MHB Question about the "E" Lie algebras

  • Thread starter Thread starter topsquark
  • Start date Start date
  • Tags Tags
    Lie algebras
topsquark
Science Advisor
Homework Helper
Insights Author
MHB
Messages
2,020
Reaction score
843
I'm in chapter 2 of my Lie algebra text. I have an interest in E_8 but I'm only going to work my way up to E_6 as it has fewer operators. Anyway for now I only have a quick question: Is E_6 defined by its Dynkin diagram or is E_6 defined by something else and the Dynkin diagram follows from that? I haven't been able to find a definition of what E_6 is in my text yet.

-Dan
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

A What is the definition of quotient Lie algebra?
Replies
15
Views
3K
A From Lie algebras to Dynkin diagrams and back again
Replies
2
Views
2K
A Questions about solvable Lie algebras
Replies
7
Views
2K
About semidirect product of Lie algebra
Replies
19
Views
3K
A About Schur's lemma in lie algebra
Replies
2
Views
2K
MHB Deriving Structure Factors for Lie Algebras | Dan's Project
Replies
0
Views
1K
A Infinite-Dimensional Lie Algebra
Replies
4
Views
2K
Insights Lie Algebras: A Walkthrough - The Basics
Replies
8
Views
3K
A Do Dynkin Diagrams Only Describe Lie Algebras?
Replies
2
Views
2K
MHB Looking for a new Lie Algebra text
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top