Recent zbMATH articles in MSC 17B30https://zbmath.org/atom/cc/17B302021-05-28T16:06:00+00:00WerkzeugComputational experiments with nilpotent Lie algebras.https://zbmath.org/1459.170252021-05-28T16:06:00+00:00"Gorbatsevich, V. V."https://zbmath.org/authors/?q=ai:gorbatsevich.vladimir-vitalevichSummary: Results of computer experiments on the study of properties of generic Lie subalgebras with two generators in the Lie algebra of nilpotent matrices whose order does not exceed 10 are presented. The calculations carried out have made it possible to formulate several statements (so-called virtual theorems) on properties of the Lie subalgebras in question. The dimensions of the lower and upper central series and of the series of commutator subalgebras and the characteristic nilpotency property of the Lie subalgebras constructed here and of generic Lie subalgebras of codimension 1 in these Lie subalgebras are studied.On characterizing pairs of non-abelian nilpotent and filiform Lie algebras by their Schur multipliers.https://zbmath.org/1459.170232021-05-28T16:06:00+00:00"Arabyani, H."https://zbmath.org/authors/?q=ai:arabyani.homayoon"Safa, H."https://zbmath.org/authors/?q=ai:safa.hesam"Saeedi, F."https://zbmath.org/authors/?q=ai:saeedi.farshidSummary: Let \(L\) be an \(n\)-dimensional non-abelian nilpotent Lie algebra. \textit{P. Niroomand} and \textit{F. G. Russo} [Commun. Algebra 39, No. 4, 1293--1297 (2011; Zbl 1250.17019)] proved that \(\dim \mathcal M(L) = \tfrac12 (n-1)(n-2) +1 - s(L)\), where \(\mathcal M(L)\) is the Schur multiplier of \(L\) and \(s(L)\) is a nonnegative integer. They also characterized the structure of \(L\), when \(s(L) = 0\). Assume that \((N,L)\) is a pair of finite dimensional nilpotent Lie algebras, in which \(L\) is non-abelian and \(N\) is an ideal in \(L\) and also \(\mathcal M(N,L)\) is the Schur multiplier of the pair \((N,L)\). If \(N\) admits a complement \(K\) say, in \(L\) such that \(\dim K = m\), then
\[
\dim \mathcal M(N,L) = \tfrac12 (n^2 + 2nm - 3n - 2m + 2) + 1 - (s(L) - t(K)),
\]
where \(t(K) = \frac12 m(m - 1) - \dim \mathcal M(K)\).
In the present paper, we characterize the pairs \((N,L)\), for which \(0\le t(K)\le s(L)\le 3\). In particular, we classify the pairs \((N,L)\) such that \(L\) is a non-abelian filiform Lie algebra
and \(0\le t(K)\le s(L) \le 17\).On certain graded representations of filiform Lie algebras.https://zbmath.org/1459.170242021-05-28T16:06:00+00:00"Bernik, Janez"https://zbmath.org/authors/?q=ai:bernik.janez"Šivic, Klemen"https://zbmath.org/authors/?q=ai:sivic.klemenSummary: Let \(G\subset\mathrm{GL}(V)\) be a connected complex linear algebraic group of the same dimension as \(V\) such that the poset of the Zariski closures of the orbits for its action coincides with a full flag of subspaces of \(V\). Using the classification of graded filiform Lie algebras, we determine the isomorphism types of the unipotent radical \(U\) of \(G\) in case \(G\) is not nilpotent and \(U\) is of maximal class. In particular, if \(\dim (G)=\dim (V)\geq 11\), there are, up to isomorphism, only two such unipotent groups.Minimal nonnilpotent Leibniz algebras.https://zbmath.org/1459.170032021-05-28T16:06:00+00:00"Bosko-Dunbar, Lindsey"https://zbmath.org/authors/?q=ai:bosko-dunbar.lindsey"Dunbar, Jonathan D."https://zbmath.org/authors/?q=ai:dunbar.jonathan-d"Hird, J. T."https://zbmath.org/authors/?q=ai:hird.john-t"Stagg Rovira, Kristen"https://zbmath.org/authors/?q=ai:rovira.kristen-staggMinimal non nilpotent Lie algebras have been classified by \textit{D. Towers} [Linear Algebra Appl. 32, 61--73 (1980; Zbl 0442.17004)]. The authors extend these results to Leibniz algebras. For such algebras, \(L\), the Leibniz algebra concepts of cyclic algebras and \(\text{Leib}(L)\), not involved in the Lie case, are highlighted in their work. They show that \(L\) is a one dimensional extension of the nilradical of \(L\). They also give several examples that illustrate the results. In particular, they contrast the complex, real and rational cases.
Reviewer: Ernest L. Stitzinger (Raleigh)Camina Lie algebras.https://zbmath.org/1459.170262021-05-28T16:06:00+00:00"Sheikh-Mohseni, S."https://zbmath.org/authors/?q=ai:sheikh-mohseni.sedigheh"Saeedi, F."https://zbmath.org/authors/?q=ai:saeedi.farshid