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.
By the first property above, in a restricted Lie algebra, the derivation
In fact, a Lie algebra is restrictable if and only if the derivation
[2] For example: For an associative algebra A over a field k of characteristic p>0, the commutator
[1] In particular, taking A to be the ring of n x n matrices shows that the Lie algebra
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
is needed to express the pth power of the sum of two matrices over k,
be the Zariski tangent space at the identity element of G. Then
is a restricted Lie algebra over k.[3] This is essentially a special case of the previous example.
is an equivalence of categories from finite group schemes G of height at most 1 over k (meaning that
for all regular functions f on G that vanish at the identity element) to restricted Lie algebras of finite dimension over k.[4] In a sense, this means that Lie theory is less powerful in positive characteristic than in characteristic zero.
have the same restricted Lie algebra, namely the vector space k with the p-mapping
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.
The corresponding Frobenius kernel is the subgroup scheme
of vector fields on X is a restricted Lie algebra over k. (If X is affine, so that
for a commutative k-algebra A, this is the Lie algebra of derivations of A over k. In general, one can informally think of
[2] In particular, this comment applies to any simple Lie algebra of characteristic p>0.
; then the restricted enveloping algebra is the quotient ring
are equivalent to modules over the restricted enveloping algebra.
The simple Lie algebras of finite dimension over an algebraically closed field of characteristic zero were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s, using root systems.
Namely, every simple Lie algebra is of type An, Bn, Cn, Dn, E6, E7, E8, F4, or G2.
Surprisingly, there are also many other finite-dimensional simple Lie algebras in characteristic p>0.
In particular, there are the simple Lie algebras of Cartan type, which are finite-dimensional analogs of infinite-dimensional Lie algebras in characteristic zero studied by Cartan.
Namely, Cartan studied the Lie algebra of vector fields on a smooth manifold of dimension n, or the subalgebra of vector fields that preserve a volume form, a symplectic form, or a contact structure.
In characteristic p>0, the simple Lie algebras of Cartan type include both restrictable and non-restrictable examples.
[9] Richard Earl Block and Robert Lee Wilson (1988) classified the restricted simple Lie algebras over an algebraically closed field of characteristic p>7.
Alexander Premet and Helmut Strade (2004) extended the classification to Lie algebras which need not be restricted, and to a larger range of characteristics.
(In characteristic 5, Hayk Melikyan found another family of simple Lie algebras.)
[10] Jacobson's Galois correspondence for purely inseparable field extensions is expressed in terms of restricted Lie algebras.