Nekomutativní Geometrie Kvantových Vlajkových Variet

• Thesis title in Czech: Nekomutativní Geometrie Kvantových Vlajkových Variet
• Thesis title in English: Noncommutative Geometry of Quantum Flag Manifolds
• Key words: kvantová grupa|nekomutativní geometrie|teorie reprezentace|BGG sekvence
• English key words: quantum groups|non commutative geometry|representation theory|BGG sequence
• Academic year of topic announcement: 2023/2024
• Thesis type: dissertation
• Department: Mathematical Institute of Charles University (32-MUUK)
• Supervisor: Dr. Re O'Buachalla, Dr.

Guidelines

The thesis aims to produce a geometric framework on quantum differential calculus extended to the higher orders, where quantum Dolbeault–Dirac operator for the full quantum flag (A type) is replaced by an operator constructed from the quantum BGG sequence of Uq(sln). This will form a significant contribution towards a solution of the Baum-Connes conjecture for the A-series Drinfeld-Jimbo quantum groups.

References

E. Beggs and S. Majid, Quantum Riemannian geometry, 1 ed., Grundlehren der mathematischen Wissenschaften, vol. 355, Springer International Publishing, 2019.

D. Huybrechts, Complex geometry: an introduction, 1 ed., Universitext, Springer– Verlag Berlin Heidelberg, 2005.

I. Heckenberger and S. Kolb, The locally finite part of the dual coalgebra of quantized irreducible flag manifolds, Proc. London Math. Soc. (3) 89 (2004), no. 2, 457–484

I. Heckenberger and S. Kolb, De Rham complex for quantized irreducible flag mani- folds, J. Algebra 305 (2006), no. 2, 704–741

I. Heckenberger and S. Kolb, Differential forms via the Bernstein-Gelfand-Gelfand resolution for quantized irreducible flag manifolds, J. Geom. Phys. 57 (2007), no. 11, 2316–2344.

R. O ́ Buachalla, Noncommutative complex structures on quantum homogeneous spaces, J. Geom. Phys. 99 (2016), 154–173

R. O ́ Buachalla, Noncommutative K ̈ahler structures on quantum homogeneous spaces, Adv. Math. 322 (2017), 892–939