Defining SO(n) and SU(n): A Search for the Right Definition

[meteor]

Ok, is my blame for not wanting to buy a Group theory book, but anyway:
I'm searching the correct definition for the Lie groups SO(n) and SU(n), and these are the definitions that I've found until now:

SO(n): -Is the Lie group of n*n real orthogonal matrices with
determinant equal to one
-Is the Lie group of n*n special real orthogonal matrices
-Is the Lie group of n*n orthogonal matrices with
determinant equal to one

SU(n): -Is the Lie group of special complex unitary n*n matrices
-Is the Lie group of n*n unitary matrices with determinant
equal to one

Well, what's the correct definition?? Or, if they are wrong, what's the exact definition of SO(n) and SU(n)?

Thanks

 

[mathman]

I'm a little rusty on the definitions, but here goes. Special means determinant = 1. Orthogonal matrices are usually discussed as real matrices, while unitary matrices are always complex. In that case, all the definitions for SO are the same, as is true for SU.

 

[M98Ranger]

for your question about defining SO(n) and SU(n). It can be confusing to understand the exact definitions of these Lie groups, especially since they are closely related and have similar properties. Let's take a closer look at each of these definitions to better understand their differences.

First, let's start with SO(n). This stands for "special orthogonal group" and is the Lie group of n*n real orthogonal matrices with determinant equal to one. This means that the matrices in this group are square matrices with real entries that are also orthogonal, meaning their transpose is equal to their inverse, and have a determinant of one. Additionally, the term "special" means that the matrices have a further restriction, such as being symmetric or having certain diagonal elements equal to one. This is why you may see different variations of the definition, but they all refer to the same group.

On the other hand, SU(n) stands for "special unitary group" and is the Lie group of n*n complex unitary matrices with determinant equal to one. This means that the matrices in this group are square matrices with complex entries that are also unitary, meaning their conjugate transpose is equal to their inverse, and have a determinant of one. Similar to SO(n), the term "special" indicates a further restriction, such as being Hermitian or having certain diagonal elements equal to one.

So, to summarize, the main difference between SO(n) and SU(n) is the type of matrices they consist of - real orthogonal for SO(n) and complex unitary for SU(n). Additionally, the term "special" indicates a further restriction on the matrices in each group.

In conclusion, there is not one "correct" definition for SO(n) and SU(n) as they can be defined in different ways depending on the context. However, the main idea to remember is that SO(n) is the Lie group of real orthogonal matrices and SU(n) is the Lie group of complex unitary matrices, both with a determinant of one and a further restriction denoted by the term "special". I hope this clarifies the definitions for you.