## Ideals

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)

Originally posted on 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…

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