Question

Can anyone tell me the German term for "restricted Lie algebra" ? Many thanks in advance ! Kind regards, Stephan Kroneck.

Answer 1

This is called: Lie-$p$-Algebra

See for example, Strade "Einige Vereinfachungen in der Theorie der modularen Lie Algebren" Abh. Math. Sem Hamburg 1984, starting:

1937 hat N. Jacobson den Begriff der Lie-$p$-Algebra (englisch: restricted Lie algebra) über Körpern der Characteristik $p>0$ eingeführt.

My translation: In 1937 N. Jacobson introduced the notion 'Lie-$p$-Algebra' (in English: restricted Lie algebra) for fields of characteristic $p>0$.

Also see this recent Master thesis written in German (Nolte 'Lie-$p$-Algebren und die Berechnung ihrer $p$-Darstellungen'): in particular Definition 3.3 on page 17. This could also be useful as a general 'dictionary' for the subject as various notions are recalled at the start.

Answer 2

The correct answer to the narrow question asked about terminology has been given by quid, but I'm tempted to add some broader semi-historical remarks as well. As usual, the development of mathematical notation and terminology is somewhat arbitrary but complicated.

Jacobson was probably inspired to study Lie algebras (originally infinitesimal groups in Lie group theory) by purely algebraic methods over general fields after writing up notes of lectures Weyl gave at IAS. Jacobson's invented term restricted Lie algebra isn't at all informative and omits mention of the underlying prime characteristic $p$. Even so, the related notions of restricted homomorphism (representation, universal enveloping algebra, etc.) followed at once. But when he constructed a finite dimensional quotient of the usual universal enveloping algebra $U(\mathfrak{g})$ by truncating at $p$th powers, he called it (unhelpfully) the u-algebra. Except for this last notion, much of this language continues in use up to now in English, though as early as 1967 Jacobson's student (and my thesis adviser) George Seligman published a note Some results on Lie $p$-algebras.

There was relatively little further activity until after 1950, though the early history isn't easy to trace. Gradually some Russians (including Kostrikin, Shafarevich, Manin) began to study problems in the area such as the classification of simple Lie algebras, whereas French mathematicians (including Cartier, Dieudonne, Bourbaki) went in other directions. In French one has $p$-algebre de Lie and in Russian a shorter Cyrillic version. In Germany this became Lie-$p$-Algebra or perhaps $p$-Lie-Algebra. Similar conventions developed around representation theory, where the restricted enveloping algebra $u(\mathfrak{g})$ or $V(\mathfrak{g})$ became die universell $p$-einhullende Algebra with varying notations involving the prime.

Behind the variation of language and notation lurk the more interesting questions of structure, representations, classification which were taken over from classical Lie theory but immediately showed different features. Beyond this there are the close analogues for quantum groups at a root of unity, including the "small" quantum group.