In mathematics, a restricted Lie algebra (or p-Lie algebra) is a Lie algebra over a field of characteristic p>0 together with an additional "pth power" operation. Most naturally occurring Lie algebras in characteristic p come with this structure, because the Lie algebra of a group scheme over a field of characteristic p is restricted.

Let ${\displaystyle {\mathfrak {g}}}$ be a Lie algebra over a field k of characteristic p>0. The adjoint representation of ${\displaystyle {\mathfrak {g}}}$ is defined by ${\displaystyle ({\text{ad }}X)(Y)=[X,Y]}$ for ${\displaystyle X,Y\in {\mathfrak {g}}}$. A p-mapping on ${\displaystyle {\mathfrak {g}}}$ is a function from ${\displaystyle {\mathfrak {g}}}$ to itself, ${\displaystyle X\mapsto X^{[p]}}$, satisfying:

Nathan Jacobson (1937) defined a restricted Lie algebra over k to be a Lie algebra over k together with a p-mapping. A Lie algebra is said to be restrictable if it has at least one p-mapping. By the first property above, in a restricted Lie algebra, the derivation ${\displaystyle (\mathrm {ad} \;X)^{p}}$ of ${\displaystyle {\mathfrak {g}}}$ is inner for each ${\displaystyle X\in {\mathfrak {g}}}$. In fact, a Lie algebra is restrictable if and only if the derivation ${\displaystyle (\mathrm {ad} \;X)^{p}}$ of ${\displaystyle {\mathfrak {g}}}$ is inner for each ${\displaystyle X\in {\mathfrak {g}}}$.

For example:

For an associative algebra A over a field k of characteristic p>0, the commutator ${\displaystyle [X,Y]:=XY-YX}$ and the p-mapping ${\displaystyle X^{[p]}:=X^{p}}$ make A into a restricted Lie algebra. In particular, taking A to be the ring of n x n matrices shows that the Lie algebra ${\displaystyle {\mathfrak {gl}}(n)}$ of n x n matrices over k is a restricted Lie algebra, with the p-mapping being the pth power of a matrix. This "explains" the definition of a restricted Lie algebra: the complicated formula for ${\displaystyle (X+Y)^{[p]}}$ is needed to express the pth power of the sum of two matrices over k, ${\displaystyle (X+Y)^{p}}$, given that X and Y typically do not commute.

Let A be an algebra over a field k. (Here A is a possibly non-associative algebra.) Then the derivations of A over k form a Lie algebra ${\displaystyle {\text{Der}}_{k}(A)}$, with the Lie bracket being the commutator, ${\displaystyle [D_{1},D_{2}]:=D_{1}D_{2}-D_{2}D_{1}}$. When k has characteristic p>0, then iterating a derivation p times yields a derivation, and this makes ${\displaystyle {\text{Der}}_{k}(A)}$ into a restricted Lie algebra. If A has finite dimension as a vector space, then ${\displaystyle {\text{Der}}_{k}(A)}$ is the Lie algebra of the automorphism group scheme of A over k; that indicates why spaces of derivations are a natural way to construct Lie algebras.

Let G be a group scheme over a field k of characteristic p>0, and let ${\displaystyle \mathrm {Lie} (G)}$ be the Zariski tangent space at the identity element of G. Then ${\displaystyle \mathrm {Lie} (G)}$ is a restricted Lie algebra over k. This is essentially a special case of the previous example. Indeed, each element X of ${\displaystyle \mathrm {Lie} (G)}$ determines a left-invariant vector field on G, and hence a left-invariant derivation on the ring of regular functions on G. The pth power of this derivation is again a left-invariant derivation, hence the derivation associated to an element ${\displaystyle X^{[p]}}$ of ${\displaystyle \mathrm {Lie} (G)}$. Conversely, every restricted Lie algebra of finite dimension over k is the Lie algebra of a group scheme. In fact, ${\displaystyle G\mapsto \mathrm {Lie} (G)}$ is an equivalence of categories from finite group schemes G of height at most 1 over k (meaning that ${\displaystyle f^{p}=0}$ for all regular functions f on G that vanish at the identity element) to restricted Lie algebras of finite dimension over k.

In a sense, this means that Lie theory is less powerful in positive characteristic than in characteristic zero. In characteristic p>0, the multiplicative group ${\displaystyle G_{m}}$ (of dimension 1) and its finite subgroup scheme ${\displaystyle \mu _{p}=\{x\in G_{m}:x^{p}=1\}}$ have the same restricted Lie algebra, namely the vector space k with the p-mapping ${\displaystyle a^{[p]}=a^{p}}$. More generally, the restricted Lie algebra of a group scheme G over k only depends on the kernel of the Frobenius homomorphism on G, which is a subgroup scheme of height at most 1. For another example, the Lie algebra of the additive group ${\displaystyle G_{a}}$ is the vector space k with p-mapping equal to zero. The corresponding Frobenius kernel is the subgroup scheme ${\displaystyle \alpha _{p}=\{x\in G_{a}:x^{p}=0\}.}$