Lie algebra Definition and 163 Threads

I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups. Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e, Lgh = gh where g,h ∈ G Which corresponds to a translation by g. Is this an example of a module over a ring...

Hello! I am a bit confused by some definitions. We have that a Lie algebra is abelian if ##[a,b]=0## for all ##a,b \in L## and ##L'## is an invariant subalgebra of ##L## if ##[a,b]=0## for all ##a \in L'## and ##b \in L##. From here I understand that ##L'## is abelian. Then they define a...

I am trying to decompose some exponential operators in quantum optics. The interesting thing is that the operators includes operators from Su(1,1) algebra $$ [K_+,K_-]=-2K_z \quad,\quad [K_z,K_\pm]=\pm K_\pm.$$ For example this one: $$ (K_++K_-)^2.$$ But as you can see they are squares of it. I...

Good Day I have been having a hellish time connection Lie Algebra, Lie Groups, Differential Geometry, etc. But I am making a lot of progress. There is, however, one issue that continues to elude me. I often read how Lie developed Lie Groups to study symmetries of PDE's May I ask if someone...

Homework Statement How can you find all inequivalent (non-isomorphic) 2D Lie algebras just by an analysis of the commutator? Homework Equations $$[X,Y] = \alpha X + \beta Y$$ The Attempt at a Solution I considered three cases: ##\alpha = \beta \neq 0, \alpha = 0## or ##\beta = 0, \alpha =...

Homework Statement Given for one-dimensional Galilean symmetry the generators ##K, P,## and ##H##, with the following commutation relations: $$[K, H] = iP$$ $$[H,P] = 0$$ $$[P,K] = 0$$ Homework Equations Show that the Lie algebra for the generators ##K, P,## and ##H## is isomorphic to the...

I would like to know where one may operate with tensor quantities in quantum field theory: Minkowski tensors, spinors, effective lagrangians (for example sigma models or models with four quark interaction), gamma matrices, Grassmann algebra, Lie algebra, fermion determinants and et cetera. I...

Part 1. The second part of my journey to the manifold ##SU(2)## deals with some representations. We start with some bases and cite the classification theorem of representations of the three-dimensional, simple Lie algebra. Continue reading ...

Hello! I am reading some representation theory/Lie algebra stuff and at a point the author says "the states of the adjoint representation correspond to generators". I am not sure I understand this. I thought that the states of a representation are the vectors in the vector space on which the...

Hello. I will be attending a course on Group theory and the book that the professor suggests is Georgi's Lie Algebras in Particle Physics. As I liked Zee's book on General Relativity, I thought that it would be a blast to also use his Group theory textbook for the course. Problem is that I don't...

IN Srednicki's QFT he seems to make two different choices for normalizing the generators of lie algebras. In chapter 24 (eqn 24.5) he chooses Tr (TaTb) = 2 δab and in chapter 69 (eqn 69.8) he chooses Tr (TaTb) = (1/2) δab Is there a reason for this? Is there any particular reason to make one...

Hello! I read that the for the lie algebra of the Lorentz group we can parametrize the generators as an antisymmetric tensor ##J^{\mu \nu}## and the parameters as an another antisymmetric tensor ##\omega_{\mu \nu}## and a general transformation would be ##\Lambda = exp(-\frac{i}{2} \omega_{\mu...

Homework Statement The group ##G = \{ a\in M_n (C) | aSa^{\dagger} =S\}## is a Lie group where ##S\in M_n (C)##. Find the corresponding Lie algebra. Homework EquationsThe Attempt at a Solution As far as I've been told the way to find these things is to set ##a = exp(tA)##, so...

Homework Statement Show the (real) dimension of su(n) is ##n^2-1##. Homework EquationsThe Attempt at a Solution ##su(n) = \{ A \in M_n(\mathbb{C}) | A+A^T = 0, tr(A) =0 \}## Maybe the solution is obvious, because I can't find a thing online about how to do this. But I can't see how to do it! I...

Hi! I'm studying Lie Algebras and Lie Groups. I'm using Brian Hall's book, which focuses on matrix lie groups for a start, and I'm loving it. However, I'm really having a hard time connecting what he does with what physicists do (which I never really understood)... Here goes one of my questions...

Hello! Is there any rule to do sums and products like the one in the attached picture (Lie.png) without going through all the math theory behind? I understand the first (product) and last (sum) terms, but I am not sure I understand how you go from one to another. Thank you!

Hello! I am sorry that this questions is not actually directly related to physics, but, can anyone recommend me a good book about abstract algebra (basically lie algebra, representation theory etc.) used in physics? I have tried for a long time to find something online but I haven't find a...

Hello! I am a diploma student at HEP section. I am going to have an interview for PhD within a week. I've finished the course and learned a lot about Lie algebra, quantum field theory, general relativity, standard model, etc. How can I review everything as soon as possible? For example, Mark...

No question this time. Just a simple THANK YOU For almost two years years now, I have been struggling to learn: differential forms, exterior algebra, calculus on manifolds, Lie Algebra, Lie Groups. My math background was very deficient: I am a 55 year old retired (a good life) professor of...

I recently got confused about Lie group products. Say, I have a group U(1)\times U(1)'. Is this group reducible into two U(1)'s, i.e. possible to resepent with a matrix \rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}}...

Hello, As you might discern from previous posts, I have been teaching myself: Calculus on manifolds Differential forms Lie Algebra, Group Push forward, pull back. I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost...

The Lie Algebra is equipped with a bracket notation, and this bracket produces skew symmetric matrices. I know that there exists Lie Groups, one of which is SO(3). And I know that by exponentiating a skew symmetric matrix, I obtain a rotation matrix. ----------------- First, can someone edit...

Hi everybody, Let G a four dimmensionnal Lie group with g as lie algebra. Let T1 ... T4 the four generator. I would like to find à lorentzian scalar product (1-3 Signature) on it and left invariant. A classical algebra take tr (AB^t) as scalar product but I don't find à lorentzian équivalent...

A textbook or even better some openly available pdf would be preferred. I have no idea where to start a search and I normally prefer recommendations over just picking any book from some publisher. Thank you.

Elements of Lie algebra are generators. So for example Pauli matrices are generators of rotation and the elements of Lie algebra. And multiplication in Lie algebra is commutator. Right? What about if there is only one generator. As in case in rotation in plane. What is Lie algebra product in...

Hi y'all, This is more of a maths question, however I'm confident there are some hardcore mathematical physicists out there amongst you. It's more of a curiosity, and I'm not sure how to address it to convince myself of an answer. I have a Lie group homomorphism \rho : G \rightarrow GL(n...

I've heard it said that the commutation relations of the generators of a Lie algebra determine the multiplication laws of the Lie group elements. I would like to prove this statement for ##SO(3)##. I know that the commutation relations are ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##. Can you...

The generators ##(A_{ab})_{st}## of the ##so(n)## Lie algebra are given by: ##(A_{ab})_{st} = -i(\delta_{as}\delta_{bt}-\delta_{at}\delta_{bs}) = -i\delta_{s[a}\delta_{b]t}##, where ##a,b## label the number of the generator, and ##s,t## label the matrix element. Now, I need to prove the...

Homework Statement Consider a semi simple lie algebra. Show that if ##T_a## are the generators of a semi simple Lie algebra then ##\text{Tr}T_a=0##. 2. Homework Equations The commutation relations of the generators and the cyclic properties of the trace. The Attempt at a Solution $$[T_a...

This is only a minor question. I watched an on-line video recently on su2 and how it applies to Physics. Now, one of the first things the instructor did was to change the base to sl2. Fine and all, but she called sl2 "su2" for the whole video. Since the two Lie algebras have different...

In quantum mechanics, a physical quantity is expressed as an operator G, then the unitary transformation coresponding to the physical quantity is expressed as exp(-iG/ħt), being also an operator, where t is the tranformation parameter. G is actually the conservative quantity corresponding to the...

Right now I'm using "Affine Lie Algebras and Quantum Groups" by Fuchs. I'm getting sick of it. As a Physicist the structure and focus of the text is attractive to me. But, to give one example... The text constructs two vector spaces: L and Lw. Lw is dual to L. The text goes on to state...

Hi, this is a rather mathematical question. The inner product between generators of a Lie algebra is commonly defined as \mathrm{Tr}[T^a T^b]=k \delta^{ab} . However, I don't understand why this trace is orthogonal, i.e. why the trace of a multiplication of two different generators is always zero.

In quantum mechanics, a Hilbert space always means (in mathematical terms) a Hilbert space together with a distinguished irreducible unitary representation of a given Lie algebra of preferred observables on a common dense domain. Two Hilbert spaces are considered (physically) different if this...

I was wondering about the following Λ=I+iT T are the generators and Λ a continuous LT transformation, thus it is real. Therefore T needs to be imaginary. And we can find two sets one being the generators for SO(3) J_i and the other for boosts K_i, which are both imaginary. Now I am wondering...

Is the following correct? We begin with a set. Then, we specify a certain collection of subsets and thereby create a topology. This endows the set with certain properties, one of which is “nearness” and “boundedness.” Then we specify that the topology be smooth. In so doing, our topology...

First off: I am not complaining about any of the members here or elsewhere. I do not post questions expecting help. When I post a question I would like an answer but I do not require it. Getting an answer depends on who is around and who is willing to help when I post the question. Still...

Homework Statement The question is how to get the mass term for the W Gauge bosons within the Left Right Symmetric Model (LRSM). My main struggle is with the Left-Right Mixing part. So the LRSM has the gauge group $SU(2)_L\times SU(2)_R \times U(1)_{B-L}$ and with an additional discrete...

Hello everybody, in Schwartz' QFT book it says (p. 483 - 484) In Problem 25.3 this is repeated asking the reader for a proof. I wonder though if this is really true. I know this can be proven for Lie algebras of compact Lie groups (or to be precise, every representation is equivalent to a...

On one hand, in reading Georgi's book in group theory, I comprehend the invariant tensor as a special "tensor", which is unchanged under the action of any generators. On the other hand, CG decomposition is to decompose the product of two irreps into different irreps. Now it is claimed that...

As I understand it, the symplectic Lie group Sp(2n,R) of 2n×2n symplectic matrices is generated by the matrices in http://en.wikipedia.org/wiki/Symplectic_group#Infinitesimal_generators . Does this mean that sl(n,R) is a subalgebra of the corresponding lie algebra, since in that formula we can...

Homework Statement Show that the members of the Lie algebra of SO(n) are anti-symmetric nxn matrices. To start, assume that the nxn orthogonal matrix R which is an element of SO(n) depends on a single parameter t. Then differentiate the expression: R.RT= I with respect to the parameter t...

Warning: This is going to be a bit long. (Apparently my post was too long so it wouldn't render at all. I've split this into two threads.) I worked out some basic Algebraic properties of a Lie Algebra. This is similar to my previous thread about SU(2) but as I don't know this example I'm...

In the way of defining the adjoint representation, \mathrm{ad}_XY=[X,Y], where X,Y are elements of a Lie algebra, how to determine the components of its representation, which equals to the structure constant?

In Srednicki's text on quantum field theory, he has a chapter on quantum Lorentz invariance. He presents the commutation relations between the generators of the Lorentz group (equation 2.16) as follows: $$[M^{\mu\nu},M^{\rho\sigma}] =...

Hi I'm learning about Lie Groups to understand gauge theory (in the principal bundle context) and I'm having trouble with some concepts. Now let a and g be elements of a Lie group G, the left translation L_{a}: G \rightarrow G of g by a are defined by : L_{a}g=ag which induces a map L_{a*}...

Hi, I am trying to work through a proof/argument to show that the adjoint representation of a semisimple Lie algebra is completely reducible. Suppose S denotes an invariant subspace of the Lie algebra, and we pick Y_i in the invariant subspace S. The rest of the generators X_r are such that...

I've been skimming my String Theory text and I've been having a hard time understanding a couple of things. For now I'm simply going to ask...What is the Lie Algebra E8? (Not to be confused with E(8), 8 dimensional Euclidian space.) I've read the Wiki article and another on a different site...

How can we deform a given Lie algebra? In particular, in the attachment file how can we arrive at the commutation relations (20) by starting from the commutation relation (19)?

While studying Yang-Mills theory, I've come across the statement that there exists a positive-definite inner product on the lie algebra ##\mathfrak g## iff the group ##G## is compact and simple. Why is this true, and how it is proved?