A Lie superalgebra over a field of characteristic zero is a vector space 
with an even-odd decomposition
and a bilinear graded bracket respecting this decomposition. For homogeneous elements
of parities 
,
the bracket satisfies
![[x,y]=-(-1)^(|x||y|)[y,x]](/images/equations/LieSuperalgebra/NumberedEquation1.svg) |
and the graded Jacobi identity
![(-1)^(|x||z|)[x,[y,z]]+(-1)^(|y||x|)[y,[z,x]]+(-1)^(|z||y|)[z,[x,y]]=0.](/images/equations/LieSuperalgebra/NumberedEquation2.svg) |
In particular, the bracket of an odd element with itself need not vanish (Kac 1977). A graded derivation 
of parity 
satisfies ![D([u,v])=[D(u),v]+(-1)^(|D||u|)[u,D(v)]](/images/equations/LieSuperalgebra/Inline5.svg)
for homogeneous 
.
The free Lie superalgebra on graded generators is characterized by the property that
every assignment of those generators to elements of the same parity in another Lie
superalgebra extends uniquely to a parity-preserving homomorphism.
The odd Grothendieck-Teichmüller
Lie algebra is an example arising from infinitesimal symmetry relations (Naef
and Willwacher 2026).
