How Is the Killing Metric Normalized for Compact Simple Groups?

center o bass
Messages
545
Reaction score
2
The killing form can be defined as the two-form ##K(X,Y) = \text{Tr} ad(X) \circ ad(Y)## and it has matrix components ##K_{ab} = c^{c}_{\ \ ad} c^{d}_{\ \ bc}##. I have often seen it stated that for a compact simple group we can normalize this metric by
##K_{ab} = k \delta_{ab}## for some proportionality constant ##k##. How is this statement proved?
 
Last edited:
Physics news on Phys.org
I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
 
center o bass said:
The killing form can be defined as the two-form ##K(X,Y) = \text{Tr} ad(X) \circ ad(Y)## and it has matrix components ##K_{ab} = c^{c}_{\ \ ad} c^{d}_{\ \ bc}##. I have often seen it stated that for a compact simple group we can normalize this metric by
##K_{ab} = k \delta_{ab}## for some proportionality constant ##k##. How is this statement proved?
Can you give a source for where you "often" see it? I'm having trouble understanding what you want. Are you asking if the Killing form is a metric?
 

Thread '##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?'

I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
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

Any biinvariant metric proportional to Killing metric
Replies
7
Views
4K
A How to investigate a transformation that might form a Lie group?
Replies
27
Views
3K
I Meaning of "passage to the quotient"?
Replies
3
Views
429
I Convergence not defined by any metric
Replies
17
Views
969
I Meaning of "identify" & variations in different math context
Replies
5
Views
490
A Structure of modulo-integer matrix group GL(r,Z(n))?
Replies
17
Views
6K
I Principal and Gaussian curvature of the FRW metric
Replies
8
Views
1K
I No structure in ##(d,x)## in ##\textbf{Met*}(X)## is admissible
Replies
0
Views
1K
A How to Express the Schwarzschild Metric in Fermi Normal Coordinates?
Replies
13
Views
2K
I Adjoint Representation Confusion
Replies
3
Views
3K

Hot Threads

Recent Insights

Back
Top