Ne-komutativni geometrie kvantovych grup a jeji aplikacemi

• Thesis title in Czech: Ne-komutativni geometrie kvantovych grup a jeji aplikacemi
• Thesis title in English: Non-commutative geometry of quantum groups and applications
• Key words: Nekomutativni geometrie|Kvantove grupy|C^*-algebry
• English key words: Non-commutative geometry|Quantum groups|C^*-algebras
• Academic year of topic announcement: 2021/2022
• Thesis type: dissertation
• Department: Mathematical Institute of Charles University (32-MUUK)
• Supervisor: doc. RNDr. Petr Somberg, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 28.07.2021
• Date of assignment: 28.07.2021
• Confirmed by Study dept. on: 21.01.2022
• Date and time of defence: 30.09.2024 00:00
• Advisors: Dr. Re O'Buachalla, Dr.

Guidelines

The student become acquainted and familiar in the field
of non-commutative geometry, and starts his scientific work
on specific problems mentioned and described in the annotation.

References

Quantum Riemannian Geometry by Beggs, Edwin, Majid, Shahn;
Quantum Groups and Their Representations by Klimyk, Anatoli, Schmüdgen, Konrad;
C*-algebras and Operator Theory by Gerard J Murphy.

Preliminary scope of work

Jiz od pocatku 80 let, kdy doslo k objeveni kvantovych grup Drinfeldem
a Jimbem, se nekomutativni geometrie stala subjektem intenzivniho studia
souvislosti mezi funkcionalne-analytickym programem A. Connese a subjektem
kvantovych grup. I pres pomerne velke mnozstvi dulezitych prispevku v
poslednich 30 letech je tento subjekt stale vzdaleny od vhodne formulovane
teorie. Jednou z tridou geometrickych objektu, ktere nabizi soucasny
stav poznani, jsou kvantove vlajkove variety, viz series soucasnych clanku
Ó Buachally, Somberga, and Strung.

Predlozeny PhD program navrhuje a smeruje ke studiu a budovani detailnejsich
aspektu kvantove verze homogennich geometrii, konkretne parabolickych geometrii.
Priklady specifickych problemu zahrnuji klasifikaci kovariatnich differencialnich
kalkulu pro neireducibilni kvantove vlajkove variety, verzi Kostantova theoremu
cohomologii Lieovych algeber a zobecneny Borel-Weil teorem.

Dalsi zpusob popisu nekomutativnich zobecneni klasickych geometrickych konstruktu
jsou C^*-algebry vektorovych svazku nad nekomutativnimi varietami. Zde lze uvazovat
konstrukce C^*-korespondenci, Cuntz-Pimsner C^*-algebry odpovidajici svazkum vyssich
ranku, ekvivariatni K-teorie kategorie modulu, atp.

Preliminary scope of work in English

Since the emergence of Drinfeld-Jimbo quantum groups in the 1980's, the
study of their noncommutative differential geometry has been the subject
of intense investigation, offering as it does the possibility to construct
a bridge between quantum group theory and Alain Connes' celebrated body of
work. Despite a large number of very important contributions over the last
thirty years, we are still very far from a well-formed theory. However, in
recent years it has become increasingly clear that the quantum flag manifolds
have a central role to play in this story, as evidenced, for example, by a
number of recent papers of Ó Buachalla, Somberg, and Strung.

The PhD project proposes to build on this work, aiming towards a final theory
of quantum parabolic geometry. Examples of possible concrete projects include
the classification of covariant differential calculi for non-irreducible quantum
flag manifolds, and the formulation of a quantum generalization of Kostant's work
on Lie Algebra cohomology and the generalized Borel-Weil theorem.

Another way how to view non-commutative generalizations of classical geometrical
constructs are C*-algebras of vector bundles over noncommutative spaces. For a
C*-algebra A and a something called a C*-correspondence over A, one can construct
a so-called Cuntz—Pimsner algebra. A C*-correspondence over A is basically an
A-A-bimodule with a right A-valued inner product. If A = C(X) is commutative, then
a finitely generated projective module is an example of a C*-correspondence. Of
course, by Serre—Swan, such a module necessarily comes from a vector bundle over
the space. If one does the Cuntz-Pimsner construction corresponding to a line bundle
over X, one gets a commutative C*-algebra whose spectrum is the associated circle
bundle. For higher rank bundles, one gets a locally trivial continuous field of Cuntz
algebras over X. These have been studied by Dadarlat, for example, where he has
done things like looked at the K-theory class of the vector bundle to determined
whether the field of Cuntz algebras is trivial or not. Since we have a good notion
of what a higher rank vector bundle is over something like a quantum homogeneous
space, one could study these Cuntz-Pimsner algebras using thee commutative case
as a guide.