# Restricted Lie algebra

In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "p operation."

## Definition

Let L be a Lie algebra over a field k of characteristic p>0. A p operation on L is a map ${\displaystyle X\mapsto X^{[p]}}$ satisfying

• ${\displaystyle \mathrm {ad} (X^{[p]})=\mathrm {ad} (X)^{p}}$ for all ${\displaystyle X\in L}$,
• ${\displaystyle (tX)^{[p]}=t^{p}X^{[p]}}$ for all ${\displaystyle t\in k,X\in L}$,
• ${\displaystyle (X+Y)^{[p]}=X^{[p]}+Y^{[p]}+\sum _{i=1}^{p-1}{\frac {s_{i}(X,Y)}{i}}}$, for all ${\displaystyle X,Y\in L}$, where ${\displaystyle s_{i}(X,Y)}$ is the coefficient of ${\displaystyle t^{i-1}}$ in the formal expression ${\displaystyle \mathrm {ad} (tX+Y)^{p-1}(X)}$.

If the characteristic of k is 0, then L is a restricted Lie algebra where the p operation is the identity map.

## Examples

For any associative algebra A defined over a field of characteristic p, the bracket operation ${\displaystyle [X,Y]:=XY-YX}$ and p operation ${\displaystyle X^{[p]}:=X^{p}}$ make A into a restricted Lie algebra ${\displaystyle \mathrm {Lie} (A)}$.

Let G be an algebraic group over a field k of characteristic p, and ${\displaystyle \mathrm {Lie} (G)}$ be the Zariski tangent space at the identity element of G. Each element of ${\displaystyle \mathrm {Lie} (G)}$ uniquely defines a left-invariant vector field on G, and the commutator of vector fields defines a Lie algebra structure on ${\displaystyle \mathrm {Lie} (G)}$ just as in the Lie group case. If p>0, the Frobenius map ${\displaystyle x\mapsto x^{p}}$ defines a p operation on ${\displaystyle \mathrm {Lie} (G)}$.

## Restricted universal enveloping algebra

The functor ${\displaystyle A\mapsto \mathrm {Lie} (A)}$ has a left adjoint ${\displaystyle L\mapsto U^{[p]}(L)}$ called the restricted universal enveloping algebra. To construct this, let ${\displaystyle U(L)}$ be the universal enveloping algebra of L forgetting the p operation. Letting I be the two-sided ideal generated by elements of the form ${\displaystyle x^{p}-x^{[p]}}$, we set ${\displaystyle U^{[p]}(L)=U(L)/I}$. It satisfies a form of the PBW theorem.