0

Trying to understand some basic representation theory I came across the following saying.

"Since the representation of $sl(2;C)$ has to be finite dimensional, there must exist an integer $n \in N_0$ with $(J_+)^{n+1}|u> = 0$, and $(J_+)^n|u> \ne 0$."

Key for this argument is that the representation is finite dimensional. However, it is not obvious to me what it has to be, and I would greatly appreciate any explanation.

YuiTo Cheng
  • 4,705
B. Brekke
  • 151
  • Seconding what Matthew said. Just adding that in many contexts (such as applications fo quantum mechanics) the finite dimensional representations are the most interesting. They are also a natural collection of objects in pure algebra. – Jyrki Lahtonen Mar 06 '19 at 10:13
  • This is from somewhat informal course notes in a course called Group theory in physics. Thanks for the comments, I now consider the question answered. – B. Brekke Mar 06 '19 at 10:42
  • It is just plain wrong to say that only finite dimensional representations are physically interesting. https://en.m.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group – Moishe Kohan Mar 06 '19 at 13:08

1 Answers1

1

There are well-known infinite-dimensional modules for the Lie algebra $L=\mathfrak{sl}_2(\Bbb C)$. We start with an action of $L$ on the polynomial ring $\Bbb C[X,Y]$ and pass to the subspace of homogeneous polynomials.

References:

Infinite dimensional $sl(2,\mathbb{C})$-modules

Proof check - infinite-dimensional $\mathfrak{sl}(2, \mathbb{F})$-module

Dietrich Burde
  • 130,978