General Process for finding elements of a group

[Traceless]

Hi,

I'm trying to understand the process of finding the elements of a given group, such as SE(2). What I do understand is limited to finding elements of very simple symmetry groups, such as those corresponding to rotations/reflections of shapes. My overall knowledge of groups is also pretty limited. I am in a graduate Mathematical Methods of Physics course, and they didn't go into too much detail on this subject. Let me know if I'm being a little too vague.

Thanks a bunch.

 

[Stephen Tashi]

I think you are being a little too vague. You want to find the elements of a group - but given what information or objective?

 

[Traceless]

Hi Stephen, thanks for responding. As an example, there is a problem that asks us to find the elements of the group SE(2) (and another requesting the same for the group SO(2,1), followed by finding the generators and Casimir operators, with no additional information given. Given this problem as an example, I am trying to understand how I should view groups when I don't know their elements. Can I assume this is a continuous (Lie) group? Do the elements simply consist of the rotation matrices? I've also seen somewhere that translation is also a part of this group, but I don't fully understand how to represent that subgroup.

Thank you very much again.

 

[Stephen Tashi]

I'm not an authority on the notations for continuous groups. I'm sure you can look up the definitions of SE(2) and SO(2,1) as well I can!

The meaning of "finding the elements" of a group is still not clear. If I have a group, like the group of rotations about the origin in the 2-D plane, I can think of the group abstractly and denote one of it's elements with a symbol or describe it's geometric effect in words. I can also "represent" the group by writing a matrix containing trig functions of a parameter. Perhaps your question is how to find "representations" of a given group. A representation is, technically, a function that associates each individual group element with an individual linear transformation of some vector space into itself. The function must be a homomorphism. A group can have several different representations. I suppose a typical representation can be described by writing a matrix of functions containing various variables. Is that the kind of thing you want?

 

[Traceless]

Thanks again, Stephen. You may be right. It could be that the question is asking for representations and simply isn't as descriptive as it perhaps should be. That may also be why I was so confused about how to find the "elements", as I'm assuming you must have the proper representation of the operations in order to find the generators and Casimir Operators. So, in the case of SE(2), if I wanted to find the representations so that I may continue in finding the generators and so on, I would find the proper matrices that represent the rotations in the space, as well as the representation of the translations (whatever those may be; matrices?), call each of these sets as subgroups SO(2) and T(2), then proceed to finding the generators? Now, are the generators always given as (1/i)*(dE/dq)|q=0, where E is the matrix and q is the argument (or coordinate), or is this an over-simplification? I apologize if I seem to be a little scattered. If I am misunderstanding anything on a fundamental level, feel free to refer me to a decent textbook or tutorial that covers group theory from the viewpoint of a physicist, as I haven't yet been able to find one. Thanks very much.

 

[Stephen Tashi]

I don't know about applications of Lie Groups to physics. I only know the elementary definitions. The physics sections of the forums would be the better place to get help. Try quoting the question exactly as it was given to you.

Translations can be represented as matrices if we use "projective" coordinates. For example, in 2-D, the point (x,y) in cartesian coordinates is written in projective coordinates as as (x,y,1). Translation is accomplished by

\begin{pmatrix} 1 & 0 & a \\ 0 & 1 & b \\0&0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\1 \end{pmatrix} = \begin{pmatrix} x + a \\ y + b \\1 \end{pmatrix}

 

[Traceless]

Thank you very much for all of your help, Stephen. As you suggested, I'll try to continue the discussion in the physics section.