Thursday, August 9, 2012

Lie Algebras Revisited

Lie Algebras Revisited:
Well it’s been quite a while, but I think I can carve out the time to move forwards again. I was all set to start with Lie algebras today, only to find that I’ve already defined them over a year ago. So let’s pick up with a recap: a Lie algebra is a module — usually a vector space over a field \mathbb{F} — called L and give it a bilinear operation which we write as [x,y]. We often require such operations to be associative, but this time we impose the following two conditions:
\displaystyle\begin{aligned}{}[x,x]&=0\\ [x,[y,z]]+[y,[z,x]]+[z,[x,y]]&=0\end{aligned}
Now, as long as we’re not working in a field where 1+1=0 — and usually we’re not — we can use bilinearity to rewrite the first condition:
\displaystyle\begin{aligned}0&=[x+y,x+y]\\&=[x,x]+[x,y]+[y,x]+[y,y]\\&=0+[x,y]+[y,x]+0\\&=[x,y]+[y,x]\end{aligned}
so [y,x]=-[x,y]. This antisymmetry always holds, but we can only go the other way if the character of \mathbb{F} is not 2, as stated above.
The second condition is called the “Jacobi identity”, and antisymmetry allows us to rewrite it as:
\displaystyle[x,[y,z]]=[[x,y],z]+[y,[x,z]]
That is, bilinearity says that we have a linear mapping x\mapsto[x,\underline{\hphantom{X}}] that sends an element x\in L to a linear endomorphism in \mathrm{End}(L). And the Jacobi identity says that this actually lands in the subspace \mathrm{Der}(L) of “derivations” — those which satisfy something like the Leibniz rule for derivatives. To see what I mean, compare to the product rule:
\displaystyle\frac{d}{dt}\left(fg\right)=\frac{df}{dt}g+f\frac{dg}{dt}
where f takes the place of y, g takes the place of z, and \frac{d}{dt} takes the place of x. And the operations are changed around. But you should see the similarity.
Lie algebras obviously form a category whose morphisms are called Lie algebra homomorphisms. Just as we might expect, such a homomorphism is a linear map \phi:L\to L' that preserves the bracket:
\displaystyle\phi\left([x,y]\right)=\left[\phi(x),\phi(y)\right]
We can obviously define subalgebras and quotient algebras. Subalgebras are a bit more obvious than quotient algebras, though, being just subspaces that are closed under the bracket. Quotient algebras are more commonly called “homomorphic images” in the literature, and we’ll talk more about them later.
We will take as a general assumption that our Lie algebras are finite-dimensional, though infinite-dimensional ones absolutely exist and are very interesting.
And I’ll finish the recap by reminding you that we can get Lie algebras from associative algebras; any associative algebra (A,\cdot) can be given a bracket defined by
\displaystyle [x,y]=x\cdot y-y\cdot x
The above link shows that this satisfies the Jacobi identity, or you can take it as an exercise.



DIGITAL JUICE
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