# lie algebra

1. ### (Physicist version of) Taylor expansions

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)...
2. ### Lorentz algebra elements in an operator representation

1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: Please see image  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...
3. ### Other Textbooks for tensors and group theory

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...
4. ### A Highest weight of representations of Lie Algebras

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...

8. ### Insights Lie Algebras: A Walkthrough The Representations - Comments

Greg Bernhardt submitted a new blog post Lie Algebras: A Walkthrough The Representations Continue reading the Original Blog Post.
9. ### Insights Lie Algebras: A Walkthrough The Structures - Comments

Greg Bernhardt submitted a new blog post Lie Algebras: A Walkthrough The Structures Continue reading the Original Blog Post.
10. ### Insights Lie Algebras: A Walkthrough the Basics - Comments

Greg Bernhardt submitted a new blog post Lie Algebras: A Walkthrough the Basics Continue reading the Original Blog Post.
11. ### Left invariant vector field under a gauge transformation

1. 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? 2. Homework Equations 3. The Attempt at a...
12. ### I Rings, Modules and the Lie Bracket

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...
13. ### I Can we construct a Lie algebra from the squares of SU(1,1)

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...
14. ### A Example of how a rotation matrix preserves symmetry of PDE

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...
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18. ### Operation with tensor quantities in quantum field theory

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...
19. ### Insights A Journey to The Manifold SU(2) - Part II - Comments

Greg Bernhardt submitted a new PF Insights post A Journey to The Manifold SU(2) - Part II Continue reading the Original PF Insights Post.
20. ### Applied Zee and Georgi Group Theory books

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...
21. ### Generators of Lie Groups and Angular Velocity

I envision the three fundamental rotation matrices: R (where I use R for Ryz, Rzx, Rxy) I note that if I take (dR/dt * R-transpose) I get a skew-symmetric angular velocity matrix. (I understand how I obtain this equation... that is not the issue.) Now I am making the leap to learning about...
22. ### A Matrix Lie groups and its Lie Algebra

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...
23. ### Lie algebra for particle physics

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...
24. ### Praise Just Simply: Thank you

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...
25. ### I About Lie group product ($U(1)\times U(1)$ ex.)

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}}...
26. ### A Integrating the topics of forms, manifolds, and algebra

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...
27. ### A Lie Algebras and Rotations

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...
28. ### The relation between Lie algebra and conservative quantities

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...
29. ### Orthogonality of inner product of generators

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.
30. ### When are isomorphic Hilbert spaces physically different?

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...