Solvable Algebraic Groups Research Articles

Moduli spaces of morphisms into solvable algebraic groups

We construct (in significant generality) moduli spaces representing the functor of morphisms from a scheme into a solvable algebraic group.

Twisted conjugacy in linear algebraic groups II

Let G be a linear algebraic group over an algebraically closed field k and Autalg(G) the group of all algebraic group automorphisms of G. For every φ∈Autalg(G) let R(φ) denote the set of all orbits of the φ-twisted conjugacy action of G on itself (given by (g,x)↦gxφ(g−1), for all g,x∈G). We say that G has the algebraic R∞-property if R(φ) is infinite for every φ∈Autalg(G). In [1] we have shown that this property is satisfied by every connected non-solvable algebraic group. From a theorem due to Steinberg it follows that if a connected algebraic group G has the algebraic R∞-property, then Gφ (the fixed-point subgroup of G under φ) is infinite for all φ∈Autalg(G). In this article we show that the condition is also sufficient. We also show that a Borel subgroup of any semisimple algebraic group has the algebraic R∞-property and identify certain classes of solvable algebraic groups for which the property fails.

Real Liouvillian Extensions of Partial Differential Fields

In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally p-adic differential fields with a p-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally p-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.

PETERZIL–STEINHORN SUBGROUPS AND -STABILIZERS IN ACF

AbstractWe consider G, a linear algebraic group defined over $\Bbbk $ , an algebraically closed field (ACF). By considering $\Bbbk $ as an embedded residue field of an algebraically closed valued field K, we can associate to it a compact G-space $S^\mu _G(\Bbbk )$ consisting of $\mu $ -types on G. We show that for each $p_\mu \in S^\mu _G(\Bbbk )$ , $\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$ is a solvable infinite algebraic group when $p_\mu $ is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of $\mathrm {Stab}\left (p_\mu \right )$ in terms of the dimension of p.

Homogeneous varieties under split solvable algebraic groups

We present a modern proof of a theorem of Rosenlicht, asserting that every variety as in the title is isomorphic to a product of affine lines and punctured affine lines.

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.

BOREL DENSITY FOR APPROXIMATE LATTICES

We extend classical density theorems of Borel and Dani–Shalom on lattices in semisimple, respectively solvable algebraic groups over local fields to approximate lattices. Our proofs are based on the observation that Zariski closures of approximate subgroups are close to algebraic subgroups. Our main tools are stationary joinings between the hull dynamical systems of discrete approximate subgroups and their Zariski closures.

Character sheaves on neutrally solvable groups

Let G G be an algebraic group over an algebraically closed field k \mathtt {k} of characteristic p > 0 p>0 . In this paper we develop the theory of character sheaves on groups G G such that their neutral connected components G ∘ G^\circ are solvable algebraic groups. For such algebraic groups G G (which we call neutrally solvable) we will define the set CS ⁡ ( G ) \operatorname {CS}(G) of character sheaves on G G as certain special (isomorphism classes of) objects in the category D G ( G ) \mathscr {D}_G(G) of G G -equivariant Q ¯ ℓ \overline {\mathbb {Q}}_{\ell } -complexes (where we fix a prime ℓ ≠ p \ell \neq p ) on G G . We will describe a partition of the set CS ⁡ ( G ) \operatorname {CS}(G) into finite sets known as L \mathbb {L} -packets and we will associate a modular category M L \mathscr {M}_L with each L \mathbb {L} -packet L L of character sheaves using a truncated version of convolution of character sheaves. In the case where k = F ¯ q \mathtt {k}=\overline {\mathbb {F}}_q and G G is equipped with an F q \mathbb {F}_q -Frobenius F F we will study the relationship between F F -stable character sheaves on G G and the irreducible characters of (all pure inner forms of) G F G^F . In particular, we will prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for neutrally solvable groups and that these almost characters coincide with the “trace of Frobenius” functions associated with F F -stable character sheaves. We will also prove that the matrix relating the irreducible characters and almost characters is block diagonal where the blocks on the diagonal are parametrized by F F -stable L \mathbb {L} -packets. Moreover, we will prove that the block in this transition matrix corresponding to any F F -stable L \mathbb {L} -packet L L can be described as the crossed S-matrix associated with the auto-equivalence of the modular category M L \mathscr {M}_L induced by F F .

Commensurators of solvable S-arithmetic groups

We show that the abstract commensurator of an S-arithmetic subgroup of a solvable algebraic group over Q is isomorphic to the Q-points of an algebraic group, and compare this with examples of nonlinear abstract commensurators of S-arithmetic groups in positive characteristic. In particular, we include a description of the abstract commensurator of the lamplighter group.

Homogeneous Spaces of Nonreductive Type That Do Not Model Any Compact Manifold

We give necessary conditions for the existence of a compact manifold locally modeled on a given homogeneous space, which generalize some earlier results, in terms of relative Lie algebra cohomology. Applications include both reductive and nonreductive cases. For example, we prove that there does not exist a compact manifold locally modeled on a positive-dimensional coadjoint orbit of a real linear solvable algebraic group.

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

This paper is the first part of a two part paper which introduces a method for determining the vanishing topology of nonisolated matrix singularities. A foundation for this is the introduction in this first part of a method for obtaining new classes of free divisors from representations V of connected solvable linear algebraic groups G. For equidimensional representations where dimG = dimV , with V having an open orbit, we give sufficient conditions that the complement E of this open orbit, the “exceptional orbit variety”, is a free divisor (or a slightly weaker free* divisor). We do so by introducing the notion of a “block representation”which is especially suited for both solvable groups and extensions of reductive groups by them. This is a representation for which the matrix representing a basis of associated vector fields on V defined by the representation can be expressed using a basis for V as a block triangular matrix, with the blocks satisfying certain nonsingularity conditions. We use the Lie algebra structure of G to identify the blocks, the singular structure, and a defining equation for E. This construction naturally fits within the framework of towers of Lie groups and representations yielding a tower of free divisors which allows us to inductively represent the variety of singular matrices as fitting between two free divisors. We specifically apply this to spaces of matrices including m×m symmetric, skew-symmetric or general matrices, where we prove that both the classical Cholesky factorization of matrices and a further “modified Cholesky factorization”which we introduce are given by block representations of solvable group actions. For skew-symmetric matrices, we further introduce an extension of the method valid for a representation of a nonlinear infinite dimensional solvable Lie algebras. In part II, we shall use these geometric decompositions and results for computing the vanishing topology for nonlinear sections of free divisors to compute the vanishing topology for matrix singularities for all of the classes.

Integrality and arithmeticity of solvable linear groups

We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group G is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of G. We also provide a simple new algorithm for integrality testing of finitely generated solvable-by-finite linear groups over the rational field. The algorithms have been implemented in Magma.

Nonarchimedean superrigidity of solvable S-arithmetic groups

Let Γ be an S-arithmetic subgroup of a solvable algebraic group G over an algebraic number field F, such that the finite set S contains at least one place that is nonarchimedean. We construct a certain group H=(ResF/QΓ¯)charSΓ, such that if L is any local field and α:Γ→GLn(L) is any homomorphism, then α virtually extends (modulo a bounded error) to a continuous homomorphism defined on some finite-index subgroup of H. In the special case where F=Q, rankRG=0, and Γ is Zariski-dense in G, we may let H=GS.We also point out a generalization that does not require G to be solvable.

Isoperimetric profile and random walks on locally compact solvable groups

We study the large-scale geometry of a large class of amenable locally compact groups comprising all solvable algebraic groups over a local field and their discrete subgroups. We show that the isoperimetric profile of these groups is in some sense optimal among amenable groups. We use this fact to compute the probability of return of symmetric random walks, and to derive various other geometric properties.

Solvable group representations and free divisors whose complements are [formula omitted]ʼs

We apply previous results on the representations of solvable linear algebraic groups to construct a new class of free divisors whose complements are K ( π , 1 ) ʼs. These free divisors arise as the exceptional orbit varieties for a special class of “block representations” and have the structure of determinantal arrangements. Among these are the free divisors defined by conditions for the (modified) Cholesky-type factorizations of matrices, which contain the determinantal varieties of singular matrices of various types as components. These complements are proven to be homotopy tori, as are the Milnor fibers of these free divisors. The generators for the complex cohomology of each are given in terms of forms defined using the basic relative invariants of the group representation.

Actions of Solvable Algebraic Groups on Central Simple Algebras

Let k be an algebraically closed base field of arbitrary characteristic. In this paper, we study actions of a connected solvable linear algebraic group G on a central simple algebra Q. The main result is the following: Q can be split G-equivariantly by a finite-dimensional splitting field, provided that G acts “algebraically,” i.e., provided that Q contains a G-stable order on which the action is rational. As an application, it is shown that rational torus actions on prime PI-algebras are induced by actions on commutative domains.

Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups

We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties of subgroups in solvable linear algebraic groups.

Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups

We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties of subgroups in solvable linear algebraic groups.

Positivity in equivariant Schubert calculus

We prove a positivity property for the cup product in the T-equivariant cohomology of the flag variety. This was conjectured by D. Peterson and has as a consequence a conjecture of S. Billey. The result for the flag variety follows from a more general result about algebraic varieties with an action of a solvable linear algebraic group such that the unipotent radical acts with finitely many orbits. The methods are those used by S. Kumar and M. Nori.

Complexes and Vanishing of Cohomology for Group Schemes

Completely reducible homology complexes were originally introduced in the context of finite groups and used to study the question of vanishing of cohomology. In this paper we study these complexes and the vanishing of cohomology for arbitrary finite-dimensional cocommutative Hopf algebras. Applications are later provided for infinitesimal group schemes of reductive and solvable algebraic groups.