International Journal of Algebra and Computation

We show that any abelian variety that is not affine has a nontrivial strongly abelian subvariety. In later papers in this sequence, we apply this result to the study of minimal abelian varieties.

Under broad finiteness conditions, such as the existence of a greatest or least element, a semidistributive lattice has the two element lattice as a homomorphic image, and so, if it has more than two elements, is not simple. However, the existence of a simple, semidistributive lattice with more than two elements has remained in question. This paper constructs such a lattice.

To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)={\sum }_{n\in \mathbb{N}}{a}_n(G)/n^s$ is associated, where $a_n(G)={\sum }_{|G:H|=n}\mu (H,G)$ and $\mu (H,G)$ denotes the Möbius function of the lattice of open subgroups of $G.$ Its formal inverse ${(P_G(s))}^{-1}$ is the probabilistic zeta function of $G$. When $G$ is prosoluble, every coefficient of ${(P_G(s))}^{-1}$ is nonnegative. In this paper we discuss the general case and we produce a non-prosoluble finitely generated group with the same property.

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number $b_n$ of minimal varieties of given exponent $n$ is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit $\beta ={lim}_{n\to \infty }\sqrt[n]{b_n}$ exists and can be expressed as the positive solution of an equation $a(t)=0$ where $a(t)$ is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank $d$. It follows from classical results on lacunary power series that the generating function of the sequence $b_n$, $n=1,2,\dots$, is transcendental. With the same approach we construct examples of free graded semigroups $〈Y〉$ with the following property. If $d_n$ is the number of elements of degree $n$ of $〈Y〉$, then the limit $\delta ={lim}_{n\to \infty }\sqrt[n]{d_n}$ exists and is transcendental.

In the last years, the sequence of codimensions of PI-algebras has been studied by several authors and the classification of unitary algebras, up to equivalence, with at most cubic codimension growth was given by Giambruno, La Mattina and Petrogradsky in 2007. In this paper, we establish a new approach by studying the possibilities of specific proper codimensions of a unitary algebra with growth $n^4$ in order to present a complete list of varieties generated by unitary algebras with polynomial growth $n^4$. Also, we classify, up to PI-equivalence, the unitary algebras with growth $n^5$ whose leading coefficient of the polynomial describing the codimension sequence achieves the largest and the smallest possible value.

Let $Q$ be a finite group acting on a group $G$ as a group automorphisms, $C_{\ast }(G)$ the bar complex, $H_{\ast }^Q(G,A)$ the homology of invariant group chains and $H_Q^{\ast }(G,A)$ the cohomology invariant, both defined in Knudson’s paper “The homology of invariant group chains”. In this paper, we define the Tate homology of invariants ${Ĥ}_{\ast }^Q(G,A)$ and the Tate cohomology of invariants ${Ĥ}_Q^{\ast }(G,A)$. When the coefficient $A$ is the abelian group of the integers, we proved that these groups are isomorphics, ${Ĥ}_Q^i(G,\mathbb{Z})\cong {Ĥ}_{i-1}^Q(G,\mathbb{Z})$. Further, we prove that the homology and cohomology of invariant group chains are duals, $H_Q^i(G,\mathbb{Z})\cong H_{i-1}^Q(G,\mathbb{Z})$, $i\ge 2$.

Let $A^{\prime }$ be the Auslander algebra of a finite-dimensional basic connected Nakayama algebra $A$ with radical cube zero and $n$ simple modules. Then the cardinality $\#\mathrm{\text{tilt}}{A}^{\prime }$ of the set consisting of isomorphism classes of basic tilting $A^{\prime }$-modules is

We study the representation theory of finite-dimensional $\omega$-Lie algebras over the complex field. We derive an $\omega$-Lie version of the classical Lie’s theorem, i.e., any finite-dimensional irreducible module of a soluble $\omega$-Lie algebra is 1-dimensional (1D). We also prove that indecomposable modules of some 3D $\omega$-Lie algebras could be parametrized by the complex field and nilpotent matrices. We introduce the notion of a tailed derivation of a nonassociative algebra $𝔤$ and prove that if $𝔤$ is a Lie algebra, then there exists a one-to-one correspondence between tailed derivations of $𝔤$ and 1D $\omega$-extensions of $𝔤$.

We continue the algebraic study of almost inner derivations of Lie algebras over a field of characteristic zero and determine these derivations for free nilpotent Lie algebras, for almost abelian Lie algebras, for Lie algebras whose solvable radical is abelian and for several classes of filiform nilpotent Lie algebras. We find a family of $n$-dimensional characteristically nilpotent filiform Lie algebras ${𝔣}_n$, for all $n\ge 13$, all of whose derivations are almost inner. Finally, we compare the almost inner derivations of Lie algebras considered over two different fields $K\supseteq k$ for a finite-dimensional field extension.