Complex Representations: Real vs. Complex Lie Algebras

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Discussion Overview

The discussion revolves around the definitions and examples of complex representations in the context of Lie algebras, particularly focusing on the distinctions between real and complex forms. Participants explore the implications of these definitions in both mathematical and physical frameworks, as well as the confusion arising from varying terminologies across disciplines.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant questions the criteria for calling a representation complex and seeks examples, linking this to the classification of real and complex forms of Lie algebras.
  • Another participant notes the differing definitions of "complex representation" in physics and mathematics, suggesting that context is crucial for understanding.
  • A participant discusses the nature of matrices in representations, indicating that complex entries can exist in real forms, as seen in the case of su(2) being a real form of sl(2,C).
  • Concerns are raised about the clarity of terminology, particularly regarding Weyl spinor representations and what constitutes a complex representation, highlighting the ambiguity in definitions.
  • A later reply emphasizes the importance of context, suggesting that definitions may vary between mathematicians and physicists, and that informal usage can lead to confusion.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of complex representations, indicating that there is no consensus on the terminology and its applications across fields.

Contextual Notes

Participants note that definitions may depend on the specific context in which they are used, and that varying interpretations can lead to confusion, particularly when discussing complex versus real representations in Lie algebras.

Lapidus
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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
 
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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!
 
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.
 

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