Lists of asteroids typically refer to the various catalogs and databases that document the asteroids discovered in our solar system. These lists can include a wide range of information about each asteroid, such as its designation number, name, size, orbital characteristics, and sometimes other data such as composition and surface features. Some of the notable lists and catalogs include: 1. **Main Belt Asteroids**: A list of asteroids primarily located in the asteroid belt between Mars and Jupiter.

Full set of all possible special relativity symmetries:

In simple and concrete terms. Suppose you observe N particles following different trajectories in Spacetime.

There are two observers traveling at constant speed relative to each other, and so they see different trajectories for those particles:Note that the first two types of transformation are exactly the non-relativistic Galilean transformations.

The Poincare group is the set of all matrices such that such a relationship like this exists between two frames of reference.

Subset of Galilean transformation with speed equals 0.

This is a good and simple first example of Lie algebra to look into.

Take the group of all Translation in ${\mathbb{R}}^1$.

Let's see how the generator of this group is the derivative operator:

The way to think about this is:

So let's take the exponential map:and we notice that this is exactly the Taylor series of $f(x)$ around the identity element of the translation group, which is 0! Therefore, if $f(x)$ behaves nicely enough, within some radius of convergence around the origin we have for finite $x_0$:

This example shows clearly how the exponential map applied to a (differential) operator can generate finite (non-infinitesimal) Translation!

Physics from Symmetry by Jakob Schwichtenberg (2015) page 66 shows one in terms of 4x4 complex matrices.

More importantly though, are the representations of the Lie algebra of the Lorentz group, which are generally also just also called "Representation of the Lorentz group" since you can reach the representation from the algebra via the exponential map.

Bibliography:

One of the representations of the Lorentz group that show up in the Representation theory of the Lorentz group.

This was getting very hot as of 2022 for some reason. Would be good to understand why besides the awesome name.

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