Suppose we want to construct a semisimple Lie algebra $L$ of dimension $n$.
We know that if we can write $L = L_1 \oplus L_2 \oplus \cdots \oplus L_s$ as a direct sum of simple ideals, then $L$ must be semisimple.
I am not quite sure of what simple ideals we could use here to get a Lie algebra for a given dimension, any suggestions? Thanks.
Best Answer
We have $\dim(A_1)=\dim \mathfrak{sl}_2(\Bbb C)=3$ and $\dim(A_2)=\dim \mathfrak{sl}_3(\Bbb C)=8$. By the Frobenius coin problem we know that we can obtain every integer $n\ge 3\cdot 8-3-8=13$ as a linear combination $3x+8y=n$, i.e., direct sums of $A_1$ and $A_2$ yield a semisimple Lie algebra of every dimension $n\ge 13$. For dimensions less than $13$ we can check by hand that only $n=1,2,4,5,7$ cannot arise.