Smooth embeddings of SL(2) and PGL(2)

An affine algebraic group

<p>We can view quiver representations of a fixed dimension vector as an algebraic variety over an algebraically closed field $K$. There is an action of the product of general linear groups on each of these varieties where the orbits of the action correspond to isomorphism classes of quiver representation. A $K$-algebra $A$ is said to have the dense orbit property if for each dimension vector, the product of the general linear group acts on each irreducible component of the module variety with a dense orbit. Under certain conditions, a $K$ algebra $A$ is representation finite if and only if it $A$ has the dense orbit property. The implication representation finite implies the dense orbit property is always true. The converse is not true in general, as shown by Chindris, Kinser, and Weyman in \cite{ryan}. Our main theorem of this thesis builds on their work to give a family of representation infinite algebras with the dense orbit property. We also give a conjectured classification of indecomposables with dense orbits. \par</p> <p>In the future, we hope the work presented here can be used to find even more examples of representation infinite algebra with the dense orbit property to then develop deeper theory to classify algebras with the dense orbit property that are representation infinite.</p>

We show that ifG is a connected Lie group andH is a closed subgroup such that the dimension ofG/H is at least 2, then there exists a nontrivial oneparameter subgroup ofG whose action onG/H has no dense orbits. This strengthens a result ofC. Scheiderer [10].

One of the most elegant classical results about nonsingular algebraic curves is Riemann’s singularity theorem, which provides a beautiful interpretation of the singularities of Riemann’s theta function in terms of the geometry of special divisors. In recent times, soliton theory has suggested generalizations of the classical geometry. Integrable partial differential equations posed with periodic or scattering boundary conditions lead to Riemann surfaces of infinite, or even “continuously infinite”, genus, and to correspondingly huge Jacobians ([McKT], [EMcK]). More special boundary conditions have been studied via the theory of infinite-dimensional Lie groups and their homogeneous spaces. In this latter approach, one finds a large family of (possibly infinite-dimensional) algebraic varieties that are natural generalizations of the classical Jacobians. The Lie-theoretic aspects of these varieties are, of course, prominent in such work ([SW], [Sch]). In another direction, soliton problems have led naturally to singular limits of smooth Riemann surfaces. While, in the abstract, singular curves and their Jacobians are a subject of current research, soliton theory provides a concrete context in which to study such algebraic varieties.

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → DerK(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.

The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received much substance and impetus from the work of Severi on commutative algebraic groups over the complex number field, that of Kolchin, Chevalley, and Borel on algebraic groups of matrices, and especially Weil's research on abelian varieties and algebraic transformation spaces. The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic groups of matrices, or actually of neither of these types.

Ideals with bounded approximate identities in the Fourier algebras on homogeneous spaces

In this paper we will be concerned with orbits of a closed subgroup Z of an algebraic group (respectively Lie group) G on a homogeneous space X for G. More precisely, let D be a closed subgroup of G and let X denote the coset space G/D. Let S be a subgroup of G and let Z denote (GS)0 the identity component of GS, the centralizer of S in G. We consider the orbits of Z on XS, the set of fixed points of S on X. We also treat the more general situation in which S is an algebraic group (respectively Lie group) which acts on G by automorphisms and acts on X compatibly with the action of G; again we consider the orbits of (GS)0 on XS.

On Algebraic Groups and Homogeneous Spaces

Universal torsors of Del Pezzo surfaces and homogeneous spaces

Let $X=G/\Gamma $ be the quotient of a semisimple Lie group G by its non-cocompact arithmetic lattice. Let H be a reductive algebraic subgroup of G acting on X . We give several equivalent algebraic conditions on H for the existence of a fixed compact set in X intersecting every H -orbit. This generalizes previous results concerning certain special reductive group action on X in this setting. When G is of real rank one, $\Gamma $ is a non-cocompact lattice of G , and $H<G$ is an algebraic group, we also obtain an algebraic condition on H which is equivalent to the return of every H -orbit to a single compact set in X . This complements our results in the case of an arithmetic lattice.

Let H be a compact subgroup of a locally compact group G, and let μ be a strongly quasi-invariant Radon measure on the homogeneous space G/H. In this article, we show that every element of G/Hˆ, the character space of G/H, determines a nonzero multiplicative linear functional on L1(G/H,μ). Using this, we prove that for all ϕ∈G/Hˆ, the right ϕ-amenability of L1(G/H,μ) and the right ϕ-amenability of M(G/H) are both equivalent to the amenability of G. Also, we show that L1(G/H,μ), as well as M(G/H), is right ϕ-contractible if and only if G is compact. In particular, when H is the trivial subgroup, we obtain the known results on group algebras and measure algebras.

Let [Formula: see text] be a [Formula: see text]-equivariant algebraic principal [Formula: see text]-bundle over a normal complex affine variety [Formula: see text] equipped with an action of [Formula: see text], where [Formula: see text] and [Formula: see text] are complex linear algebraic groups. Suppose [Formula: see text] is contractible as a topological [Formula: see text]-space with a dense orbit, and [Formula: see text] is a [Formula: see text]-fixed point. We show that if [Formula: see text] is reductive, then [Formula: see text] admits a [Formula: see text]-equivariant isomorphism with the product principal [Formula: see text]-bundle [Formula: see text], where [Formula: see text] is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal [Formula: see text]-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal [Formula: see text]-bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math. 27(14) (2016)].

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindzuk in 1964. We also prove several related hypotheses of Baker and Sprindzuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices.

The main topic of discussion in this survey is the geometric and metric properties of geodesic and horocycle flows on surfaces of constant curvature and their multidimensional analogues, the so-called G-induced actions (that is, restrictions to a suitable subgroup F of the natural action of G on a homogeneous space of it). One usually takes as F one-parameter or cyclic subgroups of G and the horospherical subgroups corresponding to them (the details are given in Sect. 1). However, in the course of the exposition of our main topic we sometimes mention results relating to the case when F is an n-parameter subgroup of G or a lattice. This class of dynamical systems consists precisely of group actions admitting an extension to a transitive action of a Lie group. We shall not be considering in our survey (or, at any rate, not systematically) the following themes which are close to our main subject matter: Topological dynamics and metric properties of general group actions (in particular, the work of Zimmer on semisimple group actions (Zimmer 1984); Group automorphisms; Affine transformations of homogeneous spaces; G-induced flows on spaces with an infinite invariant measure. The papers (Hopf 1939), (Sullivan 1981) and the recent book (Nicholls 1989) are devoted to the connections between G-induced flows (and also geodesic flows) and the theory of discrete groups of motions of a hyperbolic space; The actions of subgroups on the double cosets related to geodesic (for n > 2) and horocycle (for n > 3) flows on an n-dimensional manifold of constant negative curvature (see (Gel’fand and Fomin 1952) and, in particular, (Mautner 1957)). Flows of this type in somewhat more general form differing from geodesic and horocycle flows have not been investigated.