Tensors & Differential Geometry - What are lie groups?

[QuantumTheory]

Tensors & Differential Geometry -- What are lie groups?

I've heard a lot about "lie groups" on this section of the forum, and was wondering what they are and if someone could explain it in simple terms.

Thank you.

 

[meteor]

If you know what's a group, it's easy
A set G is called a Lie group if there is given on G a structure satisfying the following three axioms:
(1) G is a group;
(2) G is a paracompact, real analytic manifold (G need not be
connected);
(3) the mapping G x G -> G
defined by (x,y) -> xy^-1 is real analytic

(Source: Encyclopedic Dictionary of Mathematics)

With respect to their physical applications, they are important in modern physics, for example in the Standard Model

 

[Hurkyl]

Hrm, that's a little overkill. According to Wikipedia, all you need is that it's a manifold and that the group operations are continuous, which is closer to the definition I recall.

 

[mathwonk]

come on guys, just tell him it generalizes the group of all invertible matrices. another example being the family of all matrices whose inverse equals their transpose, or which preserve length. or some other quadratic form.

by the way, as to the abstract definition, do you mean lie's definition or the more recent ones? As is well known, lie included differentiability hypotheses, and hilbert's 5th problem was to decide whether these were superfluous,a s i believe they were found to be by montgomery, zippin, and gleason.

there has also been work on when smoothness hypotheses imply analytic conclusions.

 

[Hurkyl]

When I first encountered these things, I found the topological aspect difficult to see when I thought of my Lie Group as a group of matrices. It sure makes the algebra easier, though.

 

[mathwonk]

well a matrix is an n^2 tuple of numbers, so if topology makes sense in R^n it should make sense for matrices.

 

[Hurkyl]

Certainly, but arranging them in a square tends to block people's intuition.

 

[mathwonk]

you are right. that's why teaching and learning are so interesting and challenging. how to evade blocks to the intuition?

Many students will be puzzled by the equivalence of a mathematical statement with its contrapositive, but will readily agree that if their dad promises them 500 dollars for getting an A, he need not pay if they do not get it.

Or a boy may say to another hesitating on the high board " I'll jump if you will", but agree he need not jump if the other does not.

I spend hours trying to think up homely illustrations of mathematical phenomena, to offer my classes. In this case I would probably talk about unwinding the matrix into a vector, or something, like unfolding a long folded up ruler.?

To illustrate slicing a sphere by spherical coordinate planes, I used to recall those chocolate oranges that fall into slices when you strike them sharply on the table. (One of my students had never seen one, but she went out and found one as a present for me!).

 

[gvk]

Yes, Lie groups, infinitesimal generators, invariants, and other 'strange' words are the language and method of many branches of modern mathematics and all emerged from considering a special type of functions (transformations) having the geometrical properties of symmetry. Sophus Lie himself started to study a 'point' transformations something around ~1860 and then moved to the more complicated 'contact' transformations.
The point transformations concerns with a transformation of the point, let say (x,y,z)
in the 3D space by a continuous parameter a, producing the transformation
x'=X(x,y,z,a), y'=Y(x,y,z,a), z'=Z(x,y,z,a). There are many such transformations in physics(!), for example translations ( x'=x+a, y'=y, z'=z ), rotations (x'=x*cos(a)+y*sin(a), y'=-x*sin(a)+y*cos(a), z'=z), scaling ( x'=x*a, y'=y, z'=z ),
etc.,etc.,etc.
All of them have some special properties which was discovered 40 yr. before Lie by Abel and Galous. They are called group properties. The difference between the groups discovered by Abel and Galous and the groups of Lie is that the last groups are continious. But the group's properties are almost the same.

Lie found that many functions x'=X(x,a) (here x means a point in n-D space) are differentiable and have the following group properties:
(1) Two sequential transformations, x'=X(x,a) and x' '=X(x',b), can be always substituted by single transformation x' '=X(x,c), where c is the function of the a's and b's alone, i.e. c = f(a,b). It does not matter, which transformation from sequence was done first, f(a,b) = f(b,a). The function f(a,b) is called the law of composition for parameters a and b.
(2) There is a special parameter a=e, which does nothing, x=X(x,e), and a = f(a, e).
This parameter is called a 'unity' or 'identity' element.
(3) For each parameter a there is a special parameter a^(-1) (it is not the power!), which returns a point to the previous state, x=X(x', a^(-1)), and e =f (a,a^(-1)).
This parameter is called 'inverse' element.
There are many books on Lie groups and transformations. It is hard to recommend you something specific, it's depend on your level of knowledge, but based on you name (QuantumTheory) I would recommend the book by Hamermesh, Group Theory and Its Application to Physical Problems, Dover.
Good luck.

 

[mathwonk]

the most elementary book touching lie groups may be the few chapters on that topic in the book Algebra, by Michael Artin. He includes the topic in an honors sophomore abstract algebra book.

Being an expert, he is able to include brief, elementary, and useful treatments of several topics in algebra not found in most undergraduate books. some other non standard topics in his book include: number fields, finite extensions of C[X] and the Hilbert nullstellensatz.

several physicists I knew wanted to elarn about group representations, and Artin treats one of the first interesting ones, the double cover of SO(3) by SU(2).