Recent content by mma

Perhaps this is not important either, but another interesting fact for me is the multiplication table of the members of the decomposition ##M(2,\mathbb C)=\mathbb RI\oplus\mathfrak E_3\oplus su(2)\oplus i\mathbb RI##.

Of course, it isn't a direct sum. Direct sum applies only above the main diagonal. Below it, the objects originate from the green arrow.

Yes, it is a sphere in the vector space ##\mathbb H##. At the ends of the green arrows are spheres in the vector space at the beginning of that arrow.

Oh, yes, I consider them as real vector spaces. So, ##su(2)\neq isu(2).##

I regard them as elements of the (additive) vector space of 2 by 2 matrices.

I think, elements of ##su(2)## are the traceless anti-Hermitian matrices while elements of ##isu(2)## are traceless, Hermitian ones, so ##su(2)\oplus \mathfrak{E}_3=su(2)\oplus isu(2)## consists of all traceless matrices.

All the other objects in the picture have their own names, designations, and significance somewhere. I wonder if this one does too, or is it an exception?

One more question. This is the structure of the set of 2 by 2 complex matrices: Here, the only non-standard symbol is ##\mathfrak E_3=isu(2)##. It is the 3-dimensional real Euclidean space of traceless, Hermitian matrices, with the half-anticommutator as dot product. What should I write instead...

You can call it a definition, but it is still a new equation in addition to Maxwell's equations.

Thanks, I've found them here: https://archive.org/details/onmanifoldswitha0000cart/page/174/mode/2up

TL;DR Summary: I have a bad copy of book "On manifolds with an affine connection and the theory of general relativity"by Elie Cartan. Pages 36, 142 and 174 are partiy missing. I have a bad copy of Elie Cartan's book "On manifolds with an affine connection and the theory of general relativity"...

Thanks!

Yes, of course. What else?

Exactly.

It seems that FAQ is visible only when I'm not logged in. But this answer at least misses the point if not misleading there. The problem is with this answer that it describes the tangent bundle ##T\mathcal G## of the Galilean spacetime ##\mathcal G##, not ##\mathcal G## itself as a fiber...