I The Lie algebra of ##\frak{so}(3)## without complexification

redtree
Messages
335
Reaction score
15
TL;DR Summary
Can the Lie algebra of ##\frak{so}(3)## (or ##\frak{su}(2)##) be formulated without complexification utilizing the Cartan subalgebra?
All of the formulations of the Lie algebra of ##\frak{so}(3)## (or ##\frak{su}(2)##) utilizing raising/lowering operators that I have seen in the literature involve complexification to ##\frak{su}(2) + i \frak{su}(2) \cong \frak{sl}(2,\mathbb{C})##. I have found explicit derivations in a particular representation, but none from the Cartan subalgebra.

Can one derive a formulation of the Lie algebra of ##\frak{so}(3)## utilizing the Cartan subalgebra and root vectors without complexification? If so, where can I find it?
 
Physics news on Phys.org
Moderator's note: Thread moved to the linear algebra math forum.
 
$$\mathfrak{so}(3)\cong \left(\mathbb{R}^3,\times\right)=\bigl\langle U,V,W\,|\,[U,V]=W,[V,W]=U,[W,U]=V \bigr\rangle $$
We need to break that symmetry in order to get the representation via the root system that specifies the generator of the Cartan subalgebra which is not symmetric. We therefore need complex numbers for the isomorphism. You can find the basis transformations at
https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/

For a description of how the root system works, see
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-structures/
and the general basis of real orthogonal Lie algebras here:
https://www.physicsforums.com/insig...hogonal-Lie-Algebra-On-Odd-Dimensional-Spaces
 
Last edited:
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...
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...
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...
Back
Top