MU:MU08110 Theory of Groups and Algebras

MU08110 Theory of Groups and Algebras

Mathematical Institute in Opava
Summer 2012

Extent and Intensity

2/2/0. 6 credit(s). Type of Completion: zk (examination).

Guaranteed by

Course Enrolment Limitations

The course is also offered to the students of the fields other than those the course is directly associated with.

fields of study / plans the course is directly associated with

Course objectives

The structure of groups and algebras is presented with special attention to representations and further generalizations.

Syllabus

Groups, G-modules, representation of groups, tensor product, reducibility, irreducibility, complete reducibility. Representations of finite groups, characters, group algebra. Representations of symmetric groups, Young tableaux, representations of alternating groups. Lie groups and Lie algebras, representations of Lie groups and Lie algebras, representations of the algebra sl(2, C). Classical Lie algebras, semisimple Lie algebras, Killing form, Weyl group, Cartan subalgebra, roots, Cartan's classification of (semi)simple Lie algebras. Representations of algebras sl(n, C), Weyl's construction. Representations of symplectic algebras sp(2n, C). Clifford algebras, representations of orthogonal algebras so(n, C), spinors. Real, Spinor and quaternionic representations. Quantum groups and their representations, Lie superalgebras and their representations.

Literature

recommended literature

K. Erdmann, M. Wildon. Introduction to Lie algebras. Springer, 2006. info
C. Isham. Modern Differential Geometry for Physicists. Singapore, 1999. info
J. C. Jantzen. Lectures on quantum groups. Amer. Math. Soc., Providence, 1997. info
J. Fuchs, C. Schweigert. Symmetries, Lie algebras and representations. Cambridge University Press, Cambridge, 1997. info
W. Fulton, J. Harris. Representation theory: a first course. Springer, Berlin, 1996. info
J. F. Cornwell. Group theory in Physics, vol I, II, III. Academic Press, London-New York, 1989. info
A. O. Barut, R. Raczka. Theory of group representations and applications. World Scientific, Singapore, 1986. info

Language of instruction

Czech

Further comments (probably available only in Czech)

The course can also be completed outside the examination period.

Teacher's information

Oral exam; further requirements to be specified in the course of the semester.