Graded Lie algebra

Graded Lie superalgebras

A graded Lie superalgebra over a field k (not of characteristic 2) consists of a graded vector space E over k, along with a bilinear bracket operation

${\displaystyle [-,-]:E{\otimes }_{k}E\to E}$

such that the following axioms are satisfied.

• [-, -] respects the gradation of E: $${\displaystyle [E_i,E_j]\subseteq E_{i+j}.}$$
• (Symmetry) For all x in E_i and y in E_j, $${\displaystyle [x,y]=-(-1{)}^{ij}[y,x]}$$
• (Jacobi identity) For all x in E_i, y in E_j, and z in E_k, $${\displaystyle (-1{)}^{ik}[x,[y,z]]+(-1{)}^{ij}[y,[z,x]]+(-1{)}^{jk}[z,[x,y]]=0.}$$ (If k has characteristic 3, then the Jacobi identity must be supplemented with the condition ${\displaystyle [x,[x,x]]=0}$ for all x in E_odd.)

Note, for instance, that when E carries the trivial gradation, a graded Lie superalgebra over k is just an ordinary Lie algebra. When the gradation of E is concentrated in even degrees, one recovers the definition of a (Z-)graded Lie algebra.

Examples and Applications

The most basic example of a graded Lie superalgebra occurs in the study of derivations of graded algebras. If A is a graded k-algebra with gradation

${\displaystyle A=\underset{i\in \mathbb{Z}}{⨁}{A}_i,}$

then a graded k-derivation d on A of degree l is defined by

1. ${\displaystyle dx=0}$ for ${\displaystyle x\in k}$,
2. ${\displaystyle d:A_i\to A_{i+l}}$, and
3. ${\displaystyle d(xy)=(dx)y+(-1{)}^{il}x(dy)}$ for ${\displaystyle x\in A_i}$.

The space of all graded derivations of degree l is denoted by ${\displaystyle {\mathrm{Der}}_l(A)}$, and the direct sum of these spaces,

${\displaystyle \mathrm{Der}(A)=\underset{l}{⨁}{\mathrm{Der}}_l(A),}$

carries the structure of an A-module. This generalizes the notion of a derivation of commutative algebras to the graded category.

On Der(A), one can define a bracket via:

[d, δ ] = dδ − (−1)^ijδd, for d ∈ Der_i (A) and δ ∈ Der_j (A).

Equipped with this structure, Der(A) inherits the structure of a graded Lie superalgebra over k.

Further examples:

• The Frölicher–Nijenhuis bracket is an example of a graded Lie algebra arising naturally in the study of connections in differential geometry.
• The Nijenhuis–Richardson bracket arises in connection with the deformations of Lie algebras.

Generalizations

The notion of a graded Lie superalgebra can be generalized so that their grading is not just the integers. Specifically, a signed semiring consists of a pair ${\displaystyle (\mathrm{\Gamma },ϵ)}$, where ${\displaystyle \mathrm{\Gamma }}$ is a semiring and ${\displaystyle ϵ:\mathrm{\Gamma }\to \mathbb{Z}/2\mathbb{Z}}$ is a homomorphism of additive groups. Then a graded Lie supalgebra over a signed semiring consists of a vector space E graded with respect to the additive structure on ${\displaystyle \mathrm{\Gamma }}$, and a bilinear bracket [-, -] which respects the grading on E and in addition satisfies:

1. ${\displaystyle [x,y]=-(-1{)}^{ϵ(\mathrm{deg}x)ϵ(\mathrm{deg}y)}[y,x]}$ for all homogeneous elements x and y, and
2. ${\displaystyle (-1{)}^{ϵ(\mathrm{deg}x)ϵ(\mathrm{deg}z)}[x,[y,z]]+(-1{)}^{ϵ(\mathrm{deg}y)ϵ(\mathrm{deg}x)}[y,[z,x]]+(-1{)}^{ϵ(\mathrm{deg}z)ϵ(\mathrm{deg}y)}[z,[x,y]]=0.}$

Further examples:

• A Lie superalgebra is a graded Lie superalgebra over the signed semiring ${\displaystyle (\mathbb{Z}/2\mathbb{Z},ϵ)}$, where ${\displaystyle ϵ}$ is the identity map for the additive structure on the ring ${\displaystyle \mathbb{Z}/2\mathbb{Z}}$.