# Why are “restricted Lie algebras” called restricted?

Restricted Lie algebras are Lie algebras of characteristic $p$ with an additional unary operation which is like raising to $p$th power. I didn't find any motivation for this strange choice of the name in the original paper. There is a statement on the first page ("If $\mathfrak{U} = \mathfrak{L}$ is itself restricted ($ab = [ab]$) ...") that hints on calling anticommutative algebras restricted, but most probably I'm reading this sentence wrong.

In 50s it was too late to call Lie algebras with $p$th power operation Frobenius Lie algebras, since Frobenius algebras were popular and absolutely different sort of algebras, but "restricted" still strikes me as a very odd term. What is the explanation for it?

• One guess might be that the representations of these Lie algebras have an extra restriction. – Tobias Kildetoft Sep 14 '13 at 17:08
• Just guessing, I suppose not all Lie algbras in characteristic $p$ admit a $p$-th power operation. Then assuming one is given restricts the class of Lie algebras considered. Not really a convincing argument though. – Marc van Leeuwen Sep 14 '13 at 17:13

Jacobson invented the term "restricted Lie algebra" when he constructed a finite dimensional quotient of the usual universal enveloping algebra $U(\mathfrak{g})$ by truncating at pth powers - the restricted universal enveloping algebra. So this explains the word "restricted". Goerge Seligman, Jacobson's student, later tried to introduce a better word, and published a note called "Some results on Lie $p$-algebras.". Here a Lie $p$-algebra is the same as a restricted Lie algebra. The"german" word is always "Lie $p$-Algebra" now, but in english there is still the word "restricted" in use (for Lie algebras, derivations, universal enveloping algebras etc.)