What are/how to use Lie groups ?

[maddog]

I found this on Wolfram's website.

A Lie group is a differentiable manifold obeying the group properties and that satisfies the
additional condition that the group operations are differentiable.

I am familiar and have taken Differentiable Geometry in college. Anybody know some
either lower graduate level or upper undergraduate level textbooks on the subject that
don't need beyond say Diff Geom and Linear Algebra ?

Maddog

[Bo]

books on group theory are available in 2 catagories; 1 is purely mathematical, and the other is focussed on applications in physics (group representation of quantum mechanics, standard model of elementary particles).

For the first class you would probably need some knowledge on manifolds, fibres, and stuff like that.
For the second class you should need some knowledge on quantum theory
(i learned group theory with only a professor and a syllabus, no book...)

there are also quite some lecture notes on the web:
http://www.jmilne.org/math/CourseNotes/math594g.html (pure mathematics, intended for 1st year students)
http://www.maths.tcd.ie/~dwilkins/Courses/311/ (pure math, group theory and number theory)

http://www.physicsforfree.com/intro.html (group theory, focussed on elementary particles)
http://www.nbi.dk/GroupTheory/ ( it's called group theory for physics....)

Bo

[maddog]

Bo,

Thanks for the links, I will look them up. I took a course on Abstract Algebra (undergrad)
where we cover Group Theory, Rings, Fields and Vector Spaces. We didn't really touch
on Lie Groups...

What are Fibres and/or Fibre Bundles ??? I have found they are related to Manifords and
Differential Geometry. Would like to know more.

I have had an undergrad course on QM and one on QCD. I am currently studying out of two
books, on String Theory and one on QFT, both are good books.

Maddog

[Bo]

fiber bundles are basicly a map difened on a manifold; but i must admit i don't know the details (i'm following a course on this right now )

the courses on groups you take seem to be a bit mathematical, and lie groups are of no particular interst for mathematicians, they are however for physicists.

my roommate here at the university has a book 'group theory in physics', by Wu-Ki Tung, which he says is quite good...

Bo

[maddog]

Originally Posted by Bo

fiber bundles are basicly a map difened on a manifold; but i must admit i don't know the details (i'm following a course on this right now )

the courses on groups you take seem to be a bit mathematical, and lie groups are of no particular interst for mathematicians, they are however for physicists.

my roommate here at the university has a book 'group theory in physics', by Wu-Ki Tung, which he says is quite good...

Bo

Well are Lie groups, Clifford Algebras and the whole class or Normed Algebras related ?

Maddog

[sanctus]

I'm having an exam which is as well about Clifford algebras in a month. I'm not yet sure yet, but how I see it you use in defining the clifford algebra a lie group (where the Lie algebra can be seen as the generators of the group, see one of my earlier threads titled "Lie algebra"if I rember well), eg you use GL (group of linear matrix) which is a lie group.
But I'm sure BO is able to tell you more....

[maddog]

Originally Posted by sanctus

I'm not yet sure yet, but how I see it you use in defining the clifford algebra a lie group (where the Lie algebra can be seen as the generators of the group, see one of my earlier threads titled "Lie algebra"if I rember well), eg you use GL (group of linear matrix) which is a lie group.

I didn't to imply that these items were similar. I am saying I have seen both Lie Groups and
Algebras while discussing Clifford Algebras. I am attemping to clarify by contrasting the two
types as both Algebras (in this case).

Maddog

[Bo]

i must admit i don't know the mathematics behind clifford algebras... ( http://mathworld.wolfram.com/CliffordAlgebra.html gives a definition, but that doesn't help me...)
I do know an example from physics:
The dirac matrices (g_m) satisfy an anti-commuting algebra:
g_m*g_n + g_n*g_m = 2 h_mn,
(where h_mn is the metric). And my Quantum field theory book tells me that this also is a representation of the four dimensional clifford algebra C4 (that is: in 4 space time dimensions, and for a spin 1/2 particle).

Now the dirac matrices also give a 4 dimensional representation of the lorentz group; bu this is given by their commutator:

g_m*g_n - g_n*g_m = -4i S_mn (where S is are the generators of the lorentz algebra.)

Since The lorentz group is a Lie group, we have our connection.
I unfrotunatly don't know if this is a general result for Clifford and Lie algebra's.

Bo

[maddog]

Thanks a lot Bo. I will look this up. I am studying QFT out of a book by A. Zee, "Quantum
Field Theory in a Nutshell". Luv that title! I will see if some of this is in there. If not I will
look C4 up in the Atlas of Groups. Thanks a lot!

Maddog

[Bo]

if there are no dirac matrices in the book you can throw it away
(at least: if you want to have some mathematics)

Bo