Lie groups Definition and 103 Threads

Suppose θ is a differential 1 form defined on a manifold and with values in the Lie algebra of a Lie group,G. On MxG define the 1 form, ad(g)θ ,where θ is extended by letting it be zero on the tangent space to G How do you compute the exterior derivative, dad(g)θ ? BTW: For matrix...

Quantum Gravity: "Lie Groups" vs. "Banach Algebras & Spectral Theory" I'm interested in researching quantum gravity & non-commutative geometry. I am planning to take one math course outside of my physics classes this Fall to help, but can't decide between two: "Lie groups" or "Banach algebras &...

Hey guys, I'm really interested in finding out how to deal with differential equations from the point of view of Lie theory, just sticking to first order, first degree, equations to get the hang of what you're doing. What do I know as regards lie groups? Solving separable equations somehow...

I am familiar with what SO(2) means for example but am unclear what SO(1,1) refers to. This came up in a classical physics video lecture when lie groups were discussed and the significance of the notation was glossed over. Second question: is the dimensionality of such a group the same as the...

Hi, I'm a student of Nuclear Engineering (MS level) at University of Dhaka, Bangladesh. I completed my Honours and Master Degree with Mathematics. I have chosen to complete a thesis paper on "Application of Lie groups & Lie Algebras in Nuclear & Particle Physics." I need some guideline...

Hi! I was wondering why it is possible to write any proper orthochronous Lorentz transformation as an exponential of an element of its Lie-Algebra, i.e., \Lambda = \exp(X), where \Lambda \in SO^{+}(1,3) and X is an element of the Lie Algebra. I know that in case for compact...

Author: Brian Hall Title: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction Amazon link https://www.amazon.com/dp/1441923136/?tag=pfamazon01-20 Level: Grad Table of Contents: General Theory Matrix Lie Groups Definition of a Matrix Lie Group Counterexamples...

Hello! I am currently trying to get things straight about Lie group from two different perspectives. I have encountered Lie groups before in math and QM, but now I´m reading GR where we are talking about coordinate and non-coordinate bases and it seems that we should be able to find commuting...

I know that gauging a lie-goup with a kinetic term of the form: \begin{equation} \Tr{F^{\mu \nu} F_{\mu \nu} } \end{equation} Is not allowed for a non-compact lie group because it does not lead to a positive definite Hamiltonian. I was wondering if anyone knew of a general way to gauge...

Hello, I have a doubt on the definition of Lie groups that I would like to clarify. Let's have the set of functions G=\{ f:R^2 \rightarrow R^2 \; | \; < f(x),f(y)>=<x,y> \: \forall x,y \in R^2 \}, that is the set of all linear functions ℝ2→ℝ2 that preserve the inner product. Let's associate the...

Hello, let's suppose I have the following system of curvilinear coordinates in ℝ2: x(u,v) = u y(u,v) = v + e^u where one arbitrary coordinate line C_\lambda(u)=u \mathbf{e_1} + (\lambda+e^u) \mathbf{e_2} represents the orbit of some point in ℝ2 under the action of a Lie group. Now I consider...

How can you tell if two Lie groups are isomorphic to each other? If you have a set of generators, Ti, then you can perform a linear transformation: T'i=aijTj and these new generators T' will have different structure constants than T. Isn't it possible to always find a linear...

Hello. I have a question that has been on my mind for some time. I always see in mathematical physics books that they identify elements of the Lie algebra with group elements "sufficiently close" to the identity. I have never seen a real good proof of this so went on an gave a proof. Let Xi be...

Hi everybody! Ok, so from a few days I've begun a group theory class, and i have to say i love the subject. In particular i happened to like Lie groups, but there are things that are not cristal clear to me, hope you'll help to figure'em out!First of all, Lie groups are continuous group, so...

I'm trying to understand why a Lie group always has a non-vanishing vector field. I know that one can somehow generate one by taking a vector from the Lie algebra and "moving it around" using the group operations as a mapping, but the nature of this map eludes me.

For discrete groups, we can easily find the decomposition of the direct product of irreducible representations with the help of the character table. All we need to do is multiply the characters of the irreducible representations to get the characters of the direct product representation and then...

Hi all, Sorry, I'm not quite sure that I've posted this question in the proper place, but I figured field theory matches best with lie groups in this context. Anyway, my question has to do with the relationship between the fundamental forces (electromagnetism, weak, and strong) and their...

Hi, so I didn't see exactly where group theory stuff goes...but since Lie groups are also manifolds, then I guess I can ask this here? If there's a better section, please move it. I just have a simple question regarding the definition of a Lie group. My book defines it as a group which is...

http://arxiv.org/abs/1104.1106 Lecture Notes in Lie Groups Vladimir G. Ivancevic, Tijana T. Ivancevic (Submitted on 6 Apr 2011) These lecture notes in Lie Groups are designed for a 1--semester third year or graduate course in mathematics, physics, engineering, chemistry or biology. This...

Hello! Is someone aware if there are lecture notes about Lie Groups from a physics course? I would to study an exposition of this subject made by a physicist. Thank you in advance!

Why do people try against all odds to make SU(2) isometric with SO(3) when it's clear from the definition that it's actually isometric with SO(4). Either way you've got 4 variables and the same constraint between them. It's interesting to see all the dodgy tricks that go into this deception...

Out of curiosity, how does the product rule work in Lie groups? I ended up needing it because I approached a problem incorrectly and then saw that the product rule was unnecessary, but it seems to create a strange scenario. For example: Consider a Lie group G and two smooth curves \gamma_1...

I'm currently learning Lie groups/algebras and I am trying to find the infinitesimal generators of the special orthogonal group SO(n). It is the n-dimensions that throws me off. I know that the answer is n(n-1)/2 generators of the form, X_{\rho,\sigma}=-i\left(x_{\rho}\frac{\partial}{\partial...

Suppose \mathfrak{g} and \mathfrak{h} are some Lie algebras, and G=\exp(\mathfrak{g}) and H=\exp(\mathfrak{h}) are Lie groups. If \phi:\mathfrak{g}\to\mathfrak{h} is a Lie algebra homomorphism, and if \Phi is defined as follows: \Phi:G\to H,\quad \Phi(\exp(A))=\exp(\phi(A))...

Hi. I'm looking for an introductory book on Lie Groups and Lie Algebras and their applications in physics. Preferably the kind of book that emphasizes understanding, applications and examples, rather than proofs. Any suggestions? Edit: Please move this to Science Book Discussion.

A while ago I heard the following two facts about semi-simple Lie groups (though I have a feeling they may have to be restricted to connected semi-simple Lie groups): 1. That semi-simple Lie groups are classified by their weight (and co-weight) and root (and co-root) lattices; 2. That all of...

As i understand it, the commutation rules for the quantum angular momentum operator in x, y, and z (e.g. Lz = x dy - ydx and all cyclic permutations) are the same as the lie algebras for O3 and SU2. I'm not entirely clear on what the implications of this are. So I can think of Lz as generating...

Let G be a Lie group. Show that there exists a unique affine connection such that \nabla X=0 for all left invariant vector fields. Show that this connection is torsion free iff the Lie algebra is Abelian.

Can anyone expand on the relationship between Lie groups, Lie algebras, exponential maps and unitary operators in QM? I've been reading lately about Lie groups and exponential maps, and now I'm trying to tie it all together relating it back to QM. I guess I'm trying to make sense of how Lie...

1. The exponential map is a map from the lie algebra to a matrix representation of the group. For abelian groups, the group operation of matrix multiplication for the matrix rep clearly corresponds to the operation of addition in the lie algebra: \sum_a \Lambda_a t_a \rightarrow exp(\sum_a...

Homework Statement Let G be a Lie group with unit e. For every g in G let Lg: G -> G, h-> gh be left multiplication. a) Prove that the map G x TeG -> TG, (g,v) -> De Lg (v) is a diffeomorphism, and conclude G has a basis of vector fields. b) Prove for every v in TeG the is a unique...

Homework Statement Prove SOnR and SLnR are Lie groups, and determine their dimensions. SOnR = {nxn real hermitian matrices and determinant > 0} SLnR = {nxn real matrices with determinant 1} The Attempt at a Solution We can see that SLnR is level set at zero of the graph of a smooth...

Please, help me with the following questions or recommend some good books. 1) We have a simply-connected nilpotent Lie group G and a lattice H in G. Let L be a Lie algebra of G. There is a one to one correspondence between L and G via exp and log maps. a) Is it true, that to an ideal in...

Hi there, I'm reading over my Lie groups notes and in them, in the introductory section on manifolds, I've written that F_{\star} is a commonly used notation for d_{x}F and so the chain rule d_{x}{G \circ F}=d_{F(x)}G \circ d_{x}F can be written (G\circ F)_{\star}=G_{\star}\circ F_{\star} Is...

I'm reading over my Lie groups notes and in them, in the introductory section on manifolds, I've written that F_{\star} is a commonly used notation for d_{x}F and so the chain rule d_{x}{G \circ F}=d_{F(x)}G \circ d_{x}F can be written (G\circ F)_{\star}=G_{\star}\circ F_{\star} Is what I've...

I'm taking a course on Lie Groups and the Representations. We are using the book: Representations of compact Lie Groups by Bröcker and Dieck, and I find it very unorganized and sometimes sloppy. Can anybody recommend a very clear and rigorous book, where it is not prove by example, "it is easily...

Let \rho : \mathbb{H} \to \mathbb{H}; q \mapsto u^{-1}q u where u is any unit quaternion. Then \rho is a continuous automorphism of H. I'm asked to show that \rho preserves the inner product and cross product on the subspace \mathbf{i}\mathbb{R} + \mathbf{j}\mathbb{R} +...

Why are Lie groups also manifolds?

What Lie groups are also Riemann manifolds? thanks

I'm looking for a solid book on Lie groups and Lie algebras, there is too many choices out there. What is a classic text, if there is one?

What is the connection between the two if any? What kind of algebra would Lie groups be best labeled under?

Hi, everyone: I am asked to show that a group G acts by isometries on a space X. I am not clear about the languange, does anyone know what this means?. Do I need to show that the action preserves distance, i.e, that d(x,y)=d(gx,gy)?. Thanks.

It's common in theoretical physics papers/books to talk about the decomposition of Lie groups, such as the adjoint rep of E_8 decomposing as \mathbf{248} = (\mathbf{78},\mathbf{1}) + (\mathbf{1},\mathbf{8})+(\mathbf{27},3) + (\overline{\mathbf{27}},\overline{\mathbf{3}}) How is this...

Is U(t)=exp(-iH/th) a Lie group? Is it an infinite dimensional Lie group? To what 'family' of Lie groups does it belong? thank you

Let G be a 3-dimensional simply-connected Lie group. Then, G is either 1.)The unit quaternions(diffeomorphic as a manifold to S$^{3}$) with quaternionic multiplication as the group operation. 2.)The universal cover of PSL$\left( 2,\Bbb{R}\right) $ 3.)The...

Does anybody know the answer of the following problem? Show that the Lie group of Euclidean motions of R^3 has a Lie algebra g which is perfect i.e., Dg=g but g is not semisimple. By Dg I mean the commutator [g,g] and a semisimple lie algebra is one has no nonzero solvable ideals. Regards

Hi all, Anybody knowes how to find, or at least knows the reference that shows, the real lie algebra of sl(n,H)? By sl(n,H), I mean the elements in Gl(n,H) [i.e. the invertible quaternionic n by n matrices] whose real determinant is one. Many Thanks Asi

\mathbb{R}^3 has an associative multiplication \mu:\mathbb{R}^3\times \mathbb{R}^3 \rightarrow \mathbb{R}^3 given by \mu((x,y,z),(x',y',z'))=(x+x', y+y', z+z'+xy'-yx') Determine an identity and inverse so that this forms a Lie group. Well, clearly e=(0,0,0) and the inverse element is...

Hi folks. I've come across a method to determine the controllability of a quantum system that depends on the Lie group generated by the commutator of the skew-Hermetian versions of the field free and interaction (dipole) Hamiltonians. If, for an N dimensional system the dimension of the group...

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.