Carlos Castro Perelman – Lie Algebra (Lecture 1 of 6)
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Physicist Carlos Perelman gave a series of lectures on lie algebra to Quantum Gravity Research’s team of research scientists. We thought it would be useful for anyone interested in learning more about lie algebra.
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@henrikljungstrand2036
August 9, 2025 at 6:10 pm
At about 1:00:00 the Jacobi identity is necessary for our commutator algebra being a Lie algebra, which means it has an associative universal enveloping algebra.
There are generalizations though, like the Malcev algebras, that apply to commutators of alternative algebras, e.g. the Octonions.
All commutators of associative algebras support the Jacobi identity.
@henrikljungstrand2036
August 9, 2025 at 6:10 pm
About 57:00 the Quaternions are not a field, but they are a skew field, also called a division ring. You cannot do as much linear algebra over skew fields as you can over fields.
Going further, the Octonions would be an alternative field, also called a (unital) division algebra. Alternative fields are not generally skew fields, just like skew fields are not generally fields.
It is even trickier to do linear algebra over alternative fields, and you can do less of it than over skew fields.
Always we have [x + y, z] = [x, z] + [y, z] and [x, y + z] = [x, y] + [x, z].
For skew fields we have [ax, y] = ax*y – y*ax =/= ax*y – ay*x = a[x, y].
So no, it doesn't work that easily.
Perhaps we can tweak things with skew field conjugates, but it is hard. The "*" symbol is the associative operation of the universal enveloping algebra (or any quotient of it) of the Lie algebra.
We can "pull out" scalars on the left or on the right in associative algebras over skew fields, but not in Lie algebras over such skew fields.
I don't know if Lie algebras work at all over skew fields or alternative fields. Perhaps if you refer to the base field as the center of that skew field or alternative field.
@Evan2718281828
August 9, 2025 at 6:10 pm
53:45 "the bracket of your superposition is nothing but the superposition of brackets"
QGR gives away their interpretation of quantum theory
@Evan2718281828
August 9, 2025 at 6:10 pm
40:30 why is that "i", which is presumably indexing the theta on the far left side of the screen, a superscript and not a subscript? Is it an exponent, or otherwise what is that notation called?
@tommyrjensen
August 9, 2025 at 6:10 pm
I don't know if this is worthwhile. Already before 8 minutes the definition of identity element is clearly wrong, and the instructor does not remedy, even though the listeners apparently question it. Then the Monster group is not "one of the largest groups", it is the largest sporadic simple group, which makes it a little more special than that.
@QuantumGravityResearch
August 9, 2025 at 6:10 pm
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Is there an 8-dimensional "engine" behind our universe? Join Marion Kerr on a fun, visually exciting journey as she explores a mysterious, highly complex structure known simply as 'E8'–a weird, 8-dimensional object that for some, strange reason, appears to encode within it all of the particles and forces of our 3-dimensional universe.
@QuantumGravityResearch
August 9, 2025 at 6:10 pm
Keep your eyes peeled for our new E8 documentary – coming soon!
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@CandidDate
August 9, 2025 at 6:10 pm
I cannot tell a lie – physics needs some new approaches – does everything have to be 'orthogonal'? – having recently been convinced we are living inside a computer simulation casts doubts in my mind as to the continuity of space and time. I am thinking now more along the lines of discrete descriptions of reality – after all, it's all just 1s and 0s inside God's mind!
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