Graded Lie algebra

A graded Lie algebra has a graded Lie bracket preserving total degree, with $[u,v]=-(-1{)}^{|u||v|}[v,u]$ and $[u,[v,w]]=[[u,v],w]+(-1{)}^{|u||v|}[v,[u,w]]$ for homogeneous vectors in the indicated degrees. A Gerstenhaber algebra uses this sign rule after shifting degrees by one.

The bracket on a graded Lie algebra is a bilinear operation satisfying graded antisymmetry and the graded Jacobi identity. For homogeneous elements of degrees $r,s$, $[u,v]=-(-1{)}^{rs}[v,u]$. A Gerstenhaber bracket uses this convention on the shifted degrees $r=|u|-1$ and $s=|v|-1$.