Algebraic classification of higher dimensional spacetimes

A. Coley; N. Pelavas

doi:10.1007/s10714-006-0232-2

Spinor-helicity and the algebraic classification of higher-dimensional spacetimes

Classical and Quantum Gravity ◽

10.1088/1361-6382/ab03df ◽

2019 ◽

Vol 36 (6) ◽

pp. 065006 ◽

Cited By ~ 3

Author(s):

Ricardo Monteiro ◽

Isobel Nicholson ◽

Donal O’Connell

Keyword(s):

Algebraic Classification ◽

Higher Dimensional

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Algebraic classification of higher dimensional spacetimes based on null alignment

Classical and Quantum Gravity ◽

10.1088/0264-9381/30/1/013001 ◽

2012 ◽

Vol 30 (1) ◽

pp. 013001 ◽

Cited By ~ 39

Author(s):

Marcello Ortaggio ◽

Vojtěch Pravda ◽

Alena Pravdová

Keyword(s):

Algebraic Classification ◽

Higher Dimensional

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Betti numbers and pseudoeffective cones in 2-Fano varieties

Advances in Geometry ◽

10.1515/advgeom-2021-0004 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Giosuè Emanuele Muratore

Keyword(s):

Large Class ◽

Betti Numbers ◽

Fano Varieties ◽

Higher Dimensional ◽

Special Case ◽

Answering Questions

Abstract The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher-dimensional analogous properties of Fano varieties. We consider (weak) k-Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties, in analogy with the case k = 1. Then we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index at least n − 2, and we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in [2].

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Classification of one-dimensional superattracting germs in positive characteristic

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.35 ◽

2014 ◽

Vol 35 (7) ◽

pp. 2242-2268 ◽

Cited By ~ 2

Author(s):

MATTEO RUGGIERO

Keyword(s):

Positive Characteristic ◽

Closed Field ◽

Algebraically Closed Field ◽

One Dimensional ◽

Higher Dimensional ◽

Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

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Asymptotic stability of heteroclinic cycles in systems with symmetry. II

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500003693 ◽

2004 ◽

Vol 134 (6) ◽

pp. 1177-1197 ◽

Cited By ~ 59

Author(s):

Martin Krupa ◽

Ian Melbourne

Keyword(s):

Asymptotic Stability ◽

Transition Matrix ◽

Sufficient Conditions ◽

Systematic Investigation ◽

Necessary And Sufficient Conditions ◽

Sufficient Condition ◽

Heteroclinic Cycles ◽

Higher Dimensional ◽

Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

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Algebraic Classification of Field Equations

The General Theory of Relativity ◽

10.1007/978-1-4614-3658-4_7 ◽

2012 ◽

pp. 465-536

Author(s):

Anadijiban Das ◽

Andrew DeBenedictis

Keyword(s):

Field Equations ◽

Algebraic Classification

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The Algebraic and Geometric Classification of Four Dimensional Superalgebras

10.26686/wgtn.16915645.v1 ◽

2021 ◽

Author(s):

◽

Aaron Armour

Keyword(s):

Classification Problem ◽

Associative Algebras ◽

New Variety ◽

Graded Algebras ◽

Finite Representation ◽

Algebraic Classification ◽

Irreducible Components ◽

Module Algebras ◽

Sweedler’S Hopf Algebra

The algebraic and geometric classification of k-algbras, of dimension fouror less, was started by Gabriel in “Finite representation type is open” [12]. Several years later Mazzola continued in this direction with his paper “Thealgebraic and geometric classification of associative algebras of dimensionfive” [21]. The problem we attempt in this thesis, is to extend the resultsof Gabriel to the setting of super (or Z2-graded) algebras — our main effortsbeing devoted to the case of superalgebras of dimension four. Wegive an algebraic classification for superalgebras of dimension four withnon-trivial Z2-grading. By combining these results with Gabriel’s we obtaina complete algebraic classification of four dimensional superalgebras. This completes the classification of four dimensional Yetter-Drinfeld modulealgebras over Sweedler’s Hopf algebra H4 given by Chen and Zhangin “Four dimensional Yetter-Drinfeld module algebras over H4” [9]. Thegeometric classification problem leads us to define a new variety, Salgn —the variety of n-dimensional superalgebras—and study some of its properties. The geometry of Salgn is influenced by the geometry of the varietyAlgn yet it is also more complicated, an important difference being thatSalgn is disconnected. While we make significant progress on the geometricclassification of four dimensional superalgebras, it is not complete. Wediscover twenty irreducible components of Salg4 — however there couldbe up to two further irreducible components.

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