A Dolbeault complex for the contact quantised flag manifolds of type C2 and D4
Abstract
Differential calculi on non-irreducible quantised flag manifolds of the A-series have recently been investigated using Lusztig positive root vectors, revealing that several features of classical differential geometry persist in the noncommutative setting. We extend this investigation to quantised flag manifolds associated with the lowest-rank quantum groups of types C and D whose classical spaces are holomorphic contact manifolds. Building on the Heckenberger--Kolb framework, we construct covariant first order differential calculi. The maximal prolongations of calculi restricted to contact directions have lower dimensions than their classical counterparts. In type C2, however, adjoining the Reeb direction yields a three-dimensional calculus on the contact flag manifold whose maximal prolongation has classical dimension. Adjoining the remaining simple-root direction yields a four-dimensional calculus on the full flag manifold, again with a maximal prolongation of classical dimension. For both calculi, we explicitly present the exterior algebras and identify them with those of the corresponding framing calculi on the quantum group. In type D4, we show that adjoining the Reeb direction yields a nine-dimensional calculus on the contact flag. We conjecture that its maximal prolongation also has classical dimension.
Related
- The intersection of two real flag manifolds in a complex flag manifold, and the canonical form of a compact symmetric triad
- CR Orbits on the complex flag manifolds of G2
- Dolbeault cohomology of Endo-Pajitnov manifolds
- Laplace and Dolbeault operators on Sasakian manifolds
- Dolbeault-Hochster Theory of Polytopal LVM Manifolds
BibTeX
@misc{donno2026dolbeault,
title = {{A Dolbeault complex for the contact quantised flag manifolds of type $\mathrmC_2$ and $\mathrmD_4$}},
author = {Antonio Del Donno},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40076},
eprint = {2610.40076},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40076},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Dolbeault complex for the contact quantised flag manifolds of type C_2 and D_4
AU - Antonio Del Donno
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40076
UR - https://arxiv.org/abs/2610.40076
AB - Differential calculi on non-irreducible quantised flag manifolds of the A-series have recently been investigated using Lusztig positive root vectors, revealing that several features of classical differential geometry persist in the noncommutative setting. We extend this investigation to quantised flag manifolds associated with the lowest-rank quantum groups of types C and D whose classical spaces are holomorphic contact manifolds. Building on the Heckenberger--Kolb framework, we construct covariant first order differential calculi. The maximal prolongations of calculi restricted to contact directions have lower dimensions than their classical counterparts. In type C_2, however, adjoining the Reeb direction yields a three-dimensional calculus on the contact flag manifold whose maximal prolongation has classical dimension. Adjoining the remaining simple-root direction yields a four-dimensional calculus on the full flag manifold, again with a maximal prolongation of classical dimension. For both calculi, we explicitly present the exterior algebras and identify them with those of the corresponding framing calculi on the quantum group. In type D_4, we show that adjoining the Reeb direction yields a nine-dimensional calculus on the contact flag. We conjecture that its maximal prolongation also has classical dimension.
ER -