Restricted Lie algebra

Summary

A restricted Lie algebra is a modular Lie algebra with an extra operation $x \mapsto x^{[p]}$ that behaves like the $p$-th power in an associative algebra. The notion is due to Nathan Jacobson. Lie algebras of algebraic groups in characteristic $p$, and algebras of derivations, carry such an operation. The $p$-operation gives restricted enveloping algebras, which are finite-dimensional when the Lie algebra is. It is basic for the representation theory of modular Lie algebras.