Lie group Definition and 83 Threads

Hello, I have troubles formulating this question properly. So I will explain it through one example. If we consider the Lie group R=SO(2) of rotations on the plane, we know that we can find a manifold on which the group SO(2) acts regularly: this manifold is the unit circle in ℝ2. In fact...

Hello, I want to prove that the set SO(2) of orthogonal 2x2 matrices with det=1 is a Lie group. The group operation is of course assumed to be the ordinary matrix multiplication \times:SO(2)→SO(2). I made the following attempt but then got stuck at one point. We basically have to prove that...

I want to show that if G is a smooth manifold and the multiplication map m:G×G\rightarrow G defined by m(g,h)=gh is smooth, then G is a Lie group. All there is to show is that the inverse map i(g)=g^{-1} is also a smooth map. We can consider a map F:G×G\rightarrow G×G where F(g,h)=(g,gh) and...

Homework Statement Prove that in SL(2) group the matrix ## \begin{pmatrix} -1 & \lambda \\ 0 & -1 \end{pmatrix} ## can not be presented as a single exponentail but instead as product of two exponentials of ##sl(2)## algebra. ##\lambda \in \mathbb{R} ## Homework Equations I don't understand...

Hello, it is known that "Every regular G-action is isomorphic to the action of G on G given by left multiplication". Is this true also when G is a Lie group? There is an ambiguous sentence in Wikipedia that is confusing me. It says: "The above statements about isomorphisms for regular, free...

I am currently reading a paper discussing the convexity of the image of moment maps for loop groups. In particular, if G is a compact Lie group and S^1 is the circle, the paper defines the loops group to be the set of function f: S^1 \to G of "Sobolev class H^1 ." Now in the traditional...

Homework Statement Let H be a connected Lie group with Lie algebra \mathfrak h such that [\mathfrak h, \mathfrak h] = 0. Show that: \exp: \mathfrak h \rightarrow H is the covering homomorphism. --------- I am not really sure what I have to show here, specifically I don't know...

Hi! I was just going through this script on Lie groups: http://www.mit.edu/~ssam/repthy.pdf At one point the following is said: (see attachment) I've spent multiple hours trying to figure out why this is a group homomorphism. Sure, once you know the theorem is correct, this follows. But...

Being not an expert, my question might sound naive to students of mahematics. My question is how on Earth a Lie group helps to solve an ode. Can anyone explain me in simple terms?

Hello, Let's suppose that I have a Lie group G parametrized by one real scalar t and acting on ℝ2. Is it generally correct to say that the orbits of the points of ℝ2 under the group action are one-dimensional submanifolds of ℝ2, because G is parametrized by one single scalar? If so, how can I...

Hello, if I have a set of functions of the kind \{ f_t | f_t:\mathbb{R}^2 \rightarrow \mathbb{R}^2 \; ,t\in \mathbb{R} \}, where t is a real scalar parameter. The operation I consider is the composition of functions. What should I do in order to show that it forms a Lie Group?

Hi everyone, I was just wondering if anyone had any suggestions of more-mathematically-rigorous textbooks on Lie groups and Lie algebras for (high-energy) physicists than, say, Howard Georgi's book. I have been eying books such as "Symmetries, Lie Algebras And Representations: A Graduate...

Hello, I am reading Naive Lie Theory by John Stillwell, and he gives the definition of a simple Lie group as a Lie group which has no non-trivial normal subgroups. Wikipedia, on the other hand, defines it as a Lie group which has no connected normal subgroups. I was wondering, which...

Please teach me this: What is the adjoint representation in Lie group? Where is the vector space that the ''elements of the group'' act on in this representation(adjoint representation)? Thank you very much for your kind helping.

Suppose we have a simple Lie Group G, i.e, a Lie Group with a trivial center(the identity). Show that this group must be linear, i.e, we can map it to a Lie subgroup of GL(N).So far, I have that from abstract algebra we can show a group with trivial center is isomorphic to the inner...

I've been trying for many hours to wrap my head around this problem. Schutz, in his Geometrical Methods of Mathematical Physics book goes through great lengths defining a left-translation map on a Lie Group G, and then defining left-invariant vector fields on G, and then he goes on to say that...

Leech lattice is a 'lie group?" My understanding of Lie groups is non-existent. But I'm trying to understand if the Leech lattice is a 'lie group?"

Hello all, I'm attempting to find in literature a method of determining from a Lie algebra's full root system in an arbitrary basis which roots are simple. It seems there are many books, articles, etc on getting all the roots from the simple roots but none that go the other way. My task is...

What can we tell about Lie group if we know its Lie algebra. Let's consider the following example: we have three elements of Lie algebra which fulfill condition [L_i,L_j]=i \epsilon_{ijk}L_k . The corresponding Lie group is SU(2) or SO(3) (are there any other?). Does anyone know what...

Is there a name for studying a Lie "group" that doesn't use the identity matrix as a member of the group? I know it's not technically a group anymore, but is there any mathematical work pertaining to the general idea... and what is the terminology so that I can research it better?

I am trying to read through this paper on the standard model. The ideas seem straightforward enough, but as always, I'm tripping over the "physicist's math" it uses. I was wondering if I can get some clarification or general guidance...

Question is in the title. Seems a lot of people throw that statement around as if its obvious, but it isn't obvious to me. I can kind of see how it might be true. If you take a group element, differentiate it wrt the group parameters to pull down the generators, and then evaluate this...

Here is few statements that I proved but I suspect that are incorrect (but I can't find mistake), term group means Lie group same goes for algebra: 1. Noncompact group G doesn't have faithfull (ie. kernel has more that one element) unitary representation. Proof: If D(G) is faithfull unitary...

Homework Statement I'm supposed to prove, that when G is a Lie group, i:G\to G is the inverse mapping i(g)=g^{-1}, then i_{*e} v = -v\quad\quad\forall \; v\in T_e G where i_{*e}:T_e G \to T_e G is the tangent mapping. Homework Equations I'm not sure how standard the tangent mapping...

What's the correct way to state the relationship between these two Lie groups? One is the "covering group" of the other, right? Okay, then - what's that mean, to a non-expert? I know the basics, i.e. SO(3) can be represented by rotation matrices in 3-space, and U(2) does the same in a...

https://www.physicsforums.com/showthread.php?p=1277407 (main thread in Linear & Abstract Algebra) http://science.slashdot.org/science/07/03/19/117259.shtml

Comment: My question is more of a conceptual 'why do we do this' rather than a technical 'how do we do this.' Homework Statement Given a lie group G parameterized by x_1, ... x_n, give a basis of left-invariant vector fields. Homework Equations We have a basis for the vector fields...

On page 116 of Choquet-Bruhat, Analysis, Manifolds, and Physics, Lie groups are defined, and the first exercise after that asks you to prove that for a Lie group G f:G \rightarrow G; x \mapsto x^{-1} is differentiable. I know from the previous definitions that a function f on a manifold...

Hello, I seem to be having difficulty proving something. I hope you can help me. I will write del_X(Y) when I refer to the levi-chivita connection (used on Y in the direction of X). Let G be a lie group, with a bi-invariant metric , g , on G. I want to prove that del_X(Y) = 0.5 [X,Y]...

hello, i have met with a problem. please help me. A Lie group,with a left-invariant Riemannian metric, i want to compute the connection compatible with the Riemannian metric. C(ij, k) are the structure constants, g(ij) are the metric, then how to compute the Riemannian connection in terms of...

A few friends have expressed an interest in exploring the geometry of symmetric spaces and Lie groups as they appear in several approaches to describing our universe. Rather than do this over email, I've decided to bring the discussion to PF, where we may draw from the combined wisdom of its...

Hi. I'm now studying Lie Groups, and have received the following exercise to solve. I have absolutely no idea where to begin, so please give me a direction. Let U be any neighborhood of e. Prove that any element of G can be written as a finite product of elements from U (i.e., U generates G).

Does somebody know an example of a differentiable manifold which is a group but NOT a Lie group? So the additional condition: the group operations multiplication and inversion are analytic maps, is not satisfied.