Category Archive

You are currently browsing the category archive for the '[18++]Representation Theory' category.

Faithful representation and center of a group

July 14, 2009 in [18++]Representation Theory | Leave a comment

In this topic we’ll solve following problem:

Let \rho be an irreducible representation of G of degree n and character \chi; let C be the center of G and let c be its order.

(a)Show that \rho_s is a homothety for each s\in C. Deduce from this that |\chi(s)|=n for all s\in C.

(b)Prove the inequality n^2\leq \dfrac{g}{c}.

(c)Show that, if \rho is faithful, the group C is cyclic.

The problem is posted in the website mathscope.org by PnAT and solved by 2M and Vodka, they are members of the site too. This is it in detail:

(a)For each s\in C and h\in G we have \rho_h\rho_s=\rho_{hs}=\rho_{sh}=\rho_s\rho_h. Therefore by Schur’s lemma we have \rho_s is a homothety. Now for all s\in C put \rho_s=\lambda_sId_V, since \rho_{s^g}=Id_V we have \lambda_s is a root of unit, particularly |\lambda_s|=1. Final, |\chi(s)|=|Tr(\rho_s)|=|n\lambda_s|=n.

(b)This part is easy! In fact, we have g=\sum_{s\in G}|\chi(s)|^2\geq \sum_{s\in C}|\chi(s)|^2=\sum_{s\in C}n^2=cn^2 and we’re done.

(c)If \rho is faithful then by all what above we can see C as a subgroup of the group g-th roots of unit, and therefore C is cyclic.

Bài giảng về đại số Lie và nhóm Lie

October 28, 2008 in [18++]Representation Theory | Tags: , | 1 comment

Introduction to Lie Algebras and Lie Groups

Fall 2008

International Master Class

Institute of Mathematics

Vietnam Academy of Science and Technology

This course will cover the basic theory of Lie groups and Lie algebras. The prequisites include knowledge of linear algebra and group theory as covered by Algebra courses and basic notions of differential geometry (manifolds, vector fields,… etc).

TIME and PLACE

  • 13:30 – 16:00, Monday, Wednesday and Thursday at Lecture hall 301A, Building A5.

  • The first lecture will be held on Wednesday, October 29, 2008.

INSTRUCTOR

Professor Pierre Cartier, IHES

The best way to contact Professor P. Cartier is during the lecture or at his office (room 110, building A5)

CONTENTS

  • Introduction: Global and infinitesimal symmetris

  • Lie algebras: Basic definitions, enveloping algebra, Hopf lgebras, classical Lie algebras, Cartan subalgebras (roots and weights)

  • Lie groups: Classical Lie groups, Lie algebra of a Lie group, algebraic groups, maximal torus and Bruhat decomposition

  • Basic results about linear representations

  • A glimpse into modern developments: Quantum groups , Lie groupoids

TEXTBOOKS

  1. A. Kirillov Jr., Introduction to Lie Groups and Lie Algebras, Cambridge University Press, 2002

  2. N. Bourbaki, Lie groups and Lie algebras Chapter 1-3 ISBN 3-540-64242-0, Chapters 4-6 ISBN 3-540-42650-7, Chapters 7-9 ISBN 3-540-43405-4

  3. J. P. Serre, Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University, LNM 1500, Springer

  4. R. Carter et al., Lectures on Lie Groups and Lie Algebras, LMS Student Texts Series, 1995

  5. J. E. Humphreys, Introduction to Lie Algebras and Representaion theory, Springer 1978

Note: Almost all of these textbooks are available at the library of the Institute of Mathematics. Some of them are available electronically also.

FINAL EXAM

The final exam will be posted here.

P.S. Cái này mình copy trên trang của Viện Toán, bác nào rảnh thì đi nghe nhá!

 

December 2009
M T W T F S S
« Nov    
 123456
78910111213
14151617181920
21222324252627
28293031  

Archives