Methods of group theory in Leibniz algebras: some compelling results

I.Ya Subbotin

doi:10.15421/242108

Abstract

The theory of Leibniz algebras has been developing quite intensively. Most of the results on the structural features of Leibniz algebras were obtained for finite-dimensional algebras and many of them over fields of characteristic zero. A number of these results are analogues of the corresponding theorems from the theory of Lie algebras. The specifics of Leibniz algebras, the features that distinguish them from Lie algebras, can be seen from the description of Leibniz algebras of small dimensions. However, this description concerns algebras over fields of characteristic zero. Some reminiscences of the theory of groups are immediately striking, precisely with its period when the theory of finite groups was already quite developed, and the theory of infinite groups only arose, i.e., with the time when the formation of the general theory of groups took place. Therefore, the idea of using this experience naturally arises. It is clear that we cannot talk about some kind of similarity of results; we can talk about approaches and problems, about application of group theory philosophy. Moreover, every theory has several natural problems that arise in the process of its development, and these problems quite often have analogues in other disciplines. In the current survey, we want to focus on such issues: our goal is to observe which parts of the picture involving a general structure of Leibniz algebras have already been drawn, and which parts of this picture should be developed further.

Highlights

Introduction and some remarksLet L be an algebra over a eld F with the binary operations + and [, ]. L is called a Leibniz algebra if it satises the Leibniz identity [[a, b], c] = [a, [b, c]] − [b, [a, c]]for all a, b, c ∈ L.Another form of this identity is: [a, [b, c]] = [[a, b], c] + [b, [a, c]].An algebra R over a eld F is called right Leibniz if it satises the Leibniz identity [a, [b, c]] = [[a, b], c] − [[a, c], b]Note at once that the classes of left Leibniz algebras and right Leibniz algebras are dierent
Note that a very large part of articles concerned Leibniz algebras dealt with only nite-dimensional Leibniz algebras, and in most of these articles the algebras were considered over a eld of characteristic 0
For the case of the eld of complex numbers, the description of cyclic nitedimensional Leibniz algebras was obtained in the paper [24]; it does not show the structure of cyclic Leibniz algebras

Summary

METHODS

Áiëüøiñòü ðåçóëüòàòiâ ùîäî ñòðóêòóðíèõ îñîáëèâîñòåé àëãåáð Ëåéáíiöà áóëè îòðèìàíi äëÿ ñêií÷åííîâèìiðíèõ àëãåáð, i áàãàòî ç íèõ íàä ïîëÿìè íóëüîâîõàðàêòåðèñòèêè. ×àñòèíà öèõ ðåçóëüòàòiâ 1 àíàëîãàìè âiäïîâiäíèõ òåîðåì ç òåîðiàëãåáð Ëi. Ñïåöèôiêó àëãåáð Ëåéáíiöà, îñîáëèâîñòi, ùî âiäðiçíÿþòüõ âiä àëãåáð Ëi, ìîæíà ïîáà÷èòè ç îïèñó àëãåáð Ëåéáíiöà ìàëèõ âèìiðíîñòåé. Îäíàê öåé îïèñ ñòîñó1òüñÿ àëãåáð íàä ïîëÿìè íóëüîâîõàðàêòåðèñòèêè. Ùî íå ìîæíà ãîâîðèòè ïðî ÿêóñü ïîäiáíiñòü ðåçóëüòàòiâ; ìè ìîæåìî ãîâîðèòè ïðî ïiäõîäè òà çàäà÷i, ïðî çàñòîñóâàííÿ ôiëîñîôiòåîðiãðóï. Êîæíà òåîðiÿ ìà íèçêó ïðèðîäíèõ ïðîáëåì, ÿêi âèíèêàþòü ó ïðîöåñi ̈ ðîçâèòêó, i öi ïðîáëåìè äîñèòü ÷àñòî ìàþòü àíàëîãè â iíøèõ äèñöèïëiíàõ. ÷àñòèíè ìàëþíêà iç çàãàëüíîþ ñòðóêòóðîþ àëãåáð Ëåéáíiöà âæå áóëè íàìàëüîâàíi, à ÿêi ÷àñòèíè öi1 ̈ êàðòèíè ñëiä ðîçâèâàòè äàëi. Êëþ÷îâi ñëîâà: àëãåáðà Ëåéáíiöà, àëãåáðà Ëi, öèêëi÷íà ïiäàëãåáðà, ëiâèé öåíòð, ïðàâèé öåíòð, öåíòð àëãåáðè Ëåéáíiöà, íiëüïîòåíòíà ïiäàëãåáðà, àáåëåâà ïiäàëãåáðà, åêñòðàñïåöiàëüíà ïiäàëãåáðà

Introduction and some remarks

Nilpotency of Leibniz algebras

A Leibniz algebra L is said to be locally nilpotent if every nite subset of