Lie algebra Definition and 163 Threads

As follow up of this thread in Special and General Relativity subforum, I'd like to better investigate the following topic. Consider the 4d euclidean space in which there are 10 ##\mathbb R##-linear independent KVFs. Their span at each point is 4 dimensional (i.e. at any point they span the...

##SO(3)## is a Lie group of dimension 3. It is the set of 3x3 matrices ##R## with the following properties: $$RR^T = R^TR=I, \text{det}(R)=+1$$ There exists a parametrization of ##SO(3)## that maps it on the sphere in ##\mathbb R^3## of radius ##\pi## where the antipodal points are identified...

A spherically symmetric manifold has, by definition, a set of 3 independent Killing Vector Fields (KVFs) satisfying: $$\begin{align}[R,S] &=T \nonumber \\ [S,T] &=R \nonumber \\ [T,R] &=S \nonumber \end{align}$$ These 3 KVFs define a linear subspace of the (infinite dimensional) vector space...

Hi, I'm currently taking a class in Classical Field Theory. We've covered topics such as relativity, Poincaré and Lorentz groups, tensor algebra and calculus, as well as Lie algebras and groups. I would like to review for my exam and was wondering if anyone has practice questions or past exams...

All of the formulations of the Lie algebra of ##\frak{so}(3)## (or ##\frak{su}(2)##) utilizing raising/lowering operators that I have seen in the literature involve complexification to ##\frak{su}(2) + i \frak{su}(2) \cong \frak{sl}(2,\mathbb{C})##. I have found explicit derivations in a...

Hi, consider the group ##SL(n,\mathbb R)##. It is a subgroup of ##GL(n,\mathbb R)##. To show it is a Lie group we must assign a differential structure turning it into a differential manifold, proving further that multiplication and taking the inverse are actually smooth maps. With the...

Consider ##\mathbb R^2## as the Euclidean plane. Since it is maximally symmetric it has a 3-parameter group of Killing vector fields (KVFs). Pick orthogonal cartesian coordinates centered at point P. Then the 3 KVFs are given by: $$K_1=\partial_x, K_2=\partial_y, K_3=-y\partial_x + x...

Dear @fresh_42 , Hope you are well. Please, I have a question if you do not mind, about Lie Algebra, In page 2 in the book of Lie algebra, written by Humphreys, Classical Lie algebras, ##A, B, C## and ##D##, I did not get it well, especially, symplectic and orthogonal.. Could you please...

I have some clarifications on the discussion of adjoint representation in Group Theory by A. Zee, specifically section IV.1 (beware of some minor typos like negative signs). An antisymmetric tensor ##T^{ij}## with indices ##i,j = 1, \ldots,N## in the fundamental representation is...

Hi, I found on some lectures the following parametrization of ##SU(2, \mathbb C)## group elements \begin{pmatrix} e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\ ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}}...

Please, I have a question about this: The Universal enveloping algebra of a finite dimensional Lie algebra is Noetherian. How we can prove it? Please..

Please, I have a question about Schur's Lemma ; Let $\phi: L \rightarrow g I((V)$ be irreducible. Then the only endomorphisms of $V$ commuting with all $\phi(x)(x \in L)$ are the scalars. Could you explain it, and please, how we can apply this lemma on lie algebra ##L=\mathfrak{s l}(2)##thanks...

Please, I have a question about universal enveloping algebra: Let ##U=U(\mathfrak{g})## be the quotient of the free associative algebra ##\mathcal{F}## with generators ##\left\{a_i: i \in I\right\}## by the ideal ##\mathcal{I}## generated by all elements of the form ##a_i a_j-a_j a_i-\sum_{k \in...

in the Proof of Engel's Theorem. (3.3), p. 13: please, how we get this step: ##L / Z(L)## evidently consists of ad-nilpotent elements and has smaller dimension than ##L##. Using induction on ##\operatorname{dim} L##, we find that ##L / Z(L)## is nilpotent. Thanks in advance,

Please, How we can solve this: $$ \mathfrak{h}=\mathbb{K} H \text { is a Cartan subalgebra of } \mathfrak{s l}_2 \text {. } $$ it is abelian, but how we can prove it is self-normalizer, please:Dear @fresh_42 , if you could help, :heart: 🥹

Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.12, I have a question: Proposition. (3.2). Let ##L## be a Lie algebra. (c) If ##L## is nilpotent and nonzero, then ##Z(L) \neq 0##. how we prove this, Thanks in advance,

Please, in the book of Introduction to Lie Algebras and Representation Theory J. E. Humphreys p.11, I have a question: Proposition. Let ##L## be a Lie algebra. (a) If ##L## is solvable, then so are all subalgebras and homomorphic images of ##L##. (b) If ##I## is a solvable ideal of ##L## such...

Please, in the definition of quotient Lie algebra If ##I## is an ideal of ##\mathfrak{g}##, then the vector space ##\mathfrak{g} / I## with the bracket defined by: $$[x+I, y+I]=[x, y]+I, for all x, y \in \mathfrak{g}$$, is a Lie algebra called the quotient Lie algebra of ##\mathfrak{g}## by...

Please, I am looking for a simple example of derivation on ##sl_2##, if possible, I try to use identity map, but not work with me, A derivation of the Lie algebra ##\mathfrak{g}## is a linear map ##\delta: \mathfrak{g} \rightarrow \mathfrak{g}## such that ##\delta([x, y])=[\delta(x), y]+[x...

Please, I need some clarifications about second direction, in the file attached, $$ \text { Then ad } x \text { ad } y \text { maps } L \rightarrow L \rightarrow I \text {, and }(\text { ad } x \text { ad } y)^2 \text { maps } L \text { into }[I I]=0 \text {. } $$Thank you in advance,

Homework Statement: About semidirect product of Lie algebra Relevant Equations: ##\mathfrak{s l}_2=## ##\mathbb{K} F \oplus \mathbb{K} H \oplus \mathbb{K} E## Hi, Please, I have a question about the module of special lie algebra: Let ##\mathbb{K}## be a field. Let the Lie algebra...

Homework Statement: please, could you help me to know hoe I compute the Casimir element of lie algebra sl(2), I know the basis and their relations, but i could not find the book explain in details how we get the Casimir element.. I think it is related to killing form, but also I could not find...

This is the defining generator of the Lorentz group which is then divided into subgroups for rotations and boosts And I then want to find the commutation relation [J_m, J_n] (and [J_m, K_n] ). I'm following this derivation, but am having a hard time to understand all the steps: especially...

The full Lorentz group includes discontinuous transformations, i.e., time inversion and space inversion, which characterize the non-orthochronous and improper Lorentz groups, respectively. However, these groups are excluded from the Poincare group, in which only the proper, orthochronous...

My intuition about the Lie algebra is that it tries to capture how infinitestimal group generators fails to commute. This means ##[a, a] = 0## makes sense naturally. However the Jacobi identity ##[a,[b,c]]+[b,[c,a]]+[c,[a,b]] = 0## makes less sense. After some search, I found this article...

I’m reading Weinberg’s QFT books, and stacking how to solve problem 15.4. Weinberg says there is no simple lie algebra with just four generators, but I have no idea how to approach this problem. If the number of generators are only one or two, it can easy to say there is not such a simple lie...

1.) The rule for the global ##SO(3)## transformation of the gauge vector field is ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## for ##\omega \in SO(3)##. The proof is by direct calculation. First, if ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## then ##F^i_{\mu \nu} \to \omega_{ij}F^j_{\mu\nu}##, so...

Let ##\mathfrak{A}:=\operatorname{span}\left\{D_n:=x^n\dfrac{d}{dx}\, : \,n\in \mathbb{Z}\right\}## and ##\mathfrak{B}:=\operatorname{span}\left\{E_n:=x^n\dfrac{d}{dx}\, : \,n\in \mathbb{N}_0\right\}## with the usual commutation rule. My question is: How can we prove or disprove the Lie algebra...

I want to show that ##[C, a_{r}] = 0##. This means that: $$ Ca_{r} - a_{r}C = \sum_{i,j} g_{ij}a_{i}a_{j}a_{r} - a_{r}\sum_{i,j} g_{ij}a_{i}a_{j} = 0$$ I don't understand what manipulating I can do here. I have tried to rewrite ##g_{ij}## in terms of the structure...

Hello everyone, I am new here, so please let me know if I am doing something wrong regarding the formatting or the way I am asking for help. I did not really know how to start off, so first I tried to just write out all the ##\mu \nu \rho \sigma## combinations for which ##\epsilon \neq 0## and...

I'm following the lecture notes by https://www.thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf. On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM. The fundamental representation makes sense to me. For example, for ##SU(3)##, we define the...

"The group given by ## H = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i a \theta } } \end{array} \right) | \theta \in \mathbb { R } \right\} \subset \mathbb { T } ^ { 2 } = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0...

Posting for my son (who does not have an account here): He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants...

I am just starting a QM course. I hope these are reasonable questions. I have been given my first assignment. I can answer the questions so far but I do not really understand what's going on. These questions are all about so(N) groups, Pauli matrices, Lie brackets, generators and their Lie...

I have confusions about representation theory. In the following questions, I will try to express it as best as possible. For this thread say representation is given as ρ: L → GL(V) where L is the Lie group(or symmetry group for a physicist) GL(V) is the general linear...

I'm solving these problems concerning the SU(4) group and I've reached the point where I have determined the Cartan matrix of SU(4), its inverse and the weight schemes for (1 0 0) and (0 1 0) highest weight states. How do I decompose the (1 0 0) and (0 1 0) into irreps of SU(3) x U(1) using...

In Quantum Mechanics, by Wigner's theorem, a symmetry can be represented either by a unitary linear or antiunitary antilinear operator on the Hilbert space of states ##\cal H##. If ##G## is then a Lie group of symmetries, for each ##T\in G## we have some ##U(T)## acting on the Hilbert space and...

This is one problem from Robin Ticciati's Quantum Field Theory for Mathematicians essentially asking us to find Cartan subalgebras for the matrix algebras ##\mathfrak{u}(n), \mathfrak{su}(n),\mathfrak{so}(n)## and ##\mathfrak{so}(1,3)##. The only thing he gives is the definition of a Cartan...

3) Taylor expansion question in the context of Lie algebra elements: Consider some n-dimensional Lie group whose elements depend on a set of parameters \alpha =(\alpha_1 ... \alpha_n) such that g(0) = e with e as the identity, and that had a d-dimensional representation D(\alpha)=D(g( \alpha)...

1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: Please see image [2] below. I aim to consider the product L^0{}_0(\Lambda_1\Lambda_2). Consider the following notation L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu. How then, does...

Hello, I am an undergraduate who has taken basic linear algebra and ODE. As for physics, I have taken an online edX quantum mechanics course. I am looking at studying some of the necessary math and physics needed for QFT and particle physics. It looks like I need tensors and group theory...

Is it correct saying that the Exponential limit is an exact solution for passing from a Lie Algebra to a Lie group because a differential manifold with Lie group structure is such that for any point of the transformation the tangent space is by definition the Lie algebra: is that the underlying...

Hello there, Given a Lie Algebra ##\mathfrak{g}##, its Cartan Matrix ##A## and a finite representation ##R##, is there a way of determining its highest weight ##\Lambda## in a simple way? In my course, we consider ##\mathfrak{g}=A_2= \mathfrak{L}_{\mathbb{C}}(SU(3))##. It is stated that the...

Let us for a moment look a field transformations of the type $$\phi(x)\longmapsto \exp\left(\frac{1}{2}\omega_{\mu\nu}S^{\mu\nu}\right)\phi(x),$$ where ##\omega## is anti-symmetric and ##S^{\mu\nu}## satisfy the commutation relations of the Lorentz group, namely $$\left[S_{\mu \nu}, S_{\rho...

Given a representation of a Lie Group, is there a equivalence between possible electric charges and projections of the roots? For instance, in the standard model Q is a sum of hypercharge Y plus SU(2) charge T, but both Y and T are projectors in root space, and so a linear combination is. But I...

Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group SO(3). For example: $$\vec{\mathbf{A}} \times \vec{\mathbf{B}} = (A^T \cdot J_x \cdot B) \>\> \hat{i} + (A^T \cdot J_y \cdot B)...

Frobenius began in ##1896## to generalize Weber's group characters and soon investigated homomorphisms from finite groups into general linear groups ##GL(V)##, supported by earlier considerations from Dedekind. Representation theory was born, and it developed fast in the following decades. The...

Lie algebra theory is to a large extend the classification of the semisimple Lie algebras which are direct sums of the simple algebras listed in the previous paragraph, i.e. to show that those are all simple Lie algebras there are. Their counterpart are solvable Lie algebras, e.g. the Heisenberg...

This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems, and a brief overview of the subject. Physicists usually call the elements of Lie algebras generators, as for them they are merely differentials of trajectories, tangent vector fields...

Homework Statement For a left invariant vector field γ(t) = exp(tv). For a gauge transformation t -> t(xμ). Intuitively, what happens to the LIVF in the latter case? Is it just displaced to a different point in spacetime or something else? Homework EquationsThe Attempt at a Solution