Related papers: Algebraic groups over the field with one element
We prove that the number of unimodular integral matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group is of strictly lower order of magnitude than the number of all such matrices in…
Let $\mathrm{R}$ be a real closed field and $\mathrm{C}$ the algebraic closure of $\mathrm{R}$. We give an algorithm for computing a semi-algebraic basis for the first homology group, $\mathrm{H}_1(S,\mathbb{F})$, with coefficients in a…
We prove the Tits-Weiss conjecture for Albert division algebras over fields of arbitrary characteristics in the affirmative. The conjecture predicts that every norm similarity of an Albert division algebra is a product of a scalar homothety…
The main goal of this note is to provide a First-Order Logic with Betweenness (FOLB) axiomatization of the main classes of graphs occurring in Metric Graph Theory, in analogy to Tarski's axiomatization of Euclidean geometry. We provide such…
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of…
To every finite-dimensional $\mathbb C$-algebra $\Lambda$ of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by "$u$ variables" satisfying "$u$-equations", first…
We introduce a unified theory of Cartan subgroups and maximal toroids - defined as connected multiplicative type subgroups that are maximal amongst all such subgroups - which holds for all affine algebraic groups over a field, regardless of…
The aim of this paper is to study the group of isomorphism classes of torsors of finite flat group schemes of rank 2 over a commutative ring $R$. This, in particular, generalises the group of quadratic algebras (free or projective), which…
We describe in this note a torsor structure arising on the affine scheme defined by a system of rationnal algebraic relations between polyzetas at roots of unity (values of hyperlogarithmic functions on a fixed finite group of complex roots…
Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying the condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ of the elements $x_1,\ldots,x_n$ are equal…
Generators and relations are given for the subalgebra of cocommutative elements in the quantized coordinate rings of the classical groups, where the deformation parameter q is transcendental. This is a ring theoretic formulation of the well…
Let $G$ be a simply connected and simple algebraic group defined and split over a finite prime field $\mathbb{F}_p$ of $p$ elements. In this paper, using an $\mathbb{F}_p$-linear map splitting Frobenius endomorphism on a hyperalgebra…
Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in GL_n(Z). Assume that the characteristic polynomial of M is prime to all polynomials X^m-1. We show that any finite equivariant map from another algebraic dynamics…
In 1960 Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In this paper we address…
In Proposition I of "Memoire sur les conditions de resolubilite des equations par radicaux", Galois established that any intermediate extension of the splitting field of a polynomial with rational coefficients is the fixed field of its…
The aim of this paper is to solve a problem proposed by Dominique Bourn: to provide a categorical-algebraic characterisation of groups amongst monoids and of rings amongst semirings. In the case of monoids, our solution is given by the…
An excellent ring of prime characteristic for which the Frobenius map is pure is also Frobenius split in many commonly occurring situations in positive characteristic commutative algebra and algebraic geometry. However, using a fundamental…
Ehresmann's introduction of differentiable groupoids in the 1950s may be seen as a starting point for two diverging lines of research, many-object Lie theory (the study of Lie algebroids and Lie groupoids) and sketch theory. This thesis…
We construct a local deformation problem for residual Galois representations $\bar{\rho}$ valued in an arbitrary reductive group $\hat{G}$ which we use to develop a variant of the Taylor-Wiles method. Our generalization allows Taylor-Wiles…
When considering the unit group of $\mathcal{O}_F G$ ($\mathcal{O}_F$ the ring of integers of an abelian number field $F$ and a finite group $G$) certain components in the Wedderburn decomposition of $FG$ cause problems for known generic…