Hyperpolygon spaces and their cores

Given an -tuple of positive real numbers , Konno (2000) defines the hyperpolygon space , a hyperkähler analogue of the Kähler variety parametrizing polygons in with edge lengths . The polygon space can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural -action, and the union of the precompact orbits is called the core. We study the components of the core of , interpreting each one as a moduli space of pairs of polygons in with certain properties. Konno gives a presentation of the cohomology ring of ; we extend this result by computing the -equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.

     

Adjacency preservers,symmetric matrices,and cores

It is shown that the graph Γ _n that has the set of all n×n symmetric matrices over a finite field as the vertex set, with two matrices being adjacent if and only if the rank of their difference equals one, is a core if n≥3. Eigenvalues of the graph Γ _n are calculated as well.     

K-Loops, gyrogroups and symmetric spaces

In this paper we present a new class of K-Loops consisting of homogeneous symmetric spaces of the noncompact type. The most famous physical example of a representative of this class is found in special relativity: The set of all relativistically admissible velocities, R _c ³:= {υ ∈ R3: ¦v¦< c} (c the speed of light), together with the relativistic velocity composition law forms a K-Loop, as was already shown by A. Ungar in 1988. The set of the corresponding boosts turns out to be the effective homogeneous symmetric space SO (1,3)⁰/SO(3), where SO(1,3)⁰ denotes the proper orthochronous Lorentz group and SO(3) the subgroup of proper rotations in three dimensions, which is also found to be the set of fixed points of an automorphism σ: SO(1,3)⁰ → SO(1,3)⁰, whose action on a matrix A in the defining representation is given by σ(A) = (AT)?1. Moreover, the triple (so(l,3),so(3),s) consisting of the Lie algebras of SO(1,3) and SO(3) and the induced automorphism s = (dσ)e is an orthogonal symmetric Lie algebra of the noncompact type. This property of the Lie algebra makes SO(l, 3)/SO(3) an (effective) homogeneous symmetric space of the noncompact type. The method of exact decomposition allows endowing each homogeneous symmetric space of the noncompact type with the structure of a K-Loop in a similar manner. Finally, we treat some alternative approaches which lead to K-Loops that are isomorphic to the one described above.     

Invariants,orbits and symmetric spaces

Lavoro svolto nell'ambito del G.N.S.A.G.A. del C.N.R. con contributo del M.P.I. fondi 40%.     

Invariant differential operators on Hermitian symmetric spaces and their eigenvalues

Let be the invariant Cauchy Riemann operator and the corresponding invariant Laplacians on a bounded symmetric domain. We calculate the eigenvalues ofM _m on spherical functions. In particular we prove that for a symmetric domain of rank two the operatorsM ₁,M ₃ generate all invariant differential operators. We also find the eigenvalues of the generators introduced by Shimura.     

Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

In this paper we define a distinguished boundary for the complex crowns of non-compact Riemannian symmetric spaces . The basic result is that affine symmetric spaces of can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.

     

Weakly symmetric spaces and bounded symmetric domains

LetG be a connected, simply-connected, real semisimple Lie group andK a maximal compactly embedded subgroup ofG such thatD=G/K is a hermitian symmetric space. Consider the principal fiber bundleM=G/K _s G/K, whereK _s is the semisimple part ofK=K _s ·Z _K ⁰ andZ _K ⁰ is the connected center ofK. The natural action ofG onM extends to an action ofG ¹=G×Z _K ⁰ . We prove as the main result thatM is weakly symmetric with respect toG ¹ and complex conjugation. In the case whereD is an irreducible classical bounded symmetric domain andG is a classical matrix Lie group under a suitable quotient, we provide an explicit construction ofM=D×S ¹ and determine a one-parameter family of Riemannian metrics onM invariant underG ¹. Furthermore,M is irreducible with respect to . As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf. [M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rank-one case.Research partially supported by an NSF grant. The author wishes to thank the International Erwin Schroedinger Institute for its hospitality during the preparation of this paper.     

Totally geodesic submanifolds in compact Riemannian symmetric spaces and their stability

In this paper, we determine a large class of totally geodesic submanifolds of a compact Riemannian symmetric space. The stability of these submanifolds in their ambient space is also determined.     

Marked length rigidity of rank one symmetric spaces and their product

In this paper we show that if two Zariski dense representations, from a group G into Iso(X) where X is rank one symmetric space, have the proportional marked length spectrum, then they are conjugate. As a generalization we show that a Zariski dense representation into the isometry group of the product of rank one symmetric spaces is determined by the marked cross ratio.     

Parabolic symmetric spaces

We study here systems of symmetries on |1|-graded parabolic geometries. We are interested in smooth systems of symmetries, and we discuss non-flat homogeneous |1|-graded geometries. We show the existence of an invariant admissible affine connection under quite weak condition on the system.     

Quantum symmetric spaces

Let G be a semisimple Lie group, g its Lie algebra. For any symmetric space M over G we construct a new (deformed) multiplication in the space A of smooth functions on M. This multiplication is invariant under the action of the Drinfeld-Jimbo quantum group U_hg and is commutative with respect to an involutive operator . Such a multiplication is unique. Let M be a kählerian symmetric space with the canonical Poisson structure. Then we construct a U_hg-invariant multiplication in A which depends on two parameters and is a quantization of that structure.     

Projectively symmetric spaces

Sunto In questo lavoro l'autore estende la nozione di simmetria al caso di una varietà reale C dotata di una connessione proiettiva normale, introducendo la classe degli spazi proiettivamente simmetrici. L'autore prova che ogni spazio proiettivamente simmetrico p-iperbolico è proiettivamente equivalente ad uno spazio Riemanniano simmetrico di Einstein con curvatura negativa. Infine l'autore prova che una varietà Riemanniana completa n-dimensionale (n>/2) localmente affinemente simmetrica, proiettivamente simmetrica in senso proprio e proiettivamente omogenea è proiettivamente equivalente alla sfera Sⁿ oppure allo spazio proiettivo realeRP ⁿ.     

Maximal antipodal sets of compact classical symmetric spaces and their cardinalities I

We classify and explicitly describe maximal antipodal sets of some compact classical symmetric spaces and those of their quotient spaces by making use of suitable embeddings of these symmetric spaces into compact classical Lie groups. We give the cardinalities of maximal antipodal sets and we determine the maximum of the cardinalities and maximal antipodal sets whose cardinalities attain the maximum.     

Singular integrals on symmetric spaces, II

We extend some of our earlier results on boundedness of singular integrals on symmetric spaces of real rank one to arbitrary noncompact symmetric spaces. Our main theorem is a transference principle for operators defined by -bi-invariant kernels with certain large scale cancellation properties. As an application we prove boundedness of operators defined by Fourier multipliers that satisfy singular differential inequalities of the Hörmander-Michlin type.

     

On certain properties of growth points and their application to symmetric spaces

We prove the nonemptiness of a certain type of growth points and on the basis of this we construct an example of a discrete symmetric space (of sequences) which is not interval-complete, i.e., contains an order-bounded Cauchy sequence which does not have a limit. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 57–63, January, 1977. The author thanks G. Ya. Lozanovskii and V. A. Geiler for useful discussions.     

Rotations and Hermitian symmetric spaces

On an almost Hermitian manifold (M, g, J) one considers the naturally defined field of local diffeomorphismsj _m =exp _m J _m exp _m ^–1 ,mM, and in particular, one studies isometric, harmonic, holomorphic and symplecticj _m . This leads to some characterizations of special classes of almost Hermitian manifolds, including the class of Hermitian symmetric spaces. In addition, one treats some intrinsic and extrinsic geometrical properties of geodesic spheres relating to these local diffeomorphisms.Supported by grant 203.01.50 of the C.N.R., Italy.