific quantum twist automorphisms is also verified.
arXiv commentv3: 57 pages. Major revision. We have detailed our explanation of the classical counterpart of the De Concini-Procesi isomorphisms and the proof of Theorem 7.25. The organization of the paper has been changed. The main results are unchanged v4: 53 pages. Minor corrections. Final version
References in corpus (16)
- Cluster structures for 2-Calabi-Yau categories and unipotent groups
- Kac-Moody groups and cluster algebras
- Cluster structures on quantum coordinate rings
- Semicanonical bases and preprojective algebras
- Cluster algebra structures and semicanonical bases for unipotent groups
- Triangular bases in quantum cluster algebras and monoidal categorification conjectures
- Some quantum analogues of solvable Lie groups
- Quantum cluster algebras and quantum nilpotent algebras
- Quantum cluster algebra structures on quantum nilpotent algebras
- Semicanonical bases and preprojective algebras II: A multiplication formula
- Quantum Unipotent Subgroup and dual canonical basis
- Generic bases for cluster algebras and the Chamber Ansatz
- Invariant prime ideals in quantizations of nilpotent Lie algebras
- Quantum cluster characters of Hall algebras
- Quantum twist maps and dual canonical bases
- The Chamber Ansatz for quantum unipotent cells