Graded Lie algebra

For example, the Lie algebra ${\displaystyle {\mathfrak {sl}}(2)}$ of trace-free 2 × 2 matrices is graded by the generators:

${\displaystyle X={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad Y={\begin{pmatrix}0&0\\1&0\end{pmatrix}},\quad {\textrm {and}}\quad H={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}$

These satisfy the relations ${\displaystyle [X,Y]=H}$, ${\displaystyle [H,X]=2X}$, and ${\displaystyle [H,Y]=-2Y}$. Hence with ${\displaystyle {\mathfrak {g}}_{-1}={\textrm {span}}(X)}$, ${\displaystyle {\mathfrak {g}}_{0}={\textrm {span}}(H)}$, and ${\displaystyle {\mathfrak {g}}_{1}={\textrm {span}}(Y)}$, the decomposition ${\displaystyle {\mathfrak {sl}}(2)={\mathfrak {g}}_{-1}\oplus {\mathfrak {g}}_{0}\oplus {\mathfrak {g}}_{1}}$ presents ${\displaystyle {\mathfrak {sl}}(2)}$ as a graded Lie algebra.

The free Lie algebra on a set X naturally has a grading, given by the minimum number of terms needed to generate the group element. This arises for example as the associated graded Lie algebra to the lower central series of a free group.

If ${\displaystyle \Gamma }$ is any commutative monoid, then the notion of a ${\displaystyle \Gamma }$-graded Lie algebra generalizes that of an ordinary (${\displaystyle \mathbb {Z} }$-) graded Lie algebra so that the defining relations hold with the integers ${\displaystyle \mathbb {Z} }$ replaced by ${\displaystyle \Gamma }$. In particular, any semisimple Lie algebra is graded by the root spaces of its adjoint representation.

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

${\displaystyle A=\bigoplus _{i\in \mathbb {Z} }A_{i},}$

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

1. ${\displaystyle dx=0}$ for ${\displaystyle x\in k}$,
2. ${\displaystyle d\colon A_{i}\to A_{i+l}}$, 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 ${\displaystyle \operatorname {Der} _{l}(A)}$, and the direct sum of these spaces,

${\displaystyle \operatorname {Der} (A)=\bigoplus _{l}\operatorname {Der} _{l}(A),}$

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.

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 ${\displaystyle (\Gamma ,\epsilon )}$, where ${\displaystyle \Gamma }$ is a semiring and ${\displaystyle \epsilon \colon \Gamma \to \mathbb {Z} /2\mathbb {Z} }$ 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 ${\displaystyle \Gamma }$, and a bilinear bracket [-, -] which respects the grading on E and in addition satisfies:

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

Further examples:

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