in Turing’s on permutations paper, he refers (upon editing) to normalizers, idealizers etc. We can get a feel for what these are:- (the ideal is a bit like the 0, in Hs = 0 for LDPCs)

The Unapologetic Mathematician

As we said, a homomorphism of Lie algebras is simply a linear mapping between them that preserves the bracket. I want to check, though, that this behaves in certain nice ways.

First off, there is a Lie algebra $latex 0$. That is, the trivial vector space can be given a (unique) Lie algebra structure, and every Lie algebra has a unique homomorphism $latex L\to0$ and a unique homomorphism $latex 0\to L$. This is easy.

Also pretty easy is the fact that we have kernels. That is, if $latex \phi:L\to L’$ is a homomorphism, then the set $latex I=\left\{x\in L\vert\phi(x)=0\in L’\right\}$ is a subalgebra of $latex L$. Indeed, it’s actually an “ideal” in pretty much the same sense as for rings. That is, if $latex x\in L$ and $latex y\in I$ then $latex [x,y]\in I$. And we can check that

$latex \displaystyle\phi\left([x,y]\right)=\left[\phi(x),\phi(y)\right]=\left[\phi(x),0\right]=0$

proving that $latex \mathrm{Ker}(\phi)\subseteq L$ is an…

View original post 361 more words


Computer Programmer who often does network administration with focus on security servers. Very strong in Microsoft Azure cloud!
This entry was posted in coding theory. Bookmark the permalink.