Kvantová BGG sekvence a nekomutativní geometrie plných kvantových vlajkových variet
| Thesis title in Czech: | Kvantová BGG sekvence a nekomutativní geometrie plných kvantových vlajkových variet |
|---|---|
| Thesis title in English: | The quantum BGG sequence and the noncommutative geometry of the full quantum flag manifolds |
| Key words: | kvantová grupa|nekomutativní geometrie|teorie reprezentace|sekvence BG|Lusztigův kořenový vektor |
| English key words: | quantum group|noncommutative geometry|representation theory|BGG sequence|Lusztig root vectors |
| Academic year of topic announcement: | 2024/2025 |
| Thesis type: | dissertation |
| Thesis language: | čeština |
| Department: | Mathematical Institute of Charles University (32-MUUK) |
| Supervisor: | Dr. Re O'Buachalla, Dr. |
| Author: | hidden - assigned and confirmed by the Study Dept. |
| Date of registration: | 30.09.2024 |
| Date of assignment: | 30.09.2024 |
| Confirmed by Study dept. on: | 03.10.2024 |
| Guidelines |
| The thesis will examine the noncommutative differential geometry of the full A-series quantum flag manifolds. This will be done using two complementary points of view. The first is the use of Lusztig's quantum root vectors, following the recent approach of Ó Buachalla and Somberg. The second is the dual BGG approach of Heckenberger and Kolb. Unifying these two points of view is the principal aim of the thesis, and it will look to classical parabolic geometry for inspiration on how to do this. It is expected that this will form a significant contribution towards a solution of the Baum-Connes conjecture for the A-series Drinfeld-Jimbo quantum groups.
The mathematical structures underlying the project will be the theory of Hopf-Galois extensions, noncommutative differential calculi, quantum principal bundles, bimodule connections, the representation theory of the Drinfeld-Jimbo quantum groups, Lusztig's quantum root vectors, and quantum Schubert cells. Moreover, from the analytic side it will use C*-algebras, K-theory, and Connes' theory of spectral triples. |
| References |
| E. Beggs and S. Majid, Quantum Riemannian geometry, 1 ed., Grundlehren der mathematischen Wissenschaften, vol. 355, Springer International Publishing, 2019.
A. Čap and J. Slovák, Parabolic Geometries I, Background and General Theory, Providence, RI, USA: American Mathematical Society, 628 s. Mathematical Surveys and Monographs, 154, 2009 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. Ó Buachalla, Noncommutative complex structures on quantum homogeneous spaces, J. Geom. Phys. 99 (2016), 154–173 R. Ó Buachalla, Noncommutative K ̈ahler structures on quantum homogeneous spaces, Adv. Math. 322 (2017), 892–939 |