Action Of Algebraic Group Research Articles

Tautological systems and free divisors

We introduce tautological systems defined by prehomogeneous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor satisfying a certain finiteness condition, we show that these systems underly mixed Hodge modules. A dimensional reduction is considered and gives rise to one-dimensional differential systems generalizing the quantum differential equation of projective spaces.

Flexibility of normal affine horospherical varieties

We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only constant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety with only constant invertible functions is flexible.

On Conjugacy of Stabilizers of Reductive Group Actions

It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf–Ness Theorem, arXiv:1811.07195v1 (17 Nov 2018), is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems.

Small quantum groups associated to Belavin-Drinfeld triples

For a simple Lie algebra g \mathsf {g} of type A A , D D , E E we show that any Belavin-Drinfeld triple on the Dynkin diagram of g \mathsf {g} produces a collection of Drinfeld twists for Lusztig’s small quantum group u q ( g ) u_q(\mathsf {g}) . These twists give rise to new finite-dimensional factorizable Hopf algebras, i.e., new small quantum groups. For any Hopf algebra constructed in this manner, we identify the group of grouplike elements, identify the Drinfeld element, and describe the irreducible representations of the dual in terms of the representation theory of the parabolic subalgebra(s) in g \mathsf {g} associated to the given Belavin-Drinfeld triple. We also produce Drinfeld twists of u q ( g ) u_q(\mathsf {g}) which express a known algebraic group action on its category of representations and pose a subsequent question regarding the classification of all twists.

Invariants of algebraic group actions from differential point of view

In this paper we suggest an approach to study actions of semisimple (or reductive) algebraic groups in their algebraic complex representations. We use differential-geometric methods instead of classical algebraic constructions. Namely, according to Borel–Weil–Bott theorem, every algebraic representation of semisimple algebraic group is isomorphic to the action of this group on the module of holomorphic sections of some reductive bundle over homogeneous space. Using this, we give a complete description of the field of differential invariants for this action and obtain a criterion, which separates regular orbits.

On base sizes for almost simple primitive groups

Let G⩽Sym(Ω) be a finite almost simple primitive permutation group with socle G0. A subset of Ω is a base for G if its pointwise stabilizer is trivial; the base size of G, denoted b(G), is the minimal size of a base. We say that G is standard if G0=An and Ω is an orbit of subsets or partitions of {1,…,n}, or if G0 is a classical group and Ω is an orbit of subspaces (or pairs of subspaces) of the natural module for G0. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have b(G)⩽7 for every non-standard group G, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. In this paper, we extend this result by classifying the non-standard groups with b(G)=6. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type.

Higher algebraic K-theory and representations of algebraic groups

This paper is concerned with Higher Algebraic K-theory and actions of algebraic groups G on such ‘nice’ categories as the category of algebraic vector bundles on a scheme X. Such ‘nice’ categories are examples of ‘exact’ categories with the observation that the category of actions on G on such exact categories also form an exact category called equivariant exact categories on which one can do higher Algebraic K-theory (of Quillen) called equivariant higher Algebraic K-theory—the higher dimensional generalizations of classical equivariant K-theory which belongs to the field of representation theory. Thus, for an Algebraic group G over a number field or p-adic field F, we present constructions and computations of equivariant higher K-groups as well as ‘profinite’ or ‘continuous’ higher K-groups for some G-Scheme X. In particular, we present explicit l-completeness (l a rational prime) and finiteness computations for higher K-groups and profinite higher K-groups for twisted flag varieties.

Twists of quantum Borel algebras

We classify Drinfeld twists for the quantum Borel subalgebra uq(b) in the Frobenius–Lusztig kernel uq(g), where g is a simple Lie algebra over C and q an odd root of unity. More specifically, we show that alternating forms on the character group of the group of grouplikes for uq(b) generate all twists for uq(b), under a certain algebraic group action. This implies a simple classification of finite-dimensional Hopf algebras whose categories of representations are tensor equivalent to that of uq(b). We also show that cocycle twists for the corresponding De Concini–Kac algebra are in bijection with alternating forms on the aforementioned character group.

On the involution fixity of exceptional groups of Lie type

The involution fixity [Formula: see text] of a permutation group [Formula: see text] of degree [Formula: see text] is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if [Formula: see text] is the socle of such a group, then either [Formula: see text], or [Formula: see text] and [Formula: see text] is a Suzuki group in its natural [Formula: see text]-transitive action of degree [Formula: see text]. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with [Formula: see text]. This extends recent work of Liebeck and Shalev, who established the bound [Formula: see text] for every almost simple primitive group of degree [Formula: see text] with socle [Formula: see text] (with a prescribed list of exceptions). Finally, by combining our results with the Lang–Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.

Degeneration of Horospheres in Spherical Homogeneous Spaces

Horospheres for an action of a semisimple algebraic group G on an affine variety X are the generic orbits of a maximal unipotent subgroup U ⊂ G or, equivalently, the generic fibers of the categorical quotient of the variety X by the action of U, which is defined by the values of the highest weight functions. The remaining fibers of this quotient (which we call degenerate horospheres) for a certain class of spherical G-varieties containing all simply connected symmetric spaces are studied.

Invariant generalized functions supported on an orbit

We study the space of invariant generalized functions supported on an orbit of the action of a real algebraic group on a real algebraic manifold. This space is equipped with the Bruhat filtration. We study the generating function of the dimensions of the filtras, and give some methods to compute it. To illustrate our methods we compute those generating functions for the adjoint action of GL3(C). Our main tool is the notion of generalized functions on a real algebraic stack, introduced recently in [Sak16].

Algebraic group actions on normal varieties

Let $G$ be a connected algebraic $k$-group acting on a normal $k$-variety, where $k$ is a field. We show that $X$ is covered by open $G$-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a $G$-linearized vector bundle on an abelian variety, quotient of $G$. This generalizes a classical result of Sumihiro for actions of smooth connected affine algebraic groups.

Geometric aspects of representation theory for DG algebras: answering a question of Vasconcelos

AbstractWe apply geometric techniques from representation theory to the study of homologically finite differential graded (DG) modules over a finite dimensional, positively graded, commutative DG algebra . In particular, in this setting we prove a version of a theorem of Voigt by exhibiting an isomorphism between the Yoneda Ext group and a quotient of tangent spaces coming from an algebraic group action on an algebraic variety. As an application, we answer a question of Vasconcelos from 1974 by showing that a local ring has only finitely many semidualizing complexes up to shift‐isomorphism in the derived category .

On base sizes for algebraic groups

Let G be a permutation group on a set \Omega . A subset of \Omega  is a base for G if its point-wise stabilizer is trivial; the base size of G is the minimal cardinality of a base. In this paper we initiate the study of bases for algebraic groups defined over an algebraically closed field. In particular, we calculate the base size for all primitive actions of simple algebraic groups, obtaining the precise value in almost all cases. We also introduce and study two new base measures, which arise naturally in this setting. We give an application concerning the essential dimension of simple algebraic groups, and we establish several new results on base sizes for the corresponding finite groups of Lie type. The latter results are an important contribution to the classical study of bases for finite primitive permutation groups. We also indicate some connections with generic stabilizers for representations of simple algebraic groups.

Almost algebraic actions of algebraic groups and applications to algebraic representations

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 [2].

Equivalences of derived factorization categories of gauged Landau–Ginzburg models

For a given Fourier–Mukai equivalence of bounded derived categories of coherent sheaves on smooth quasi-projective varieties, we construct Fourier–Mukai equivalences of derived factorization categories of gauged Landau–Ginzburg (LG) models.As an application, we obtain some equivalences of derived factorization categories of K-equivalent gauged LG models. This result is an equivariant version of the result of Baranovsky and Pecharich, and it also gives a partial answer to Segal's conjecture. As another application, we prove that if the kernel of the Fourier–Mukai equivalence is linearizable with respect to a reductive affine algebraic group action, then the derived categories of equivariant coherent sheaves on the varieties are equivalent. This result is shown by Ploog for finite groups case.

INVERTIBLE BIMODULES, MIYASHITA ACTION IN MONOIDAL CATEGORIES AND AZUMAYA MONOIDS

In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita’s classical result and sheds light on other applications of geometric nature which cannot be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.

Topology of exceptional orbit hypersurfaces of prehomogeneous spaces

We consider the topology for a class of hypersurfaces with highly nonisolated singularities which arise as ‘exceptional orbit varieties’ of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements, and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse-type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a ‘complex geometry’ resulting from a transitive action of an appropriate algebraic group, yielding a compact ‘model submanifold’ for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2-torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. Unlike isolated singularities, the cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the ‘stable range’ as the stable homotopy groups of the associated infinite-dimensional symmetric spaces. Lastly, we combine the preceding with a Theorem of Oka to obtain a class of ‘formal linear combinations’ of exceptional orbit hypersurfaces which have Milnor fibers that are homotopy equivalent to joins of the compact model submanifolds. It follows that Milnor fibers for all of these hypersurfaces are essentially never homotopy equivalent to bouquets of spheres (even allowing differing dimensions).

A remark on [formula omitted]-valued cohomology groups of algebraic group actions

We prove that for a weakly mixing algebraic action σ:G↷(X,ν), the nth cohomology group Hn(G↷X;T), after quotienting out the natural subgroup Hn(G,T), contains Hn(G,Xˆ) as a natural subgroup for n=1. If we further assume the diagonal actions σ2, σ4 are T-cocycle superrigid and H2(G,Xˆ) is torsion free as an abelian group, then the above also holds true for n=2. Applying it for principal algebraic actions when n=1, we show that H2(G,ZG) is torsion free as an abelian group when G has property (T) as a direct corollary of Sorin Popa's cocycle superrigidity theorem; we also use it (when n=2) to answer, negatively, a question of Sorin Popa on the 2nd cohomology group of Bernoulli shift actions of property (T) groups.

Topology of exceptional orbit hypersurfaces of prehomogeneous spaces

We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a complex geometry resulting from a transitive action of an appropriate algebraic group, yielding a compact model submanifold for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2 torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. The cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the stable range as the stable homotopy groups of the associated infinite dimensional symmetric spaces. Applying a Theorem of Oka we obtain a class of formal linear combinations of exceptional orbit hypersurfaces which have Milnor fibers which are homotopy equivalent to joins of the compact model submanifolds.