New geometric examples of Anosov actions

We study a new class of Anosov actions (in the sense of Hirsch, Pugh and Shub) of reductive Lie groups, which are related to Riemannian symmetric spaces of non-compact type. The orbits of these actions can be identified with unions of parallel geodesics and the resulting orbit spaces are symplectic manifolds. For symmetric spaces of rank 1 all actions coincide with the geodesic flow.     

Wonderful varieties of rank two

LetG be a complex connected reductive group. Well known wonderfulG-varieties are those of rank zero, namely the generalized flag varietiesG/P, those of rank one, classified in [A], and certain complete symmetric varieties described in [DP] such as the famous space of complete conics. Recently, there is a renewed interest in wonderful varieties of rank two since they were shown to hold a keystone position in the theory of spherical varieties, see [L], [BP], and [K]. The purpose of this paper is to give a classification of wonderful varieties of rank two. These are nonsingular completeG-varieties containing four orbits, a dense orbit and two orbits of codimension one whose closuresD ₁ andD ₂ intersect transversally in the fourth orbit which is of codimension two. We have gathered our results in tables, including isotropy groups, explicit basis of Picard groups, and several combinatorial data in relation with the theory of spherical varieties.     

Complexity of Homogeneous Spaces and Growth of Multiplicities

The complexity of a homogeneous space G/H under a reductive group G is by definition the codimension of general orbits in G/H of a Borel subgroup B\subseteq G. We give a representation-theoretic interpretation of this number as the exponent of growth for multiplicities of simple G-modules in the spaces of sections of homogeneous line bundles on G/H. For this, we show that these multiplicities are bounded from above by the dimensions of certain Demazure modules. This estimate for multiplicities is uniform, i.e., it depends not on G/H, but only on its complexity.     

Complexity and rank of homogeneous spaces

We study an algebraic varieties with the action of a reductive group G. The relation is elucidated between the notions of complexity and rank of an arbitrary G-variety and the structure of stabilizers of general position of some actions of G itself and its Borel subgroup. The application of this theory to homogeneous spaces provides the explicit formulas for the rank and the complexity of quasiaffine G/H in terms of co-isotropy representation of H. The existence of Cartan subspace (and hence the freeness of algebra of invariants) for co-isotropy representation of a connected observable spherical subgroup H is proved.     

Some characteristics of complex behavior of orbits in dynamical systems

We present a brief review of mathematical notions of complexity based on instability of orbits. We show that the complexity as a function of time may grow exponentially in chaotic situations or polynomially for systems with zero topological entropy. At the end we discuss the class of nonchaotic systems for which all orbits are stable but nevertheless behavior of orbits is complex. We introduce a new notion of complexity for such a kind of systems.     

Separated orbits for certain non-reductive subgroups

In this paper, we extend central portions of the geometric invariant theory for reductive groups G to nonreductive subgroups H satisfying the codimension 2 condition on G/H. First, the separated orbits for such subgroups are described using a one-parameter subgroup criterion. Second, the desired theorems concerning quotient varieties for spaces of separated orbits are proved.     

Homoclinic orbits to a saddle-center in a fourth-order differential equation

We study homoclinic orbits to a saddle-center of a fourth-order ordinary differential equation, which is invariant under the transformation x→−x, involving an eigenvalue parameter q and an odd, piece-wise, cubic-type nonlinearity. It is found that for a sequence of eigenvalues which tends to infinity, homoclinic orbits exist whose complexity increases as the eigenvalue becomes larger. These orbits are found to be embedded in branches of homoclinic orbits to periodic orbits as x→±∞.     

Normalizers of subsystem subgroups in finite groups of Lie type

Finite groups of Lie type form the greater part of known finite simple groups. An important class of subgroups of finite groups of Lie type are so-called reductive subgroups of maximal rank. These arise naturally as Levi factors of parabolic groups and as centralizers of semisimple elements, and also as subgroups with maximal tori. Moreover, reductive groups of maximal rank play an important part in inductive studies of subgroup structure of finite groups of Lie type. Yet a number of vital questions dealing in the internal structure of such subgroups are still not settled. In particular, we know which quasisimple groups may appear as central multipliers in the semisimple part of any reductive group of maximal rank, but we do not know how normalizers of those quasisimple groups are structured. The present paper is devoted to tackling this problem. Supported by RFBR (grant No. 05-01-00797) and by SB RAS (Young Researchers Support grant No. 29 and Integration project No. 2006.1.2). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 3–30, January–February, 2008.     

Dual pairs and Kostant-Sekiguchi correspondence. II. Classification of nilpotent elements

We classify the homogeneous nilpotent orbits in certain Lie color algebras and specialize the results to the setting of a real reductive dual pair. For any member of a dual pair, we prove the bijectivity of the two Kostant-Sekiguchi maps by straightforward argument. For a dual pair we determine the correspondence of the real orbits, the correspondence of the complex orbits and explain how these two relations behave under the Kostant-Sekiguchi maps. In particular we prove that for a dual pair in the stable range there is a Kostant-Sekiguchi map such that the conjecture formulated in [6] holds. We also show that the conjecture cannot be true in general.     

Local geometry of orbits for an ordinary classical lie supergroup

In this paper we identify a real reductive dual pair of Roger Howe with an Ordinary Classical Lie supergroup. In these terms we describe the semisimple orbits of the dual pair in the symplectic space, a slice through a semisimple element of the symplectic space, an analog of a Cartan subalgebra, the corresponding Weyl group and the corresponding Weyl integration formula.     

Leavitt path algebras are Bézout

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity.We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity.As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. Furthermore, using a theorem by Basmajian, we get a bound on the number of mapping class group orbits on a given hyperbolic surface that can contain short curves. For a fixed length, this bound is polynomial in the signature of the surface.The arguments we use are based on counting embeddings of ribbon graphs.     

Locally finite Lie algebras¶with unitary highest weight representations

In this paper we essentially classify all locally finite Lie algebras with an involution and a compatible root decomposition which permit a faithful unitary highest weight representation. It turns out that these Lie algebras have many interesting relations to geometric structures such as infinite-dimensional bounded symmetric domains and coadjoint orbits of Banach–Lie groups which are strong K?hler manifolds. In the present paper we concentrate on the algebraic structure of these Lie algebras, such as the Levi decomposition, the structure of the almost reductive and locally nilpotent part, and the structure of the representation of the almost reductive algebra on the locally nilpotent ideal. Received: 2 August 2000 / Revised version: 10 January 2001     

Relative equidimensionality and stability of actions¶of a reductive algebraic group

In order to inquire into invariants of non-semisimple groups, we introduce and study relative versions of equidimensionality and stabilty, which are called relative quasi-equidimensionality and relative stability, of actions of affine algebraic groups, especially of reductive groups, on affine varieties. As an application of our results, for complex reductive groups of semisimple rank one, we characterize, respectively, relatively stable representations and relatively equidimensional representations and, consequently, show that every equidimensional representation is cofree. Received: 23 October 1998     

Complexity of multiplication in commutative group algebras over fields of characteristic 0

Let rkA denote the bilinear complexity (also known as rank) of a finite-dimension associative algebra A. Algebras of minimal rank are widely studied from the point of view of bilinear complexity. These are the algebras A for which the Alder-Strassen inequality is satisfied as an equality, i.e., rkA = 2dimA ? t, where t is the number of maximum two-sided ideals in A. It is proved in this work that an arbitrary commutative group algebra over a field of characteristic 0 is an algebra of minimal rank. The structure and precise values of the bilinear complexity of commutative group algebras over a field of rational numbers are obtained.     

Classification of two-orbit varieties

We obtain the classification of two-orbit varieties, i.e. the normal complete complex algebraic varieties on which a reductive complex algebraic group acts with two orbits. We prove also Luna's conjecture saying that these varieties are spherical, i.e. admit a dense orbit of a Borel subgroup. Received: September 1, 2000     

Equivariant completions of homogenous algebraic varieties by homogenous divisors

Complete smooth complex algebraic varieties with an almost transitive action of a linear algebraic group are studied. They are classified in the case, when the complement of the open orbit is a homogeneous hypersurface. If the group and the isotropy subgroup at a generic point are both reductive, then there exists a natural one-to-one correspondence between these two-orbit varieties and compact riemannian symmetric spaces of rank one.     

About Knop's Action of the Weyl Group on the Set of Orbits of a Spherical Subgroup in the Flag Manifold

Let G be a connected reductive algebraic group defined on an algebraically closed field k of characteristic different from 2. Let B denote the flag variety of G. Let H be a spherical subgroup of G. F. Knop defined an action of the Weyl group W of G on the finite set of the H-orbits in B. Here, we define an invariant, namely the type, separating the orbits of W.     

Mixing chiral polytopes

An abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. Examples of chiral polytopes have been difficult to find. A ??mixing?? construction lets us combine polytopes to build new regular and chiral polytopes. By using the chirality group of a polytope, we are able to give simple criteria for when the mix of two polytopes is chiral.