Lebesgue integral by Ciro Santilli 37 Updated 2025-07-16
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.
Complex dot product by Ciro Santilli 37 Updated 2025-07-16
This section is about the definition of the dot product over , which extends the definition of the dot product over .
The complex dot product is defined as:
E.g. in :
We can see therefore that this is a form, and a positive definite because:
The 3D regular convex polyhedrons are super famous, have the name: Platonic solid, and have been known since antiquity. In particular, there are only 5 of them.
The counts per dimension are:
DimensionCount
2Infinite
35
46
>43
The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:Then, the 2 3D missing ones have 4D analogues and the sixth one in 4D does not have a 3D analogue: the 24-cell. Yes, this is the kind of irregular stuff Ciro Santilli lives for.
One parameter subgroup by Ciro Santilli 37 Updated 2025-07-16
The one parameter subgroup of a Lie group for a given element of its Lie algebra is a subgroup of given by:
Intuitively, is a direction, and 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 , the rotation group.
One parameter subgroups can be seen as the continuous analogue to the cycle of an element of a group.
General linear group by Ciro Santilli 37 Updated 2025-07-16
Invertible matrices. Or if you think a bit more generally, an invertible linear map.
When the field is not given, it defaults to the real numbers.
Non-invertible are excluded "because" otherwise it would not form a group (every element must have an inverse). This is therefore the largest possible group under matrix multiplication, other matrix multiplication groups being subgroups of it.
When viewed as matrices, it is the group of all matrices that preserve the dot product, i.e.:
This implies that it also preserves important geometric notions such as norm (intuitively: distance between two points) and angles.
This is perhaps the best "default definition".
Note that:
and for that to be true for all possible and then we must have:
i.e. the matrix inverse is equal to the transpose.
Conversely, if:
is true, then
These matricese are called the orthogonal matrices.
TODO is there any more intuitive way to think about this?
Lie algebra of by Ciro Santilli 37 Updated 2025-07-16
We can reach it by taking the rotations in three directions, e.g. a rotation around the z axis:
then we derive and evaluate at 0:
therefore represents the infinitesimal rotation.
Note that the exponential map reverses this and gives a finite rotation around the Z axis back from the infinitesimal generator :
Repeating the same process for the other directions gives:
We have now found 3 linearly independent elements of the Lie algebra, and since has dimension 3, we are done.

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