obtained by the $(m,n)$-block form of its elements. More precisely, $\mathrm{Mat}_{m,n}(\cA)_\eev$ consists of matrices in $(m,n)$-block form $$\label{blockformgg}
\begin{bmatrix}
A & B \\ 
C & D%
\end{bmatrix}%$$ such that the entries of the $m\times m$ matrix $A$ and the $n\times n$ matrix $D$ belong to $\cA_\eev$, whereas the entries of the matrices $B$ and $C$ belong to $\cA_\ood$. Similarly, $\mathrm{Mat}_{m,n}(\cA)_\ood$ consists of matrices of the form such that the entries of $A$ and $D$ belong to $\cA_\ood$, whereas the entries of $B$ and $C$ belong to $\cA_%
\eev$. The supertrace of an element of $\mathrm{Mat}_{m,n}(\cA)$ is defined by $$\str\left( 
\begin{bmatrix}
A & B \\ 
C & D%
\end{bmatrix}
\right)=\mathrm{tr}(A)-\mathrm{tr}(D).$$

Using the basis $\left\{\bfe_i\,:\,i\in\mathcal{I}_{m,n}\right\}$, we can represent elements of $\g g$ by $(m+n)\times (m+n)$ matrices, with rows and columns indexed by elements of $\mathcal{I}_{m,n}$. For every $i,j\in%
\mathcal{I}_{m,n}$, let $E_{i,j}$ denote the element of $\g g$ corresponding to the matrix with a 1 in the $(i,j)$-entry and 0’s elsewhere. The standard Cartan subalgebra of $\g g$ is $$\g h:=\spn_\C\{E_{1,1},\ldots,E_{m,m},E_{\oline 1,\oline 1},\ldots, E_{%
\oline n,\oline n}\},$$ and the standard characters of $\g h$ are $\eps_1,\ldots,\eps_m,\eps_{\oline %
1},\ldots,\eps_{\oline n}\in\g h^*$, where $\eps_i(E_{j,j})=\delta_{i,j} $ for every $i,j\in\mathcal{I}_{m,n} $. The standard root system of $\g 