Complex Representations: Real vs. Complex Lie Algebras

• Lapidus
In summary, the conversation discusses the concept of complex representations in the fields of physics and mathematics. It is noted that the definitions for this term may vary slightly between the two disciplines. Examples of complex representations are given, such as the spinors in SO(3) and SU(n) with complex entries taken over the real numbers. The conversation also delves into the relationship between real and complex forms of Lie algebras and how structure constants play a role in determining the nature of the representation. The issue of confusion surrounding the term "complex" is also brought up, as it can refer to various aspects such as matrix entries, vector coefficients, or informal usage. Ultimately, understanding the context and the field of the writer is important in grasping

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

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.

1. What is the difference between real and complex Lie algebras?

Real Lie algebras are composed of real numbers, while complex Lie algebras are composed of complex numbers. This means that the structure constants and generators of a complex Lie algebra can have both real and imaginary components, while those of a real Lie algebra can only have real components.

2. How are real and complex Lie algebras related?

Real Lie algebras can be thought of as a subset of complex Lie algebras, as every real number can be written as a complex number with an imaginary component of 0. This means that every real Lie algebra can also be considered a complex Lie algebra, but the reverse is not necessarily true.

3. What is the significance of complex Lie algebras in physics?

Complex Lie algebras have a wide range of applications in theoretical physics, particularly in the study of symmetries and conservation laws. They are used to describe fundamental forces, such as electromagnetism and the strong and weak nuclear forces, as well as in the study of quantum mechanics and relativity.

4. How are complex Lie algebras classified?

Complex Lie algebras are classified using the Cartan-Killing classification, which groups them into finite-dimensional, semisimple, and solvable types. Each type has its own distinct structure and properties, making it easier to study and understand the different types of complex Lie algebras.

5. Can complex Lie algebras be visualized?

While real Lie algebras can be visualized using vector spaces and matrices, complex Lie algebras cannot be easily visualized in the same way. However, they can be represented graphically using root systems, which are useful for understanding their structure and relationships with other Lie algebras.