Complex Representations: Real vs. Complex Lie Algebras

[Lapidus]

When do we call a representation complex? What are examples of complex representations?

Also, when we say real and complex forms of Lie algebras, is that related to real and complex representation classification?

I read that spinors are complex representations of SO(3), because their components are complex and the matrices that act on them have complex entries. But then I read elswhere SU(n) with comlex entries, but taken over the real numbers is a real form.

Or does it come down to structure constants? If they are real or complex, so is the representation?

THANKS

 

[Simon Bridge]

The physics and math definitions for "complex representation" are slightly different.
http://en.wikipedia.org/wiki/Complex_representation

 

[Lapidus]

Does it just mean in the physics literature that the matrices have complex entries?

I know that su(2) is the real form of sl(2,C). That means the su(2) matrices have complex entries but are defined over the real numbers. So you can have complex matrices but still a real Lie algebra.

But then in the QFT book that I'm currently reading is written that the (1/2,0) and (0,1/2) Weyl spinor reps are comlex reps. Unfortunately, the author fails to mention why and how. Can someone explain?

That's why it is never clear to me what people mean when they say complex Lie algebras or complex representations. What is complex? Entries of matrices, of vectors, the coefficients? Highly confusing!

 

[Simon Bridge]

Check the context: i.e. is the writer a mathematician of a physicist? Does the context make sense in terms of lie algebras or is it more informal?

- you will have noticed that people do not always use the exact definitions of words, and that different fields have different definitions anyway.
It is a problem - but you get used to it. Treat as an English comprehension exercise where metaphor and implied meanings are allowed.
Usually only one meaning will make sense.

Bear in mind the content of the wikipedia entry posted above.