MMOSS Towards the induced modules in supersymmetric setup

Faculty of Science
Spring 2025
Extent and Intensity
2/0/2. 6 credit(s). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
doc. Dr. Ioannis Chrysikos (lecturer)
Andrey Krutov, Ph.D. (lecturer)
prof. RNDr. Jan Slovák, DrSc. (lecturer)
Guaranteed by
prof. RNDr. Jan Slovák, DrSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 17. 2. to Sat 24. 5. Wed 10:00–12:50 MS1,01016
Prerequisites
Some background in differential geometry and classical Lie theory is expected.
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The aim of this interactive course is to prepare the participants to individual study and research in the area of super-manifolds, super-algebras and their applications in diverse areas of the CaLiForNIA project (https://site.unibo.it/california-msca-se/en).
Learning outcomes
The students will gain:
Basic overview of the structure of Lie (super) groups and Lie (super) algebras;
understanding of the algebraic approach to invariant differential operators via Verma module homomorphisms;
understanding of the latter concepts in the super-algebra setup.
Syllabus
  • 1. Structure theory of (super) Lie groups and (super) Lie algebras (based mainly on the SL examples);
  • 2. Homogeneous (super) spaces and (super) bundles;
  • 3. Invariant differential operators and Verma modules, both for Klein and Cartan geometries;
  • 4. Invariant linear super differential operators.
Literature
  • Carmeli, Claudio; Caston, Lauren; Fioresi, Rita; Mathematical foundations of supersymmetry; EMS Ser. Lect. Math. European Mathematical Society (EMS), Zürich, 2011, xiv+287 pp.
  • V. Serganova, Representations of Lie superalgebras, Perspectives in Lie theory, Springer INdAM Ser., vol. 19, Springer, Cham, 2017, pp. 125–177.
  • D. A. Leı̆tes, Introduction to the theory of supermanifolds, Uspekhi Mat. Nauk 35 (1980), no. 1(211), 3–57, 255. (In English)
  • P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, and E. Witten (eds.), Quantum fields and strings: a course for mathematicians, vol. 1-2, AMS, Providence, RI
  • Y. I. Manin, Introduction to the theory of schemes, Moscow Lectures, vol. 1, Springer, Cham, 2018, Translated from the Russian, edited and with a preface by Dimitry Leites.
  • F. A. Berezin, Introduction to superanalysis, Mathematical Physics and Applied Mathematics, D. Reidel Publishing Co., Dordrecht–Boston, MA, 1987.
  • I. M. Musson, Lie superalgebras and enveloping algebras, Graduate Studies in Mathematics, vol. 131, American Mathematical Society, Providence, RI, 2012.
  • Y. I. Manin, Gauge field theory and complex geometry, second ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 289, Springer-Verlag, Berlin, 1997.
  • D. Leites (ed.), Seminar on supersymmetries (v. 1: Algebra and calculus on supermanifolds), MCCME, 2011, In Russian, a version in English is in preparation but available for perusal.
Teaching methods
We plan a hybrid form of lectures provided synchronously either from Brno or from Prague (in collaboration with Andrey Krutov of the Charles University), further students might be enrolled from abroad. We expect active approach of the students, including individual reading of further materials.
Assessment methods
The final colloquium in the very end of semester will consist of a common dispute, summarizing the topics and discussing further links to actual or possible applications.
Náhradní absolvování
not applicable
Language of instruction
English
Further comments (probably available only in Czech)
Study Materials
The course is taught only once.
General note: předmět je realizován ve spolupráci s A. Krutovem z Univerzity Karlovy a je nabízen v rámci doctoral network CaLiForNIA.
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