Let $I = \{1,2,\dots,n\}$ and $S \subset I$. The set $I$ will be indexing the simple roots and $S$ will be indexing the odd generators of a Lie superalgebra.
A real matrix $A=(a_{ij})_{i,j\in I}$ is said to be a generalized Cartan matrix if the following conditions are satisfied:
$A$ is symmetric;
$a_{ii}=2$ or $a_{ii}\leq 0$;
$a_{ij}\leq 0$ if $i\neq j$;
$\frac{2a_{ij}}{a_{ii}}\in\mathbb{Z}$ if $a_{ii}>0$;
$\frac{2a_{ij}}{a_{ii}}\in2\mathbb{Z}$ if $a_{ii}>0$ and $i \in S$.
$a_{ij}=0$ if and only if $a_{ji}=0$.
The Lie superalgebra $\mathfrak g = \mathfrak g(A)$ associated to the generalized Cartan matrix $A$ is the Lie superalgebra generated by $e_i, f_i, h_i, i \in I$ with the following defining relations:
$[h_i, h_j]=0$ for $i,j\in I$,
$[h_i, e_j]=a_{ij}e_j$, $[h_i, f_j]=-a_{ij}f_j$ for $i,j\in I$,
$[e_i, f_j]=\delta_{ij}h_i$ for $i, j\in I$,
$\deg h_i = 0, i \in I$,
$\deg e_i = 0 = \deg f_i$ if $i \notin S$,
$\deg e_i = 1 = \deg f_i$ if $i \in S$,
$(\text{ad }e_i)^{1-\frac{2a_{ij}}{a_{ii}}}e_j=0 = (\text{ad }f_i)^{1-\frac{2a_{ij}}{a_{ii}}}f_j$ if $a_{ii} > 0$ and $i \ne j$,
$(\text{ad }e_i)^{1-\frac{a_{ij}}{a_{ii}}}e_j=0 = (\text{ad }f_i)^{1-\frac{a_{ij}}{a_{ii}}}f_j$ if $i \in S$ and $a_{ii}>0$ and $i \ne j$,
$[e_i, e_j]= 0 = [f_i, f_j]$ if $a_{ij}=0$.
I have the following questions regarding the definition.
a) In the definition of the matrix $A$ what is the role of (5)? I have seen only condition (4) in the case of Lie algebras. Also why the powers in (7) and (8) are differed by a factor of 2?
b) How (5) in the definition of $A$ and (8) affects the structure of the root spaces of $\mathfrak g$?
c) What are some common differences between this structure and Lie algebras? For example which roots are odd roots?
Kindly explain to me with some examples. Thank you.
