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Follow-up to the Hopf algebras course from the winter semester.
Last update: Šmíd Dalibor, Mgr., Ph.D. (09.01.2025)
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In this course we study Hopf algebras, together with their representation and corepresentation categories. As a key result, we prove the Fundamental Theorem of Hopf modules, which gives an equivalence between the categories of Hopf modules and vector spaces. This will be extended to a braided monoidal equivalence of tetra modules and Yetter-Drinfel'd modules. In continuation, we discuss quantum homogeneous spaces and give a full proof of Takeuchi equivalence of Hopf modules, generalizing the previously mentioned Fundamental Theorem of Hopf modules. The crucial examples of quantum flag manifolds are examined. Last update: Weber Thomas, Ph.D. (02.02.2025)
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The final exam consists of a seminar talk of about 45 minutes, presented by the participant. For this, a number of possible seminar topics will be given at the end of the course. The material will be based on the content of the course and is meant to extend the discussed material. For interested students there might be the possibility to continue the course project in form of a master thesis. Last update: Weber Thomas, Ph.D. (02.02.2025)
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Hopf algebras:
Chapter III, IV and XI of C. Kassel, Quantum groups. Graduate Texts in Mathematics, 155. Springer-Verlag, New York, 1995.
Fundamental Theorem of Hopf modules:
P. Schauenburg, Hopf Modules and Yetter-Drinfel’d Modules, J. Algebra, 169 (1994) 874-890.
Takeuchi equivalence:
M. Takeuchi, Relative Hopf modules-equivalences and freeness conditions, J. Algebra, 60 (1979) 452-471. Last update: Weber Thomas, Ph.D. (02.02.2025)
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Blackboard class with seminars at the end of the course.
The course will also be streamed via Teams. Recordings, as well as lecture notes, will be made available on Teams and via email. Last update: Weber Thomas, Ph.D. (02.02.2025)
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1) Hopf algebras
2) Hopf algebra modules and comodules
3) Hopf modules and the Fundamental Theorem
4) Braided monoidal equivalence and Yetter-Drinfel'd modules
5) Quantum homogeneous spaces
6) An adjunction theorem for quantum homogeneous spaces
7) Takeuchi equivalence
8) Formulation via coideal subalgebras
9) Quantum flag manifolds
This schedule is preliminary. Last update: Weber Thomas, Ph.D. (02.02.2025)
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