Action Of Algebraic Group Research Articles

Generic Stabilizers in Actions of Simple Algebraic Groups

In this paper we treat faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties. By explicit calculation, we show that in each case, with essentially one exception, there is a dense open subset any point of which has stabilizer conjugate to a fixed subgroup, called the generic stabilizer. We provide tables listing generic stabilizers in the cases where they are non-trivial; in addition we decide whether or not there is a dense orbit, or a regular orbit for the action on the module.

Intrinsic Ergodicity, Generators, and Symbolic Representations of Algebraic Group Actions

Intrinsic Ergodicity, Generators, and Symbolic Representations of Algebraic Group Actions

Внутренняя эргодичность, порождающие разбиения и символические представления алгебраических действий групп

Строятся естественные символические представления внутренне эргодических, но не обязательно расширительных главных алгебраических действий счетных бесконечных аменабельных групп. С помощью этих представлений получены порождающие разбиения (с точностью до множеств меры нуль) для таких действий.

Exploring projective equivalences between closures of orbits

Motivated by results in the literature that use representations and group actions to produce nice geometric results about algebraic varieties, this article studies projective equivalence relations between closures of orbits for several complex algebraic group actions on , where is a complex representation of . In particular, we study the cases when is one of the following:, , , and . On the way, we also obtain some interesting geometric results from studying these orbits.

Bounding nonminimality and a conjecture of Borovik–Cherlin

Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The degree of nonminimality is the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by the U -rank plus 2. The Borovik–Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of U -rank plus 1 is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.

Equivariant algebraic K-theory, G-theory and derived completions

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. The main goal of the present paper is to provide a proof of this conjecture in as broad a context as possible, making use of the technique of derived completion, and to consider several of the applications.Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the equivariant algebraic G-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). Restricting to actions by split tori, we are also able to consider actions on algebraic spaces. Moreover, the restriction that the base scheme be a field is also not required often, but is put in mainly to simplify some of our exposition. These enable us to obtain a wide range of applications, some of which are briefly sketched and which we plan to explore in detail in the future. In fact, we discuss an extension of our results to equivariant homotopy K-theory along with various Riemann-Roch theorems in a sequel. A comparison of our results with previously known results, none of which made use of derived completions, shows that without the use of derived completions one can only obtain results which are indeed very restrictive.

Canonical stratification of definable Lie groupoids

Our aim is to precisely present a tame topology counterpart to canonical stratification of a Lie groupoid. We consider a definable Lie groupoid in semialgebraic, subanalytic, o-minimal over R, or more generally, Shiota's X-category. We show that there exists a canonical Whitney stratification of the Lie groupoid into definable strata which are invariant under the groupoid action. This is a generalization and refinement of results on real algebraic group action which J.N. Mather and V.A. Vassiliev independently stated with sketchy proofs. A crucial change to their proofs is to use Shiota's isotopy lemma and approximation theorem in the context of tame topology.

On models of algebraic group actions

We show that every action of a smooth algebraic group on a variety admits a normal projective model. Along the way, we present new proofs of some basic results on algebraic transformation groups, including Weil’s regularization theorem.

Relative geometric invariant theory

Relative geometric invariant theory is an invariant theory for equivariant projective morphisms between algebraic varieties endowed with an action of a reductive linear algebraic group. We will give brief accounts of the basic results of relative geometric invariant theory and present alternative proofs for recent results obtained by Halle, Hulek, and Zhang on the variation of quotients in relative geometric invariant theory.

Equivalences of (co)module algebra structures over Hopf algebras

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra A, there exists a unique universal Hopf algebra H together with an H-(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. We apply support equivalence in the study of the asymptotic behaviour of codimensions of H-identities and, in particular, to the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture, which was originally concerned with ordinary polynomial identities. As an example we prove this analogue for all unital H-module structures on the algebra $F[x]/(x^2)$ of dual numbers.

On the topology of geometric and rational orbits for algebraic group actions over valued fields, II

The aim of this paper is twofold. First, we show that if G is a smooth nilpotent group acting on an algebraic variety V defined over an admissible valued field k and then the Zariski closedness of the geometric orbit in is equivalent to the Hausdorff closedness of the rational orbit in V(k). Second, we provide some calculations for the fact that there is a bijection between the set of G(k)-orbits and the kernel of the natural map in flat cohomology. These results are obtained in the framework of studying the rational orbits.

Topological dynamics and NIP fields

We study definable topological dynamics of some algebraic group actions over an arbitrary NIP field K. We show that the Ellis group of the universal definable flow of SL2(K) is non-trivial if the multiplicative group of K is not type-definably connected, providing a way to find multiple counterexamples to the (model-theoretic) Ellis group conjecture, particularly in the case of dp-minimal fields. We also study some structure theory of algebraic groups over K with definable f-generics.

Hodge-to-de Rham Degeneration for Stacks

Abstract We introduce a notion of a Hodge-proper stack and apply the strategy of Deligne and Illusie to prove the Hodge-to-de Rham degeneration in this setting. In order to reduce the statement in characteristic $0$ to characteristic $p$, we need to find a good integral model of a stack (namely, a Hodge-proper spreading), which, unlike in the case of proper schemes, need not to exist in general. To address this problem, we investigate the property of spreadability in more detail by generalizing standard spreading out results for schemes to higher Artin stacks and showing that all proper and some global quotient stacks are Hodge-properly spreadable. As a corollary, we deduce a (noncanonical) Hodge decomposition of the equivariant cohomology for certain classes of varieties with an algebraic group action.

Very special algebraic 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.

The Square-Zero Basis of Matrix Lie Algebras

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given.

On the topology of geometric and rational orbits for algebraic group actions over valued fields

Abstract In this note, we study the relationship between Zariski and relative closedness for actions of (smooth) algebraic groups defined over valued (mainly local) fields of any characteristic. In particular, we use some recent basic results regarding the completely reducible subgroups and cocharacter-closedness due to Bate–Herpel–Röhrle–Tange and Uchiyama to construct some actions of simple algebraic groups G of the types D 4 {D_{4}} , E 6 {E_{6}} , E 7 {E_{7}} , E 8 {E_{8}} , G 2 {G_{2}} on an affine variety defined over a local function field k, and v ∈ V ⁢ ( k ) {v\in V(k)} such that the geometric orbit G . v {G.v} is Zariski closed although the corresponding relative orbit G ⁢ ( k ) . v {G(k).v} is not closed in the topology induced from k. Besides, by using an interesting result due to Gabber, Gille and Moret-Bailly, we show that this phenomenon does not appear when we consider the action of either a smooth unipotent group or a smooth commutative algebraic group, defined over an admissible valued (e.g., local) field.

A short survey on observability

The exploration of the notion of observability exhibits transparently the rich interplay between algebraic and geometric ideas in geometric invariant theory. The concept of observable subgroup was introduced in the early 1960s with the purpose of studying extensions of representations from an afine algebraic subgroup to the whole group. The extent of its importance in representation and invariant theory in particular for Hilbert's 14th problem was noticed almost immediately. An important strenghtening appeared in the mid 1970s when the concept of strong observability was introduced and it was shown that the notion of observability can be understood as an intermediate step in the notion of reductivity (or semisimplicity), when adequately generalized. More recently starting in 2010, the concept of observable subgroup was expanded to include the concept of observable action of an afine algebraic group on an afine variety, launching a series of new applications and opening a surge of very interesting activity. In another direction around 2006, the related concept of observable adjunction was introduced, and its application to module categories over tensor categories was noticed. In the current survey, we follow (approximately) the historical development of the subject introducing along the way, the definitions and some of the main results including some of the proofs. For the unproven parts, precise references are mentioned.

Wonderful compactification of character varieties

Using the wonderful compactification of a semisimple adjoint affine algebraic group G defined over an algebraically closed field k of arbitrary characteristic, we construct a natural compactification Y of the G-character variety of any finitely generated group F. When F is a free group, we show that this compactification is always simply connected with respect to the \'etale fundamental group, and when k=C it is also topologically simply connected. For other groups F, we describe conditions for the compactification of the moduli space to be simply connected and give examples when these conditions are satisfied, including closed surface groups and free abelian groups when G=PGL(n,C). Additionally, when F is a free group we identify the boundary divisors of Y in terms of previously studied moduli spaces, and we construct a family of Poisson structures on Y and its boundary divisors arising from Belavin-Drinfeld splittings of the double of the Lie algebra of G. In the appendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson structure on a quotient of a Poisson algebraic variety by the action of a reductive Poisson algebraic group.

Integration of modules I: stability

We explore the integration of representations from a Lie algebra to its algebraic group in positive characteristic. An integrable module is stable under the twists by group elements. Our aim is to investigate cohomological obstructions for passing from stability to an algebraic group action. As an application, we prove integrability of bricks for a semisimple algebraic group.

Homogeneous fibrations on log Calabi–Yau varieties

We prove a structure theorem for the Albanese maps of varieties with $$\mathbb {Q}$$-linearly trivial log canonical divisors. Our start point is the action of a nonlinear algebraic group on a projective variety.