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 satisfying
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 and p operation make A into a restricted Lie algebra.
Let G be an algebraic group over a field k of characteristic p, and be the Zariski tangent space at the identity element of G. Each element of uniquely defines a left-invariant vector field on G, and the commutator of vector fields defines a Lie algebra structure on just as in the Lie group case. If p>0, the Frobenius map defines a p operation on.

Restricted universal enveloping algebra

The functor has a left adjoint called the restricted universal enveloping algebra. To construct this, let be the universal enveloping algebra of L forgetting the p operation. Letting I be the two-sided ideal generated by elements of the form, we set. It satisfies a form of the PBW theorem.