Algebra Colloquium

Let [Formula: see text] be a graph. We say that a hypergraph [Formula: see text] is Berge-[Formula: see text] if there exists a bijection [Formula: see text] from [Formula: see text] to [Formula: see text] such that [Formula: see text] for all [Formula: see text]. For any [Formula: see text]-uniform hypergraph [Formula: see text] and a real number [Formula: see text], the [Formula: see text]-spec…

Let [Formula: see text] be a group. A weak Cayley table isomorphism is a bijection [Formula: see text] such that (i) [Formula: see text] is conjugate to [Formula: see text] for [Formula: see text], [Formula: see text] and (ii) [Formula: see text] sends conjugacy classes to conjugacy classes. The set of all such bijections forms a group [Formula: see text]. We study [Formula: see text] for 56 of t…

Recently, Estélyi et al. investigated a representation [Formula: see text] of the automorphisms of a connected graph [Formula: see text] by [Formula: see text] unimodular matrices over [Formula: see text], where [Formula: see text] is the Betti number of [Formula: see text], and classified the graphs for which the representation is unfaithful, with two problems left open: (1) What is the smallest…

We establish that every 2-local inner derivation on a symmetrizable Kac-Moody Lie algebra [Formula: see text] over the field [Formula: see text] is a derivation. In addition, we demonstrate that if [Formula: see text], where [Formula: see text] is the derived subalgebra of [Formula: see text], then [Formula: see text] admits a 2-local derivation which is not a derivation.

The comaximal graph [Formula: see text] of a commutative ring [Formula: see text] is a simple graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] of [Formula: see text] adjacent if and only if [Formula: see text], where [Formula: see text] is the ideal generated by [Formula: see text] in [Formula: see text]. In this article, the independ…

In this paper, we give a further study of strongly nil [Formula: see text]-clean rings. Some new characterizations and extension properties of strongly nil [Formula: see text]-clean rings are provided. Furthermore, motivated by the notion of strong Drazin inverses, we introduce a new type of generalized inverse which is closely related to strongly nil [Formula: see text]-clean elements, and we pr…

For a nontrivial [Formula: see text]-group [Formula: see text], a maximal chain of [Formula: see text] is a chain of subgroups [Formula: see text] with [Formula: see text] for [Formula: see text], where [Formula: see text] is an integer. We determine the structure of finite groups having [Formula: see text]-quasinormality of a maximal chain of Sylow subgroups. We obtain new characterizations of f…

Let [Formula: see text] be a finite group and [Formula: see text]. Then [Formula: see text] is called [Formula: see text]-decomposable in [Formula: see text] if it is a union of [Formula: see text] distinct [Formula: see text]-conjugacy classes, and denote [Formula: see text] by [Formula: see text]. A group [Formula: see text] is called [Formula: see text]-decomposable with [Formula: see text]. T…

A graph is edge-transitive if its automorphism group acts transitively on the set of edges of the graph. In this paper, we classify edge-transitive 8-valent graphs of order [Formula: see text] for each prime [Formula: see text].

Motivated by the theory of homomorphisms and change-of-variable polynomials (cv-polynomials) of Ore extensions, the role of double Ore extensions in the classification of Artin-Schelter regular algebras of dimension four, and the absence of inclusions between the classes of all double Ore extensions of an algebra and of all two-step iterated Ore extensions of the same algebra, we present a first …

We introduce the notions of a Kantor superpair and a Jordan-Kantor superpair, by using which we construct 5-graded Lie superalgebras. In addition, the relationship between a Jordan-Kantor superpair [Formula: see text] and the corresponding 5-graded Lie superalgebra [Formula: see text] is described. We also characterize the universal central extension of the 5-graded Lie superalgebra [Formula: see…

It is shown that the gentle one-cycle algebra [Formula: see text] has Hall polynomials. The Hall polynomials are explicitly given for all triples of indecomposable modules, and as a consequence, the Ringel–Hall Lie algebra of [Formula: see text] is shown to be isomorphic to its Riedtmann Lie algebra.

In this paper, we determine the unique graph whose least eigenvalue attains the minimum among all the connected simple graphs with given clique number.

In this paper, we describe explicitly the structure of the derivation algebra and automorphism group of the symplectic oscillator Lie algebra [Formula: see text] ([Formula: see text]), where [Formula: see text] is the symplectic Lie algebra and [Formula: see text] is the [Formula: see text]-dimensional Heisenberg algebra.

In this paper, we show that all Coleman automorphisms of wreath products of some finite groups with trivial centers by any finite group are inner. In particular, the normalizer property holds for these groups.

For any integer [Formula: see text], a spanning [Formula: see text]-ended tree is a spanning tree with at most [Formula: see text] leaves. In this paper, we provide tight spectral radius conditions for the existence of a spanning [Formula: see text]-ended tree in [Formula: see text]-connected graphs, which generalizes a result of Ao, Liu and Yuan (2023).

The classification of irreducible conformal [Formula: see text]-modules of finite rank was given by H.B. Chen for [Formula: see text]. In this paper, we use a different way to obtain the classification and improve the results, which prove that such modules must be of rank 1.

Consider the type [Formula: see text] root system with simple roots [Formula: see text], and let [Formula: see text] be the span of [Formula: see text] and [Formula: see text] over [Formula: see text]. Let [Formula: see text] be any irreducible reduced root system but not of type [Formula: see text], with rank [Formula: see text]. Namely, [Formula: see text] is of type [Formula: see text] ([Formu…

Let [Formula: see text] be a finite set of [Formula: see text] over an algebraically closed field [Formula: see text] of [Formula: see text], and [Formula: see text] be the moduli space of semistable parabolic bundles of rank two and degree [Formula: see text] on [Formula: see text] with parabolic structures determined by weights [Formula: see text]. We prove that [Formula: see text] is F-split w…

An element [Formula: see text] of a finite group [Formula: see text] is said to be real in [Formula: see text] if [Formula: see text] for some [Formula: see text] in [Formula: see text], and quasi-central if [Formula: see text] for each [Formula: see text] in [Formula: see text]. When [Formula: see text] is a 2-group, [Formula: see text] and [Formula: see text], we prove that if every real elemen…