lie groups

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. I Derivative of the Ad map on a Lie group

Hi, let $G$ be a Lie group, $\varrho$ its Lie algebra, and consider the adjoint operatores, $Ad : G \times \varrho \to \varrho$, $ad: \varrho \times \varrho \to \varrho$. In a paper (https://aip.scitation.org/doi/full/10.1063/1.4893357) the following formula is used. Let $g(t)$ be a...
4. A Lie Algebra and Lie Group

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...
5. I Commutator identity

I'm having a little trouble proving the following identity that is used in the derivation of the Baker-Campbell-Hausdorff Formula: $$[e^{tT},S] = -t[S,T]e^{tT}$$ It is assumed that [S,T] commutes with S and T, these being linear operators. I tried opening both sides and comparing terms to no...
6. I Parametrization manifold of SL(2,R)

I'm reading a book on Lie groups and one of the first examples is on SL(2,R). It says that every element of it can be written as the product of a symmetric matrix and a rotation matrix, which I can see, but It also makes the assertion that the symmetric matrix can be parameterized by a...
7. I Computation of the left invariant vector field for SO(3)

I am trying to improve my understanding of Lie groups and the operations of left multiplication and pushforward. I have been looking at these notes: https://math.stackexchange.com/questions/2527648/left-invariant-vector-fields-example...
8. I Do the SU(n) generators represent any observables?

Hey there, I've recently been trying to get my head around Yang-Mills gauge theory and was just wandering: do the Pauli matrices for su(2), Gell-Mann matrices for su(3), etc. represent any important observable quantities? After all, they are Hermitian operators and act on the doublets and...
9. A Diagonalization of adjoint representation of a Lie Group

So, we know that if g is a Lie algebra, we can take the cartan subalgebra h ⊂ g and diagonalize the adjoint representation of h, ad(h). This generates the Cartan-Weyl basis for g. Now, let G be the Lie group with Lie algebra g. Is there a way to diagonalize the adjoint representation Ad(T) of...
10. I Question about Haar measures on lie groups

I'm not sure if this question belongs to here, but here it goes Suppose you have to integrate over a lie group in the fundamental representation. If you pass to the adjoint representation of that group, does the Haar measure have to change? I think that it should not change because it is...
11. Star groups SU*(N)

I've run across a Lie group notation that I am unfamiliar with and having trouble googling (since google won't seem to search on * characters literally). Does anyone know the definition of the "star groups" notated e.g. SU*(N), SO*(N) ?? The paper I am reading states for example that SO(5,1)...
12. 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...
13. 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...
14. Prove: SO(3)/SO(2)=S^2

1. Homework Statement Take the subgroup isomorphic to SO(2) in the group SO(3) to be the group of matrices of the form \begin{pmatrix} g & & 0 \\ & & 0 \\ 0 & 0 & 1 \end{pmatrix}, g\in{}SO(2). Show that there is a one-to-one correspondence between the coset space of SO(3) by this subgroup and...
15. Controllability of non-linear systems via Lie Brackets

In http://www.me.berkeley.edu/ME237/6_cont_obs.pdf [Broken], page 65, the controllability matrix is defined as: $$C=[g_1, g_m,\dots,[g_i,g_j],[ad_{g_i}^k,g_j],\dots,[f,g_i],\dots,[ad_f^k,g_i],\dots]$$ where the systems is in general given by $$\dot{x}=f(x)+\sum_i^m{g_i(x)\mu_i}$$ Lets say you...
16. Lie Groups, Lie Algebra and Vectorfields

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*}...