19165
Seminar
WiSe 13/14: Lie groups and Lie algebras: Geometry, Representations, Combinatorics
Raman Sanyal;Christian Stump
Information for students
Prerequisits:
A solid background in analysis and (linear) algebra. Knowledge in differential geometry/topology or representation theory is helpful but not required.
A solid background in analysis and (linear) algebra. Knowledge in differential geometry/topology or representation theory is helpful but not required.
Comments
Lie groups and Lie algebras are ubiquitous in mathematics. Their study
combines in an elegant way ideas and notions from algebra, geometry, and
combinatorics. Lie groups make an appearance as symmetries of
mathematical objects such as physical systems, differential equations,
geometric objects, etc. They are best understood in terms of associated
Lie algebras and their representations.
In this seminar, we will approach Lie groups and their Lie algebras through their geometry, their representations, and their combinatorics. To emphasize the hands-on character of the seminar and to keep the prerequisits to a minimum, we will follow the book by B. Hall supplemented by the more general and more abstract perspective of Bröcker-tom Dieck, Fulton-Harris, and Humphreys. The main advantage of the approach by Hall is the focus on matrix Lie groups, i.e., closed subgroups of the general linear group. This way familiarity with differentiable manifolds is reduced to a solid background in analysis and linear algebra.
In the first half, we will cover the basic theory of (matrix) Lie groups and Lie algebras including the Baker-Campbell-Hausdorff formula. Fundamental notions are supplemented by a wealth of examples. In the second half we will focus on semisimple Lie algebras, including their Weyl groups and root systems, and their representations.
The goal of the seminar is a coherent picture of Lie theory along the lines of the book by Brian Hall. To this end, participants are expected to actively participate throughout. close
In this seminar, we will approach Lie groups and their Lie algebras through their geometry, their representations, and their combinatorics. To emphasize the hands-on character of the seminar and to keep the prerequisits to a minimum, we will follow the book by B. Hall supplemented by the more general and more abstract perspective of Bröcker-tom Dieck, Fulton-Harris, and Humphreys. The main advantage of the approach by Hall is the focus on matrix Lie groups, i.e., closed subgroups of the general linear group. This way familiarity with differentiable manifolds is reduced to a solid background in analysis and linear algebra.
In the first half, we will cover the basic theory of (matrix) Lie groups and Lie algebras including the Baker-Campbell-Hausdorff formula. Fundamental notions are supplemented by a wealth of examples. In the second half we will focus on semisimple Lie algebras, including their Weyl groups and root systems, and their representations.
The goal of the seminar is a coherent picture of Lie theory along the lines of the book by Brian Hall. To this end, participants are expected to actively participate throughout. close
Suggested reading
T. Bröcker, T. tom Dieck - Representations of Compact Lie Groups
B. Hall - Lie gropus, Lie algebras, and representations: An elementary introduction
W. Fulton, J. Harris - Representation Theory: A first course
J.E. Humphreys - Introduction to Lie Algebras and Representation Theory close
B. Hall - Lie gropus, Lie algebras, and representations: An elementary introduction
W. Fulton, J. Harris - Representation Theory: A first course
J.E. Humphreys - Introduction to Lie Algebras and Representation Theory close
16 Class schedule
Regular appointments
Wed, 2013-10-16 10:00 - 12:00
Wed, 2013-10-23 10:00 - 12:00
Wed, 2013-10-30 10:00 - 12:00
Wed, 2013-11-06 10:00 - 12:00
Wed, 2013-11-13 10:00 - 12:00
Wed, 2013-11-20 10:00 - 12:00
Wed, 2013-11-27 10:00 - 12:00
Wed, 2013-12-04 10:00 - 12:00
Wed, 2013-12-11 10:00 - 12:00
Wed, 2013-12-18 10:00 - 12:00
Wed, 2014-01-08 10:00 - 12:00
Wed, 2014-01-15 10:00 - 12:00
Wed, 2014-01-22 10:00 - 12:00
Wed, 2014-01-29 10:00 - 12:00
Wed, 2014-02-05 10:00 - 12:00
Wed, 2014-02-12 10:00 - 12:00