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

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

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

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

[E^ ij,E^ kl]=E^ ilδ jk(1) (|i||j|)(|k||l|)E^ kjδ il [ \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.