Zoran Skoda gl(m|n)

A Lie superalgebra is a nonassociative algebra with bilinear bracket $[,]$ satisfying the superskewsymmetry

$$[x,y]=-(-1)^{|x| |y|}[y,x].$$

and the super Jacobi identity is

$$(-1)^{|x||z|}[x, [y, z]] + (-1)^{|y||x|}[y, [z, x]] + (-1)^{|z||y|}[z, [x, y]] = 0.$$

If $A$ is an associative algebra, the supercommutator of two homogeneous elements $x,y$ with degrees $|x|$, $|y|$ is given by

$$[ x, y ] = x y - (-1)^{|x| |y|} y x.$$

This way we get a Lie superalgebra from an associative algebra and the following Leibniz rule mixing the original product and the supercommutator holds

$$[x, y z] = [x, y] z + (-1)^{|x| |y|} y [x, z].$$

Lie superalgebra $gl(m|n)$ has generators $\hat{E}_{i j}$, where $i,j = 1,\ldots,m+n$, and the degree of $\hat{E}_{i j}$ is equal to $|i|-|j|$, where $|i| = 0$ if $i\leq m$ and $|i| = 1$ if $j\gt m$. The defining commutators are

$$[ \hat{E}_{i j} , \hat{E}_{k l} ] = \hat{E}_{i l} \delta_{j k} - (-1)^{(|i|-|j|)(|k|-|l|)}\hat{E}_{k j}\delta_{i l}$$