Ciro Santilli @cirosantilli 37

We can almost reach the Lie algebra of any isometry group in a single go. For every $X$ in the Lie algebra we must have:because $e^{tX}$ has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".

Then:so we reach:With this relation, we can easily determine the Lie algebra of common isometries:

Bibliography:

For this sub-case, we can define the Lie algebra of a Lie group $G$ as the set of all matrices $M\in G$ such that for all $t\in \mathbb{R}$:If we fix a given $M$ and vary $t$, we obtain a subgroup of $G$. This type of subgroup is known as a one parameter subgroup.

The immediate question is then if every element of $G$ can be reached in a unique way (i.e. is the exponential map a bijection). By looking at the matrix logarithm however we conclude that this is not the case for real matrices, but it is for complex matrices.

Examples:

TODO example it can be seen that the Lie algebra is not closed matrix multiplication, even though the corresponding group is by definition. But it is closed under the Lie bracket operation.

The author seems to have uploaded the entire book by chapters at: www.physics.drexel.edu/~bob/LieGroups.html

And the author is the cutest: www.physics.drexel.edu/~bob/Personal.html.

Overview:

TODO there was one which was relly good, can't find it anymore. One day.

The term is not very clear, as it could either mean:

We define it as a linear map where the domain is the same as the image, i.e. an endofunction.

Examples:

This is the standard model.

Example: llvm/hello.ll adapted from: llvm.org/docs/LangRef.html#module-structure but without double newline.

To execute it as mentioned at github.com/dfellis/llvm-hello-world we can either use their crazy assembly interpreter, tested on Ubuntu 22.10:This seems to use puts from the C standard library.

Or we can Lower it to assembly of the local machine:which produces:and then we can assemble link and run with gcc:or with clang:hello.s uses the GNU GAS format, which clang is highly compatible with, so both should work in general.

TODO. I think this is the key point. Notably, $U(1)$ symmetry implies charge conservation.

More precisely, each generator of the corresponding Lie algebra leads to one separate conserved current, such that a single symmetry can lead to multiple conserved currents.

This is basically the local symmetry version of Noether's theorem.

Then to maintain charge conservation, we have to maintain local symmetry, which in turn means we have to add a gauge field as shown at Video "Deriving the qED Lagrangian by Dietterich Labs (2018)".

Forces can then be seen as kind of a side effect of this.

Bibliography: