In its most basic form, a graded Lie algebra is an ordinary Lie algebra ${\displaystyle {\mathfrak {g}}}$ , together with a gradation of vector spaces

${\displaystyle {\mathfrak {g}}=\bigoplus _{i\in {\mathbb {Z} }}{\mathfrak {g}}_{i},}$ 

such that the Lie bracket respects this gradation:

${\displaystyle [{\mathfrak {g}}_{i},{\mathfrak {g}}_{j}]\subseteq {\mathfrak {g}}_{i+j}.}$ 

The universal enveloping algebra of a graded Lie algebra inherits the grading.

Examples edit

sl(2) edit

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.

Free Lie algebra edit

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.

Generalizations edit

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.

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 edit

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.

Generalizations edit

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 ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ .

• Nijenhuis, Albert; Richardson Jr., Roger W. (1966). "Cohomology and deformations in graded Lie algebras". Bulletin of the American Mathematical Society. 72 (1): 1–29. doi:10.1090/s0002-9904-1966-11401-5. MR 0195995.

• Differential graded Lie algebra
• Graded (mathematics)
• Lie algebra-valued form