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Applications of Hopf algebras in physics
- Oggetto:
Applications of Hopf algebras in physics
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Academic year 2014/2015
- Teacher
- Dott. Anna Pachol (Lecturer)
- Type
- A scelta dello studente
- Course disciplinary sector (SSD)
- MAT/07 - fisica matematica
- Delivery
- Tradizionale
- Language
- Inglese
- Attendance
- Facoltativa
- Type of examination
- Orale
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Sommario del corso
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Course objectives
The course is intended to give an introduction to Hopf algebras formalism in the connection with quantum (noncommutative) spacetimes. Such spacetimes contribute to one of the approaches to the effective description of Planck scale physics.
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Program
No previous knowledge of Hopf Algebras is required and we will start with recalling basic definitions (such as: group; field, ring, vector space; algebra, Lie algebra; tensor products, direct products; coalgebras, bialgebras and Hopf-algebras) which will be supported by examples appearing in physics.
The Hopf algebras formalism will be introduced in the connection to noncommutative spacetimes. In this picture Hopf algebras are thought as generalizations of symmetry groups of the underlying spacetime and the spacetime is Hopf module algebra over the Hopf algebra of differential operators.
The subsequent lectures will cover the deformation procedure leading to noncommutative spacetimes and their deformed relativistic symmetries. Both, Hopf algebra and its module algebra can undergo a deformation procedure (e.g. by relevant twisting element). The deformed algebra of functions is the so-called noncommutative spacetime (covariant quantum space) and is represented by deformed commutators of noncommutative coordinates which play the role of generators of that algebra. Twisted deformation via Drinfel'd twist (2-cocycle) will be also discussed and some examples of deformed Hopf algebras will be presented. Different constructions within Hopf-algebraic framework leading to physically interesting cases will be shown as well.
The framework of Quantum Groups and noncommutative spacetimes found many application in physics, especially as one of the approach to Quantum Gravity. The special cases, like Moyal-Weyl- (theta-) and kappa- deformations will be discussed in more detail. Applications in Planck scale physics will be also mentioned.Suggested readings and bibliography
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- S. Majid, Foundations of quantum group theory. Cambridge University Press, Cambridge, 1995.
- A. Klimyk, K. Schmudgen, Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
- C. Kassel, Quantum groups. Graduate Texts in Mathematics, 155. Springer-Verlag, New York, 1995.
- V. Chari, A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
- J. Fuchs, C. Schweigert, Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2003.
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Note
Period: April/June 2015Hours: 10 lectures + 5 exercisesFor comments and questions please contact apachol@unito.it
Lectures schedule:
1. Giovedì 9 aprile 2015 --> 13:00-15:00 (Aula 1 – Palazzo Campana)
2. Giovedì 16 aprile 2015 --> 13:00-15:00 (Aula 1 – Palazzo Campana)
3. Giovedì 23 aprile 2015 --> 10:00-13:00 (Aula C – Palazzo Campana)
4. Giovedì 30 aprile 2015 --> 10:00-13:00 (Aula Lagrange – Palazzo Campana)
5. Lunedi 4 maggio 2015 --> 11:00-14:00 (Aula 3 – Palazzo Campana)
Discussion and exercise session will take place right after each lecture in the same room.- Oggetto:
