Can Different Bases Change the Classification of Lie Algebras?

[topsquark]

This is only a minor question.

I watched an on-line video recently on su2 and how it applies to Physics. Now, one of the first things the instructor did was to change the base to sl2. Fine and all, but she called sl2 "su2" for the whole video. Since the two Lie algebras have different structure factors how can she do this? Or was she just being a Physicist and sloppy about it?

Another example. My text is talking about the Cartan-Weyl basis of sl3, then changes the basis to a Chevalley basis. The problem here is not merely that the structure factors are different but now the simply laced root system for the Cartan basis is no longer simply laced in the Chevalley basis. And yet the text still refers to it as sl3.

What gives?

-Dan

 

[jvicens]

Hello Dan,

Thank you for bringing up this question. It is important to note that while the terms "su2" and "sl2" may be used interchangeably in some contexts, they do refer to different Lie algebras with distinct structure factors. it is crucial to be precise and accurate in our use of terminology, especially when dealing with complex mathematical concepts such as Lie algebras.

In the case of your example, it seems that the instructor may have been using a more colloquial or simplified term for the sake of brevity or ease of understanding. However, this can lead to confusion and potential misunderstandings, as you have pointed out. It is always best to use the correct terminology when discussing scientific concepts, to avoid any potential errors or misconceptions.

Regarding the change in basis from Cartan-Weyl to Chevalley in your text, it is important to note that while the structure factors and root systems may differ, the underlying algebraic structure of sl3 remains the same. This may be why the text continues to refer to it as sl3. However, it is important to fully understand and acknowledge the differences between the two bases in order to fully grasp the concepts being presented.

In conclusion, it is important for scientists to be precise and accurate in their use of terminology, especially when dealing with complex mathematical concepts. While certain terms may be used interchangeably in some contexts, it is important to fully understand and acknowledge any differences in structure or notation. I hope this helps clarify the issue for you.