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Homogeneous Einstein metrics on G_2/T
TL;DR: In this article, the authors constructed the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold $M = G 2/T, which admits at least one non-Kahler and not normal homogeneous Einstein metric.
Abstract: We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold $M=G_2/T$. By computing a Gr\"obner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly two non-K\"ahler invariant Einstein metrics. Thus $G_2/T$ turns out to be the first known example of an exceptional full flag manifold which admits at least one non-K\"ahler and not normal homogeneous Einstein metric.
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TL;DR: In this paper, the homogeneous Einstein equation for generalized flag manifolds was constructed for a compact simple Lie group whose isotropy representation decomposes into five inequivalent irreducible submodules.
Abstract: We construct the homogeneous Einstein equation for generalized flag manifolds $G/K$ of a compact simple Lie group $G$ whose isotropy representation decomposes into five inequivalent irreducible $\Ad(K)$-submodules. To this end we apply a new technique which is based on a fibration of a flag manifold over another flag manifold and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Grobner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E_6/(SU(4) x SU(2) x U (1) x U (1)) and E_7/(U(1) x U(6)) we find explicitely all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2\ell +1)/(U(1) x U (p) x SO(2(\ell -p-1)+1)) and SO(2\ell)/(U(1) x U (p) x SO(2(\ell -p-1))) we prove existence of at least two non Kahler-Einstein metrics. For small values of $\ell$ and $p$ we give the precise number of invariant Einstein metrics.
10 citations
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TL;DR: In this article, the authors give an overview of progress on homogeneous Einstein metrics on large classes of homogeneous manifolds, such as generalized flag manifolds and Stiefel manifolds.
Abstract: We give an overview of progress on homogeneous Einstein metrics on large classes of homogeneous manifolds, such as generalized flag manifolds and Stiefel manifolds. The main difference between these two classes of homogeneous spaces is that their isotropy representation does not contain/contain equivalent summands. We also discuss a third class of homogeneous spaces that falls into the intersection of such dichotomy, namely the generalized Wallach spaces. We give new invariant Einstein metrics on the Stiefel manifold $V_5\mathbb{R}^n$ ($n\ge 7$) and through this example we show how to prove existence of invariant Einstein metrics by manipulating parametric systems of polynomial equations. This is done by using Gr\"obner bases techniques. Finally, we discuss some open problems.
9 citations
TL;DR: In this article, the invariant Einstein metrics on generalized flag manifolds of exceptional groups with six isotropy summands are presented, which can be used to calculate structure constants of generalized flag manifold with any number of isotropy sumands.
Abstract: For a generalized flag manifold M = G/K of a compact connected simple Lie group G whose isotropy representation decomposes into more than five isotropy summands, there are only a few results about the homogeneous Einstein metrics on M. Finding the invariant Einstein metrics on generalized flag manifolds, there are two difficulties. One is computing the non-zero structure constants, the other is computing the Grobner basis of the system of Einstein equations. In this paper, we give a method (Theorem A) which can be used to calculate structure constants of generalized flag manifolds with any number of isotropy summands. In this direction we present invariant Einstein metrics on some flag manifolds of exceptional groups with six isotropy summands.
5 citations
TL;DR: In this article, the authors studied invariant Einstein metrics on certain compact homogeneous spaces with three isotropy summands and gave a complete classification of invariant metrics on the coset space.
Abstract: In this paper, we study invariant Einstein metrics on certain compact homogeneous spaces with three isotropy summands. We show that, if $G/K$ is a compact isotropy irreducible space with $G$ and $K$ simple, then except for some very special cases, the coset space $G\times~G/\Delta(K)$ carries at least two invariant Einstein metrics. Furthermore, in the case that $G_{1},~G_{2}$ and $K$ are simple Lie groups, with $K\subset~G_1,~K\subset~G_2$, and $G_{1}
eq~G_{2}$, such that $G_{1}/K$ and $G_{2}/K$ are compact isotropy irreducible spaces, we give a complete classification of invariant Einstein metrics on the coset space $G_{1}\times~G_{2}/\Delta(K)$.
4 citations
TL;DR: In this paper, the authors compute non-zero structure constants of the full flag manifold M = SO(7)/T with nine isotropy summands, then construct the Einstein equations.
Abstract: The authors compute non-zero structure constants of the full flag manifold M = SO(7)/T with nine isotropy summands, then construct the Einstein equations. With the help of computer they get all the forty-eight positive solutions (up to a scale ) for SO(7)/T, up to isometry there are only five G-invariant Einstein metrics, of which one is Kahler Einstein metric and four are non-Kahler Einstein metrics.
1 citations
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01 Jan 1978TL;DR: In this article, the structure of semisimplepleasure Lie groups and Lie algebras is studied. But the classification of simple Lie algesbras and of symmetric spaces is left open.
Abstract: Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.
6,321 citations
TL;DR: In the Groebner package, the most commonly used commands are NormalForm, for doing the division algorithm, and Basis, for computing a Groebners basis as mentioned in this paper. But these commands require a large number of variables.
Abstract: (here, > is the Maple prompt). Once the Groebner package is loaded, you can perform the division algorithm, compute Groebner bases, and carry out a variety of other commands described below. In Maple, a monomial ordering is called a monomial order. The monomial orderings lex, grlex, and grevlex from Chapter 2 are easy to use in Maple. Lex order is called plex (for “pure lexicographic”), grlex order is called grlex, and grevlex order is called tdeg (for “total degree”). Be careful not to confuse tdeg with grlex. Since a monomial order depends also on how the variables are ordered, Maple needs to know both the monomial order you want (plex, grlex or tdeg) and a list of variables. For example, to tell Maple to use lex order with variables x > y > z, you would need to input plex(x,y,z). The Groebner package also knows some elimination orders, as defined in Exercise 5 of Chapter 3, §1. To eliminate the first k variables from x1, . . . , xn, one can use the monomial order lexdeg([x 1,. . .,x k],[x {k+1},. . . ,x n]) (remember that Maple encloses a list inside brackets [. . .]). This order is the elimination order of Bayer and Stillman described in Exercise 6 of Chapter 3, §1. The Maple documentation for the Groebner package also describes how to use certain weighted orders, and we will explain below how matrix orders give us many more monomial orderings. The most commonly used commands in the Groebner package are NormalForm, for doing the division algorithm, and Basis, for computing a Groebner basis. NormalForm has the following syntax:
3,332 citations
950 citations
Book•
01 Jan 2005
Abstract: Preface 1. Results on topological spaces 1.1 Irreducible sets and spaces 1.2 Dimension 1.3 Noetherian spaces 1.4 Constructible sets 1.5 Gluing topological spaces 2. Rings and modules 2.1 Ideals 2.2 Prime and maximal ideals 2.3 Rings of fractions and localization 2.4 Localization of modules 2.5 Radical of an ideal 2.6 Local rings 2.7 Noetherian rings and modules 2.8 Derivations 2.9 Module of differentials 3. Integral extensions 3.1 Integral dependence 3.2 Integrally closed rings 3.3 Extensions of prime ideals 4. Factorial rings 4.1 Generalities 4.2 Unique factorization 4.3 Principal ideal domains and Euclidean domains 4.4 Polynomial and factorial rings 4.5 Symmetric polynomials 4.6 Resultant and discriminant 5. Field extensions 5.1 Extensions 5.2 Algebraic and transcendental elements 5.3 Algebraic extensions 5.4 Transcendence basis 5.5 Norm and trace 5.6 Theorem of the primitive element 5.7 Going Down Theorem 5.8 Fields and derivations 5.9 Conductor 6. Finitely generated algebras 6.1 Dimension 6.2 Noether's Normalization Theorem 6.3 Krull's Principal Ideal Theorem 6.4 Maximal ideals 6.5 Zariski topology 7. Gradings and filtrations 7.1 Graded rings and graded modules 7.2 Graded submodules 7.3 Applications 7.4 Filtrations 7.5 Grading associated to a filtration 8. Inductive limits 8.1 Generalities 8.2 Inductive systems of maps 8.3 Inductive systems of magmas, groups and rings 8.4 An example 8.5 Inductive systems of algebras 9. Sheaves of functions 9.1 Sheaves 9.2 Morphisms 9.3 Sheaf associated to a presheaf 9.4 Gluing 9.5 Ringed space 10. Jordan decomposition and some basic results on groups 10.1 Jordan decomposition 10.2 Generalities on groups 10.3 Commutators 10.4 Solvable groups 10.5 Nilpotent groups 10.6 Group actions 10.7 Generalities on representations 10.8 Examples 11. Algebraic sets 11.1 Affine algebraic sets 11.2 Zariski topology 11.3 Regular functions 11.4 Morphisms 11.5 Examples of morphisms 11.6 Abstract algebraic sets 11.7 Principal open subsets 11.8 Products of algebraic sets 12. Prevarieties and varieties 12.1 Structure sheaf 12.2 Algebraic prevarieties 12.3 Morphisms of prevarieties 12.4 Products of prevarieties 12.5 Algebraic varieties 12.6 Gluing 12.7 Rational functions 12.8 Local rings of a variety 13. Projective varieties 13.1 Projective spaces 13.2 Projective spaces and varieties 13.3 Cones and projective varieties 13.4 Complete varieties 13.5 Products 13.6 Grassmannian variety 14. Dimension 14.1 Dimension of varieties 14.2 Dimension and the number of equations 14.3 System of parameters 14.4 Counterexamples 15. Morphisms
230 citations
TL;DR: On demontre un theoreme d'existence general for des metriques d'Einstein homogenes and on presente des espaces homogenees compacts simplement connexes qui ne portent pas une metrique d'einstein homogene as discussed by the authors.
Abstract: On demontre un theoreme d'existence general pour des metriques d'Einstein homogenes et on presente des espaces homogenes compacts simplement connexes qui ne portent pas une metrique d'Einstein homogene
210 citations