# 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 [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...
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*}...