Related papers: The integral polytope group

We describe the autotopism group Atp(G) of any abelian group G as being a semidirect product of its automorphism group Aut(G) and G^2. We then provide the subgroup structure of Atp(G) when G is a finite cyclic group.

Let $k$ be an algebraically closed field of prime characteristic $p$. Let $kGe$ be a block of a group algebra of a finite group $G$, with normal defect group $P$ and abelian $p'$ inertial quotient $L$. Then we show that $kGe$ is a matrix…

In this article, we generalise a result of Pottmeyer from the multiplicative group of the algebraic numbers to almost split semiabelian varieties defined over number fields. This concerns a consequence of R\'emond's generalisation of…

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional…

We study the notion of fundamental group in the framework of descent-exact homological categories. This setting is sufficiently wide to include several categories of "algebraic" nature such as the almost abelian categories, the semi-abelian…

In this paper we look at Grothendieck's work on classifying holomorphic bundles over the complex projective line. The paper is divided into $4$ parts. The first and second part we build up the necessary background to talk about vector…

In this paper, the problem formulated in [8] is solved. We prove, that the group of automorphisms of the category of free associative algebras is generated by semi-inner and mirror automorphisms

When does Borel's theorem on free subgroups of semisimple groups generalize to other groups? We initiate a systematic study of this question and find positive and negative answers for it. In particular, we fully classify fundamental groups…

We will introduce an $\mathbb{N}$-filtration on the negative part of a quantum group of type $A_n$, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation…

In this short note, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by L.…

We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a q-deformation of one defined…

A topologically-invariant and additive homology class is mostly not a natural transformation as it is. In this paper we discuss turning such a homology class into a natural transformation; i.e., a "categorification" of it. In a general…

As the first main result of this article, we prove that if $e$ and $e'$ are idempotents of a commutative ring $A$, then there is a canonical isomorphism of $A$-modules: $$Ae\oplus Ae'\simeq Ae/Ae(1-e')\oplus Ae'/Ae'(1-e)\oplus…

We prove a version of Grothendieck's descent theorem on an `enriched' principal fiber bundle, a principal fiber bundle with an action of a larger group scheme. Using this, we prove the isomorphisms of the equivariant Picard and the class…

We obtain a classification of the finite two-generated cyclic-by-abelian groups of prime-power order. For that we associate to each such group $G$ a list $\inv(G)$ of numerical group invariants which determines the isomorphism type of $G$.…

We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn-Handel. As an application, we prove the rank…

An approach to identify the normal subgroups determined by ideals in free group rings with the help of the derived functors of non-additive functors is explored. A similar approach, i.e., via derived functors, for computing limits of…

We investigate the class of root systems $R$ obtained by extending an $A_1$-type irreducible root system by a free abelian group $G$. In this context there is a Weyl group $W$ and a group $U$ with the presentation by conjugation. Both…

We study the groups of rational points of abelian varieties defined over a finite field $ \mathbb{F}_q$ whose endomorphism rings are commutative, or, equivalently, whose isogeny classes are determined by squarefree characteristic…

In this paper we introduce a working generalization of the theory of Gr\"obner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear…