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

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

Created on May 21, 2019 at 13:13:01. See the history of this page for a list of all contributions to it.