Ciro Santilli @cirosantilli  40

Maybe the most famous one is Mycoplasma genitalium byt there are others, and notably with lower biosafety levels:

This is a good first concrete example of a Lie algebra. Shown at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4.2 "How to linearize a Lie Group" has an example.

We can use use the following parametrization of the special linear group on variables $x$, $y$ and $z$:

Every element with this parametrization has determinant 1:Furthermore, any element can be reached, because by independently settting $x$, $y$ and $z$, $M_{00}$, $M_{01}$ and $M_{10}$ can have any value, and once those three are set, $M_{11}$ is fixed by the determinant.

To find the elements of the Lie algebra, we evaluate the derivative on each parameter at 0:

Remembering that the Lie bracket of a matrix Lie group is really simple, we can then observe the following Lie bracket relations between them:

One key thing to note is that the specific matrices $M_x$, $M_y$ and $M_z$ are not really fundamental: we could easily have had different matrices if we had chosen any other parametrization of the group.

TODO confirm: however, no matter which parametrization we choose, the Lie bracket relations between the three elements would always be the same, since it is the number of elements, and the definition of the Lie bracket, that is truly fundamental.

Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4.2 "How to linearize a Lie Group" then calculates the exponential map of the vector $x{M}_x+y{M}_y+z{M}_z$ as:with:

TODO now the natural question is: can we cover the entire Lie group with this exponential? Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation" explains why not.

This makes it clear how the Lie bracket can be seen as a "measure of non-commutativity"

Because the Lie bracket has to be a bilinear map, all we need to do to specify it uniquely is to specify how it acts on every pair of some basis of the Lie algebra.

Then, together with the Baker-Campbell-Hausdorff formula and the Lie group-Lie algebra correspondence, this forms an exceptionally compact description of a Lie group.

The one parameter subgroup of a Lie group for a given element $M$ of its Lie algebra is a subgroup of $G$ given by:

Intuitively, $M$ is a direction, and $t$ is how far we move along a given direction. This intuition is especially vivid in for example in the case of the Lie algebra of $SO(3)$, the rotation group.

One parameter subgroups can be seen as the continuous analogue to the cycle of an element of a group.

Example at: Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation".

Example at: Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation".

Furthermore, the non-compact part is always isomorphic to ${\mathbb{R}}^n$, only the non-compact part can have more interesting structure.

This is Ciro Santilli's favorite laptop brand. He's been on it since the early 2010's after he saw his then-girlfriend-later-wife using it.

Ciro doesn't know how to explain it, but ThinkPads just feel... right. The screen, the keyboard, the lid, the touchpad are all exactly what Ciro likes.

The only problem with ThinkPad is that it is owned by Lenovo which is a Chinese company, and that makes Ciro feel bad. But he likes it too much to quit... what to do?

Ciro is also reassured to see that in every enterprise he's been so far as of 2020, ThinkPads are very dominant. And the same when you see internal videos from other big tech enterprises, all those nerds are running... Ubuntu on ThinkPads! And the ISS.

Those nerds like their ThinkPads so much, that Ciro has seen some acquaintances with crazy old ThinkPad machines, missing keyboard buttons or the like. They just like their machines that much.

ThinkPads are are also designed for repairability, and it is easy to buy replacement parts, and there are OEM part replacement video tutorials: www.youtube.com/watch?v=vseFzFFz8lY No visible planned obsolescence here! With the caveat that the official online part stores can be shit as mentioned at Section "Lenovo".

Further more, in 2020 Lenovo is announced full certification for Ubuntu www.forbes.com/sites/jasonevangelho/2020/06/03/lenovos-massive-ubuntu-and-red-hat-announcement-levels-up-linux-in-2020/#28a8fd397ae0 which fantastic news!

The only thing Ciro never understood is the trackpoint: superuser.com/questions/225059/how-to-get-used-of-trackpoint-on-a-thinkpad Why would you use that with such an amazing touchpad? And vimium.

Highlighted at the Origins of Precision by Machine Thinking (2017).

In leanpub you write your book in a markdown variant they call Markua, marketed as "markdown for books".

TODO is there a reference implementation that runs locally for HTML output? Or the only reference implementation is closed under leanpub?

Spec: markua.com/