Graded Lie algebra

In the sign convention used here, a graded Lie algebra over a field of characteristic zero is a vector space $L={⨁}_{r\in \mathbb{Z}}{L}_r$ with a bilinear map $[-,-]:L_r\times L_s\to L_{r+s}$ satisfyingfor $u\in L_r$, $v\in L_s$ and homogeneous $w$. Thus two odd-degree elements have a symmetric graded Lie bracket, whereas an even-degree element has the ordinary antisymmetry rule. For a graded associative algebra, $[u,v]=uv-(-1{)}^{rs}vu$ supplies an example; expanding its products proves the graded Jacobi identity. Reducing degrees modulo two gives a Lie superalgebra. A Gerstenhaber algebra uses this sign rule after shifting degrees by one.

A different common convention uses an ordinary Lie algebra with a grading satisfying $[L_r,L_s]\subseteq L_{r+s}$ and no degree signs. The loop algebra is an example of that ordinary graded convention. The degree convention must therefore be specified before applying a graded Lie bracket formula.

For a positively graded Lie algebra $L$ over $\mathbb{Q}$, finite-dimensional in each degree, its cochain algebra is free graded commutative on the graded dual of $sL$, with differential dual to the Lie bracket and its Koszul signs. The Jacobi identity gives $d^2=0$. A Lie element of degree $r$ contributes a cochain generator of degree $r+1$. Its cohomology computes ${\mathrm{Ext}}_{U(L)}(\mathbb{Q},\mathbb{Q})$ with total grading; the standard free enveloping-algebra resolution gives this identification. For a free graded Lie algebra, the tensor-algebra augmentation has a length-one free resolution, explaining the square-zero cohomology of a rational wedge sum of spheres.

For a positively graded vector space $V$, the free graded Lie algebra $\mathbb{L}(V)$ receives a linear map from $V$ and extends every graded linear map $V\to L$ uniquely to a Lie homomorphism. Over $\mathbb{Q}$, it is the Lie subalgebra generated by $V$ inside the tensor algebra $T(V)$, using $[u,v]=uv-(-1{)}^{|u||v|}vu$. Its universal enveloping algebra is $T(V)$: both have the same universal property for maps from $V$ into associative algebras. The primitives of the tensor Hopf algebra with primitive generators are exactly $\mathbb{L}(V)$.

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 $s=|v|-1$.