Claudio Bartolone, Alfonso Di Bartolo, Karl Strambach
备注：
11 pages, this paper contains the more significant part of the article with the same title that will appear in the Volume 12 of Journal of Group Theory (2009)
This paper contains the more significant part of the article with the same title that will appear in the Volume 12 of Journal of Group Theory (2009).
In this paper we determine all algebraic transformation groups $G$, defined over an algebraically closed field $\sf k$, which operate transitively, but not primitively, on a variety $Ω$, provided the following conditions are fulfilled. We ask that the (non-effective) action of $G$ on the variety of blocks is sharply 2-transitive, as well as the action on a block $Δ$ of the normalizer $G_Δ$. Also we require sharp transitivity on pairs $(X,Y)$ of independent points of $Ω$, i.e. points contained in different blocks.
We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.
This is a brief introduction to the theories of Lie groups, algebraic groups and their discrete subgroups, which is based on a lecture series given during the Summer School held in the Banach Centre in Poland in Summer 2011.
In the paper we study twisted conjugacy classes and isogredience classes for automorphisms of reductive linear algebraic groups. We show that reductive linear algebraic groups over some fields of zero characteristic possess the $R_\infty$ and $S_\infty$ properties.
For simple algebraic groups defined over algebraically closed fields of good characteristic, we give upper bounds on the covering numbers of unipotent conjugacy classes in terms of their (co)ranks and in terms of their dimensions.
The main purpose of this paper is to give a new definition for the notion of group-groupoid. Also, several basic properties of group-groupoids are established.
Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis-Zimmer super-rigidity phenomenon due to Bader and Furman.
We show that any adjoint absolutely simple linear algebraic group over a field of characteristic zero is the automorphism group of some projector on a central simple algebra. Projective homogeneous varieties can be described in these terms; in particular, we reproduce quadratic equations by Lichtenstein defining them.
The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup of a split reductive group and show how this improves computation in the reductive group itself.
The efficacy of using complexifications to understand the structure of real algebraic groups is demonstrated. In particular the following results are proved: a) If L is an algebraic subgroup of a connected real algebraic group G such that the complexification of L contains a maximal torus of the complexification of G, then L contains a Cartan subgroup of G b) Let G be a solvable real algebraic group whose eigenvalues are all real. If the complexification of G operates algebraically on a complex variety V, and some G orbit is compact, then this orbit is a point.
Let $k$ be an algebraically closed field, $G$ a linear algebraic group over $k$ and $φ\in Aut(G)$, the group of all algebraic group automorphisms of $G$. Two elements $x, y$ of $G$ are said to be $φ$-twisted conjugate if $y=gxφ(g)^{-1}$ for some $g\in G$. In this paper we prove that for a connected non-solvable linear algebraic group $G$ over $k$, the number of its $φ$-twisted conjugacy classes is infinite for every $φ\in Aut(G)$.
Let $G$ be a linear algebraic group over an algebraically closed field $k$ and $\mathrm{Aut}_{\mathrm{alg}}(G)$ the group of all algebraic group automorphisms of $G$. For every $φ\in \mathrm{Aut}_{\mathrm{alg}}(G)$ let $\mathcal{R}(φ)$ denote the set of all orbits of the $φ$-twisted conjugacy action of $G$ on itself (given by $(g,x)\mapsto gxφ(g^{-1})$, for all $g,x\in G$). We say that $G$ has the algebraic $R_\infty$-property if $\mathcal{R}(φ)$ is infinite for every $φ\in \mathrm{Aut}_{\mathrm{alg}}(G)$. In \citep{bb} we have shown that this property is satisfied by every connected non-solva…
We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.
Let $G$ be a connected complex algebraic group and $A$ a connected abelian algebraic group endowed with an algebraic action of $G$ by group automorphisms. In the present note we describe the abelian group $\Ext_{alg}(G,A)$ of algebraic group extensions of $G$ by $A$ in terms of a short exact sequence relating the ext-group to a relative second Lie algebra cohomology space and the fundamental group of the commutator group. Our second main result is an analog of the Van-Est Theorem for algebraic group cohomology, saying that for an algebraic $G$ module $\a$ and $p \geq 0$ the algebraic group coh…
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups defined over the fields which admit arbitrary cyclic extensions.
A connected algebraic group in characteristic 0 is uniquely determined by its Lie algebra. In this paper an algorithm is given for constructing an algebraic group in characteristic 0, given its Lie algebra. Using this an algorithm is presented for finding a maximal reductive subgroup and the unipotent radical of an algebraic group.
We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group.
In this paper, we attempt to develop the Quillen Suslin theory for the algebraic fundamental group of a ring. We give a surjective group homomorphism from the algebraic fundamental group of the field of the real numbers to the group of integers. At the end of the paper, we also propose some problems related to the algebraic fundamental group of some particular type of rings.
We compute the number of orbit types for simply connected simple algebraic groups over algebraically closed fields as well as for compact simply connected simple Lie groups. We also compute the number of orbit types for the adjoint action of these groups on their Lie algebras. We also prove that the genus number of a connected reductive algebraic group coincides with the genus number of its semisimple part.
We prove analogues for reductive algebraic groups of some results for finite groups due to Knoerr and Robinson which play a central role in their reformulation of Alperin's conjecture for finite groups.
We say that an algebraic group $G$ over a field is anti-affine if every regular function on $G$ is constant. We obtain a classification of these groups, with applications to the structure of algebraic groups in positive characteristics, and to the construction of many counterexamples to Hilbert's fourteenth problem.
A fundamental theorem of Barsotti and Chevalley states that every smooth algebraic group over a perfect field is an extension of an abelian variety by a smooth affine algebraic group. In 1956 Rosenlicht gave a short proof of the theorem. In this expository article, we explain his proof in the language of modern algebraic geometry.
This is a survey on appearances of reflection groups, real and complex, in algebraic geometry. We also include a brief introduction into the theory of reflection groups.
We say that a smooth algebraic group $G$ over a field $k$ is very special if for any field extension $K/k$, every $G_K$-homogeneous $K$-variety has a $K$-rational point. It is known that every split solvable linear algebraic group is very special. In this note, we show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions.
Let $G$ be a group and let $V$ be an algebraic group over an algebraically closed field. We introduce algebraic group subshifts $Σ\subset V^G$ which generalize both the class of algebraic sofic subshifts of $V^G$ and the class of closed group subshifts over finite group alphabets. When $G$ is a polycyclic-by-finite group, we show that $V^G$ satisfies the descending chain condition and that the notion of algebraic group subshifts, the notion of algebraic group sofic subshifts, and that of algebraic group subshifts of finite type are all equivalent. Thus, we obtain extensions of well-known resul…
We show that a linear algebraic group is the Galois group of a parameterized Picard-Vessiot extension of k(x), x' = 1, for certain differential fields k, if and only if its identity component has no one dimensional quotient as a linear algebraic group.
We impose a rather unknown algebraic structure called a `hyperstructure' to the underlying space of an affine algebraic group scheme. This algebraic structure generalizes the classical group structure and is canonically defined by the structure of a Hopf algebra of global sections. This paper partially generalizes the result of A.Connes and C.Consani
The existence of closed orbits of real algebraic groups on certain real algebraic spaces is established. As an application it is shown that if $G$ is a real reductive group with Iwasawa decomposition $G=KAN$, then all unipotent subgroups of $G$ are conjugate to a subgroup of $N$.
We analyze the structure of a large class of connected algebraic rings over an algebraically closed field of positive characteristic using Greenberg's perfectization functor. We then give applications to rigidity problems for representations of Chevalley groups, recovering, in particular, a rigidity theorem of Seitz.
The prisoner's dilemma (PD) is a game-theoretic model studied in a wide array of fields to understand the emergence of cooperation between rational self-interested agents. In this work, we formulate a spatial iterated PD as a discrete-event dynamical system where agents play the game in each time-step and analyse it algebraically using Krohn-Rhodes algebraic automata theory using a computational implementation of the holonomy decomposition of transformation semigroups. In each iteration all players adopt the most profitable strategy in their immediate neighbourhood. Perturbations resetting the…
