Hi I just start learning Lie algebra and there is one hw question I don't really understand how to do, hope somebody give me some hints.
$L_1,L_2$ are Lie algebras. $L=\{(x_1,x_2):x_i \in L_i\}$. Lie bracket of $L$ is $$[(x_1,x_2),(y_1,y_2)]=([x_1,y_1],[x_2,y_2])$$ Call $L$ the direct sum of $L_1,L_2$.
The question is to prove $gl(2,\mathbb C)$ is isomorphic to the direct sum of $sl(2,\mathbb C)$ with $\mathbb C$, the 1-dimensional complex abelian Lie algebra.
I have think about this for so long, but couldn't find a function that keeps the bracket structure. Can somebody help me with this. Thank you very much.
Best Answer
The $4$-dimensional Lie algebra $\mathfrak{gl}_2(\mathbb{C})$ has a basis $e_1=E_{12}$, $e_2=E_{21}$, $e_3=E_{11}-E_{22}$ amd $e_4=E_{11}+E_{22}=I_2$, where $E_{ij}$ denotes the matrix with entry $1$ at position $(i,j)$ and zero entry otherwise. The Lie bracket is given by matrix commutator. Obviously we have $[e_1,e_2]=e_3$, $[e_1,e_3]=-2e_1$ and $[e_2,e_3]=2e_2$. Hence $\langle e_1,e_2,e_3\rangle$ is an ideal isomorphic to $\mathfrak{sl}_2(\mathbb{C})$. The center $Z$ of $\mathfrak{gl}_2(\mathbb{C})$, and ideal also, is clearly given by $\langle e_4\rangle$. Since $(e_1,e_2,e_3,e_4)$ is a basis, and $e_4$ has trivial bracket with $e_1,e_2,e_3$ it follows that $$ \mathfrak{gl}_2(\mathbb{C})\cong \mathfrak{sl}_2(\mathbb{C})\oplus Z \cong \mathfrak{sl}_2(\mathbb{C})\oplus \mathbb{C}. $$