This section is about ideas that are thought to be part of an AGI system.

archive.ph/Dd3aC web.archive.org/web/20230709141533/https://desuarchive.org/g/thread/94445084/#94448535 desuarchive.org/g/thread/94445084/#94448535

Title reply because they can't Ctrl+F: How Ciro Santilli manages to write so much

Most of the thread went into pro/anti gay trashtalk due to Ciro using Gay Putin at the time on his Stack Overflow profile as a useless way to protest the Russian invasion of Ukraine.

Some comments:

Reply: it is publicly known that Putin is homophobic as fuck and hates that picture. Therefore we use it. If Putin were heterophobic, we'd post him as hetero.

The only new information:

"More complex and general" integral. Matches the Riemann integral for "simple functions", but also works for some "funkier" functions that Riemann does not work for.

Ciro Santilli sometimes wonders how much someone can gain from learning this besides the beauty of mathematics, since we can hand-wave a Lebesgue integral on almost anything that is of practical use. The beauty is good reason enough though.

The actually had decimal time systems... why that didn't win!

This was Ciro Santilli's main study/work music for several years around 2020. Tabla rules.

so what's that point of "Open" in the name anymore??

Like everything else in Lie groups, first start with the matrix as discussed at Section "Lie algebra of a matrix Lie group".

Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map called the Lie bracket, that has to also be alternating and satisfy the Jacobi identity.

Another important way to think about Lie algebras, is as infinitesimal generators.

Because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.

To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:Note that the Lie bracket can look different under different basis of the Lie algebra however. This is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) page 71 for the Lorentz group.

As mentioned at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4 "Lie Algebras", taking the Lie algebra around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.

Bibliography:

Or equivalently, the set of rows is orthonormal, and so is the set of columns. TODO proof that it is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.