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Vincent Beck

Publications and source records attributed to Vincent Beck.

7 recordsLinked to original sources
arXiv2024

Rank two Artin-Schelter regular algebras and non commuting derivations

If $\Delta$ and $\Gamma$ are two derivations of a commutative algebra $A$ such that $\Delta\Gamma-\Gamma\Delta=\Delta$ is locally nilpotent, one can endow $A$ with a new product $\ast$ whose filtered semiclassical limit is the Poisson structure $\Delta\wedge\Gamma$.In this article we first study theses (Poisson) algebras from an algebraic point of view, and when $A$ is a polynomial algebra, we investigate their homological properties. In particular, when the derivations $\Delta$ and $\Gamma$ are linear, the algebras $(A,\ast)$ provide, in each dimension at least four, new examples of multiparameter families of Artin-Schelter regular algebras. These algebras are deformations of Poisson algebras $(A,\Delta\wedge\Gamma)$ of rank $2$, thus explaining the title of the article.Assuming furthermore a technical condition on $\Gamma$, we show that the algebra $(A,\ast)$ is Calabi-Yau if and only if the trace of $\Gamma$ is equal to $1$ if and only if the Poisson algebra $(A,\Delta\wedge\Gamma)$ is unimodular.Since the trace of $\Gamma$ is a linear function of the parameters, the algebras $(A,\ast)$ also provide, in each dimension at least four, new examples of multiparameter families of Calabi-Yau algebras.

math.RA↗
arXiv2017

Torsion subgroups of quasi-abelianized braid groups

This article extends the works of Gon\c{c}alves, Guaschi, Ocampo [GGO] and Marin [MAR2] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group. We get explicit criteria for subgroups of the (complex) reflection group to lift to subgroups of this quotient. In the specific case of the classical braid group, this enables us to describe all its finite subgroups : we show that every odd-order finite group can be embedded in it, when the number of strands goes to infinity. We also determine a complete list of the irreducible reflection groups for which this quotient is a Bieberbach group.

math.GR↗
arXiv2015

Additive combinatorics methods in associative algebras

We adapt methods coming from additive combinatorics in groups to the study of linear span in associative unital algebras. In particular, we establish for these algebras analogues of Diderrich-Kneser's and Hamidoune's theorems on sumsets and Tao's theorem on sets of small doubling. In passing we classify the finite-dimensional algebras over infinite fields with finitely many subalgebras. These algebras play a crucial role in our linear version of Diderrich-Kneser's theorem. We also explain how the original theorems for groups we linearize can be easily deduced from our results applied to group algebras. Finally, we give lower bounds for the Minkowski product of two subsets in finite monoids by using their associated monoid algebras.

math.CO↗
arXiv2010

Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology

The final result of this article gives the order of the extension $$\xymatrix{1\ar[r] & P/[P,P] \ar^{j}[r] & B/[P,P] \ar^-{p}[r] & W \ar[r] & 1}$$ as an element of the cohomology group $H^2(W,P/[P,P])$ (where $B$ and $P$ stands for the braid group and the pure braid group associated to the complex reflection group $W$). To obtain this result, we describe the abelianization of the stabilizer $N_H$ of a hyperplane $H$. Contrary to the case of Coxeter groups, $N_H$ is not in general a reflection subgroup of the complex reflection group $W$. So the first step is to refine Stanley-Springer's theorem on the abelianization of a reflection group. The second step is to describe the abelianization of various types of big subgroups of the braid group $B$ of $W$. More precisely, we just need a group homomorphism from the inverse image of $N_H$ by $p$ with values in $\QQ$ (where $p : B \ra W$ is the canonical morphism) but a slight enhancement gives a complete description of the abelianization of $p^{-1}(W')$ where $W'$ is a reflection subgroup of $W$ or the stabilizer of a hyperplane. We also suggest a lifting construction for every element of the centralizer of a reflection in $W$.

math.GR↗
arXiv2009

Fonctorial Construction of Frobenius Categories

Let $\Ascr,\Bscr$ be exact categories with $\Ascr$ karoubian and $M$ be an exact functor. Under suitable adjonction hypotheses for $M$, we are able to show that the direct factors of the objects of $\Ascr$ of the form $MY$ with $Y \in \Bscr$ make up a Frobenius category which allow us to define an $M$-stable category for $\Ascr$ only by quotienting. In addition, we propose a construction of an $M$-stable category for $\Ascr,\Bscr$ triangulated categories and $M$ a triangulated functor. We illustrate this notion with a theorem of Keller and Vossieck which links the two notions of $M$-stable category.

math.CT↗
arXiv2009

Exterior Algebra Structure for Relative Invariants of Reflection Groups

Let $G$ be a reflection group acting on a vector space $V$ (over a field with zero characteristic). We denote by $S(V^*)$ the coordinate ring of $V$, by $M$ a finite dimensional $G$-module and by $χ$ a one-dimensional character of $G$. In this article, we define an algebra structure on the isotypic component associated to $χ$ of the algebra $S(V^*) \otimes Λ(M^*)$. This structure is then used to obtain various generalizations of usual criterions on regularity of integers.

math.GR↗