In this paper, we study finite-dimensional Lie algebras with respect to the length of their chief series. We examine separately the semisimple, solvable, and mixed cases over fields of characteristic 0 , and we establish several structural results describing Lie algebras of prescribed chief length. In the solvable and mixed settings, the problem is reduced to the study of irreducible modules arising from suitable quotient actions. In positive characteristic, we investigate semisimple Lie algebras by means of Block’s structural theorem and obtain several results concerning Lie algebras of chief length 2 .
This work shows that, under natural conditions, every closed soliton of the Ricci–Bourguignon flow has constant scalar curvature. In addition, we present alternative proofs of certain results on triviality.
This paper presents a survey of several classes of Heisenberg-type algebras, including classical, quantum, and generalized variants. We examine the relationships between different families, such as quantum Heisenberg algebras, generalized Heisenberg algebras, quantum generalized Heisenberg algebras, Ore extensions and skew PBW extensions. We also explore homological properties of a special class of quantum generalized Heisenberg algebras. Special attention is given to the Koszul, Artin–Schelter regular, and (skew) Calabi–Yau properties of quantum generalized Heisenberg algebras.
Let p be a prime and let R=ℤ_p[C_p^2] , where ℤ_p is the ring of p -adic integers and C_p^2 is the cyclic group of order p^2 . We study the equivalence relation on finite R -modules introduced by Greither and Kataoka through the projective-equivalence classes of first syzygies. We show that the Fitting class of a module—its Fitting ideal up to a principal fractional factor—is the determinant class of its syzygy. Using Ernst Dieterich’s classification of indecomposable R -lattices, we determine the unique minimal generating system of the syzygy-vector monoid by an independent minimal-zero-sum argument. This corrects the previously stated count p^2+p+1 to p^2+2p-1 : exactly 3p generators have rank one and p^2-p-1 have rank two, and none has larger rank. Every reducible rank-two element has a unique factorization, up to order, into two rank-one elements. Consequently, for p≥ 5 , there exist non-equivalent finite R -modules whose Fitting ideals are isomorphic as R -modules.
Let G be a finite group and h_m(G) be the harmonic mean of element orders of G. In this short note, we prove that if h_m(G)<60/23 , then G is solvable.
