Marius Sophus Lie 1842 - 1899

Teoria reprezentacji grup i algebr Liego
(Representation theory for Lie groups and algebras)

Page of the subject in USOSweb

Problems from the June exam.

Zadania do rozwiazania pisemnego będą bliźniaczo podobne do:
Preparatory problems for the exam
Część ustna oparta na
Some problems
Notes:
Lecture 1: Quaternions. Basic examples of Lie groups.
Lecture 2: Exp, Lie algebra
Lecture 3: Lie algebras II, reductive algebras, polar decomposition
Lecture 4 - Dictionary: Lie groups - Lie algebras, Killing form, semisimple algebras
Lecture 5 - Linear algebra theorems as special cases of theorems in Lie theory
Lecture 6 - Basics of representation theory
Lecture 7 - Representations of tori, SU(2) and SL_2(C)
Lecture 8 - Representations of SL_3(C) and SL_n(C)
Lecture 9 - Construction of representations of SL_n(C), highest weights, Verma modules
Lecture 10 - Young diagrams, characters, Schur functions, Weyl character formula
Lecture 11 - Pieri formula, generalities about roots
Lecture 12 - Dynkin diagrams. Representations of Sp(n)
Lecture 13 - Clifford algebras, spinors
Lecture 14 - Representations of Spin(n) and S0(n), triality
Lecture 15 - G_2
Problems
Stare zadania o grupach Lie (in Polish)
Notatki Adama Bzowskiego ABC grup Lie (in Polish)

Bibliography: (* denotes that there exists an electonic copy)

Basic sources:
• *!!! Fulton, William; Harris, Joe - Representation theory. A first course.

• *Adams, J.F. - Lectures on Lie groups.

• Br?cker, Theodor; tom Dieck, Tammo - Representations of compact Lie groups. Graduate Texts in Mathematics, 98.

• Carter, Roger; Segal, Graeme; Macdonald, Ian - Lectures on Lie groups and Lie algebras. London Mathematical Society Student Texts, 32.

• Knapp, Anthony W. - Representation theory of semisimple groups. An overview based on examples.

• Wojty?ski, Wojciech - Grupy i algebry Liego. Biblioteka Matematyczna 60. PWN, Warsaw, 1986

* Other sources to be found in Internet
• J. F. Adams - Lectures on exceptional Lie groups

• Baez, John C. The octonions. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 145-205, http://xxx.lanl.gov/ps/math/0105155

• A. Baker, Matrix groups, an introduction to Lie groups

• R. L. Bryant, Symplectic geometry and Lie groups

• Humphreys, James E. - Linear algebraic groups.

• W.Fulton, Young tableau, representation theory and geometry

• J. Gallier, Concrete introduction to classical Lie groups via the exponential map

• Hall B.C. Lie groups, Lie algebras, and representations

• Humphreys J. Introduction to Lie algebras and representation theory (GTM 9, Springer, 1972)

• Mimura, Mamoru; Toda, Hirosi - Topology of Lie groups I, II. Translated from the 1978

• N.-P. Skoruppa, A Crash Course in Lie algebras

• A. Vistoli, Notes on Clifford algebra, Spin Groups and Triality,
http://homepage.sns.it/vistoli/clifford.pdf