scholarly journals LAURENT POLYNOMIAL LANDAU–GINZBURG MODELS FOR COMINUSCULE HOMOGENEOUS SPACES

2021 ◽  
Author(s):  
PETER SPACEK
Keyword(s):  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Laurent Polynomial ◽  
Complete Intersections ◽  
Laurent Polynomials ◽  
Young Diagrams ◽  
Similar Shape ◽  
Algebraic Torus ◽  
The One ◽  
Polynomial Potentials

AbstractIn this article we construct Laurent polynomial Landau–Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau–Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.

scholarly journals Obstructions to the existence of compact Clifford–Klein forms for tangential symmetric spaces

2021 ◽  
pp. 2150074
Author(s):  
Koichi Tojo
Keyword(s):  
Homogeneous Space ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Existence Problem ◽  
Radon Number ◽  
Odd Degree ◽  
The One ◽  
Associated Pair ◽  
Zero Points ◽  
Direct Calculations

For a homogeneous space [Formula: see text] of reductive type, we consider the tangential homogeneous space [Formula: see text]. In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino’s condition by direct calculations. In this paper, some easy-to-check necessary conditions ([Formula: see text][Formula: see text]obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister’s theorem (the existence problem of common zero points of polynomials of odd degree).

PROPER ACTIONS ON SOME EXPONENTIAL SOLVABLE HOMOGENEOUS SPACES

2005 ◽  
Vol 16 (09) ◽  
pp. 941-955 ◽  
Author(s):  
ALI BAKLOUTI ◽  
FATMA KHLIF
Keyword(s):  
Maximal Subgroup ◽  
Homogeneous Space ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Intersection Property ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Simply Connected

Let G be a connected, simply connected nilpotent Lie group, H and K be connected subgroups of G. We show in this paper that the action of K on X = G/H is proper if and only if the triple (G,H,K) has the compact intersection property in both cases where G is at most three-step and where G is special, extending then earlier cases. The result is also proved for exponential homogeneous space on which acts a maximal subgroup.

Small vases from Euboean workshops

1963 ◽  
Vol 58 ◽  
pp. 14-19 ◽  
Author(s):  
A. D. Ure
Keyword(s):  
Fourth Century ◽  
Similar Shape ◽  
The One

Perhaps the most popular vase shape in Euboea in the fourth century was the lidded lekanis. A number of specimens, decorated with floral motives on the lid and simple leaves or linear patterns on the receptacles, may be seen in BSA lv, pls. 54–57 passim. They show considerable variety in the shape of the knobs and in the treatment of the handles, but the receptacles remain fundamentally the same. Akin to these are little bowls of similar shape, lidded, but without handles. Such are the vases of a kernos of unknown provenience in Athens with a vaguely anthropomorphic central handle consisting of a long loop surmounted by a moulded head and with small arm-like projections recalling those of a herm (Plate 1, 1). The lids of three of the four little vases are preserved, the two nearest the handle having horizontal rims, while the two outer vases both had lids with rims turning vertically down, as is shown by the one extant lid and by the flanges on both the receptacles. The knobs of the two lids with horizontal rims are of a shape similar to the stemmed foot of a cup or dish; the surviving lid with down-turned rim has a ring like the footring of a stemless cup to serve as a knob. Each kind of knob, stemmed or stemless, functioned as a foot when the lid was set upside down on the table and became a dish. The low footring knobs were not unknown in Athens, but they were commoner in Euboea. One was noticed in BSA lv. 212, no. 8 on a lid, not figured, belonging to the Bonn group of floral black-figure, which is undoubtedly of Euboean, and very probably of Chalcidian manufacture. We shall see more of them later on.

The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

2016 ◽  
Vol 27 (10) ◽  
pp. 1650085
Author(s):  
A. Baklouti ◽  
N. Elaloui ◽  
I. Kedim
Keyword(s):  
Homogeneous Space ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Rigidity Theorem ◽  
General Context ◽  
Affine Transformations ◽  
Discontinuous Group ◽  
Solvable Case ◽  
Local Rigidity ◽  
Discontinuous Groups

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that any discontinuous group [Formula: see text] for a homogeneous space [Formula: see text] is abelian and at most of rank 2. Then we discuss an analog of the Selberg–Weil–Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the [Formula: see text]-action on [Formula: see text] is not always effective, and thus the space of group theoretic deformations (formal deformations) [Formula: see text] could be larger than geometric deformation spaces. We determine [Formula: see text] and also its quotient modulo uneffective parts when the rank [Formula: see text]. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.

Symbols in Berezin quantization for representation operators

2021 ◽  
pp. 296-304
Author(s):  
Vladimir F. Molchanov ◽  
Svetlana V. Tsykina
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Grassmann Manifold ◽  
Homogeneous Spaces ◽  
Finite Multiplicity ◽  
Basic Notion ◽  
Berezin Quantization ◽  
Algebraic Version ◽  
Enveloping Algebra ◽  
Algebra Of Operators

The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F^♮ defined on M. These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization — we call it the polynomial quantization — is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G. In this case symbols turn out to be polynomials on the Lie algebra g. In this paper we offer a new theme in the Berezin quantization on G/H: as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G=SL(2;R), H — the subgroup of diagonal matrices, G/H — a hyperboloid of one sheet in R^3; b) G — the pseudoorthogonal group SO_0 (p; q), the subgroup H covers with finite multiplicity the group SO_0 (p-1,q -1)×SO_0 (1;1); the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G.

scholarly journals On Formality of Some Homogeneous Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1011
Author(s):  
Aleksy Tralle
Keyword(s):  
Homogeneous Space ◽  
Root System ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Homogeneous Spaces ◽  
Direct Proof ◽  
Classical Type ◽  
Sasakian Manifolds ◽  
Full Analysis

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

scholarly journals Non-Commutative Integration of the Dirac Equation in Homogeneous Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1867
Author(s):  
Alexander Breev ◽  
Alexander Shapovalov
Keyword(s):  
Dirac Equation ◽  
Exact Solutions ◽  
Homogeneous Space ◽  
Integration Method ◽  
Homogeneous Spaces ◽  
Separation Of Variables ◽  
General Structure ◽  
Three Dimensional ◽  
De Sitter ◽  
Elementary Functions

We develop a non-commutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the non-commutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the non-commutative integration method. In addition, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time AdS3 using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the non-commutative integration method.

Description of the larva of Cinygmula levanidovi Tshernova & Belov, 1982(Ephemeroptera, Heptageniidae) with redescription of the male adult from the Russian Far East

Zootaxa ◽  
2020 ◽  
Vol 4772 (2) ◽  
pp. 371-378
Author(s):  
TATIANA M. TIUNOVA
Keyword(s):  
Russian Far East ◽  
Far East ◽  
Gill Filament ◽  
Male Adult ◽  
Similar Shape ◽  
Gill Filaments ◽  
The One ◽  
Far Eastern ◽  
First Time ◽  
The Russian Far East

The larvae, male and female imagines, and eggs of Cinygmula levanidovi Tshernova & Belov 1982 are described based on reared specimens from the Russian Far East. The larvae, female imago and eggs are described and illustrated for the first time. The larva of C. levanidovi is similar to the one of C. hirasana Imanishi, 1935 and C. kurenzovi (Bajkova, 1965). However, it can be distinguished from these species and from all other Far Eastern Cinygmula by the shape of its tergalius I, which has a heart-shape and bears a single short gill filament. Tergalius I of C. hirasana and C. kurenzovi possess a similar shape, but there are no gill filaments on the first and the other tergalii. 

scholarly journals Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

2020 ◽  
Vol 20 (03) ◽  
pp. 2050015
Author(s):  
Raphaël Carroy ◽  
Andrea Medini ◽  
Sandra Müller
Keyword(s):  
Homogeneous Space ◽  
General Result ◽  
Homogeneous Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Axiom Of Determinacy ◽  
Locally Compact Spaces

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.

Naturally Reductive Homogeneous Riemannian Manifolds

1985 ◽  
Vol 37 (3) ◽  
pp. 467-487 ◽  
Author(s):  
Carolyn S. Gordon
Keyword(s):  
Homogeneous Space ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Invariant Metrics ◽  
List Type ◽  
Left Invariant Metrics ◽  
Homogeneous Riemannian Manifolds ◽  
Transitive Subgroup ◽  
Naturally Reductive

The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are(i) to study all naturally reductive homogeneous spaces of G when G is either semisimple of noncompact type or nilpotent and(ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

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