We choose a basis for the diagonal matrices as follows: denote E_{i,j} as the matrix with a 1 in the i,j’th position and 0’s everywhere else. For diagonal matrices the basis is h_i=E_{i,i}-E_{i+1,i+1}. Now we want to calculate the Killing form for these matrices, so we choose a diagonal matrix H, with diagonal entries e_i(H) for some set of diagonal entries. The commutator [H,E_{i,j}]=(e_i(H)-e_j(H))E_{i,j}, shows us that the Tr(ad_had_h) can be written as the following sum: [k(H,H)=sum_{i,j}(e_i(H)-e_j(H))^2]which we can expand the...
We choose a basis for the diagonal matrices as follows: denote E_{i,j} as the matrix with a 1 in the i,j’th position and 0’s everywhere else. For diagonal matrices the basis is h_i=E_{i,i}-E_{i+1,i+1}. Now we want to calculate the Killing form for these matrices, so we choose a diagonal matrix H, with diagonal entries e_i(H) for some set of diagonal entries.
The commutator [H,E_{i,j}]=(e_i(H)-e_j(H))E_{i,j}, shows us that the Tr(ad_had_h) can be written as the following sum:
[k(H,H)=sum_{i,j}(e_i(H)-e_j(H))^2]which we can expand the bracket and double sum to give
[k(H,H)=sum_{i,j}e_i(H)^2+e_j(H)^2-2e_i(H)e_j(H)]
Now we note the first two terms can combine to give 2nTr(H^2), and the third term in the sum has all e_j(H) terms multiplied by a factor of e_1(H)+cdots +e_n(H), but we know that H in mathfrak{sl}_n is traceless, so this term cancels to 0. Thus we can conclude (k(H,H)=2nTr(H^2)), and due to the semisimplicity of mathfrak{sl}_n this holds for all x,y in mathfrak{sl}_n so k(x,y)=2nTr(xy).
chapter{Cartan Subalgebras}
epigraph{textit{“We’re gonna need a bigger Subalgebra…”}}{Martin Brody, Jaws}
Now we have seen what a textit{semisimple} Lie algebra is, we shall define Cartan Subalgebras which will pave the way for us to examine the roots of a Lie algebra, and from this we can build up a system of classification. Note for the remainder of the report we will take the ground field to be mathbb{C}.
section{Existence and Uniqueness of Cartan subalgebras}
begin{framed}
begin{defn} A Cartan Subalgebra (mathfrak{h}) of a semisimple Lie algebra(mathfrak{g}) is the maximal Abelian subalgebra of (mathfrak{g}) such that every element of (mathfrak{h}) is semisimple.
end{defn}
end{framed}
