Irreducible holonomy algebras of Riemannian supermanifolds

Galaev, Anton S.

doi:10.1007/s10455-011-9299-4

Cited by 12 publications

(10 citation statements)

References 38 publications

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“…This gives, in particular, a description of all the irreducible representations of Lie superalgebras with a nontrivial first prolongation. This description is clearly of interest per se but we would also like to mention that some representations with nontrivial first prolongation had recently played an important rôle in a partial generalization to the case of supermanifolds of the classical result of M. Berger on the possible irreducible holonomy algebras of Riemannian manifolds (see [11,12]).…”

Section: Introductionmentioning

confidence: 90%

Homogeneous irreducible supermanifolds and graded Lie superalgebras

Alekseevsky

Santi

2016

Int Math Res Notices

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Abstract. A depth one grading g = g −1 ⊕ g 0 ⊕ g 1 ⊕ · · · ⊕ g ℓ of a finite dimensional Lie superalgebra g is called nonlinear irreducible if the isotropy representation ad g 0 | g −1 is irreducible and g 1 = (0). An example is the full prolongation of an irreducible linear Lie superalgebra g 0 ⊂ gl(g −1 ) of finite type with non-trivial first prolongation. We prove that a complex Lie superalgebra g which admits a depth one transitive nonlinear irreducible grading is a semisimple Lie superalgebra with the socle s⊗Λ(C n ), where s is a simple Lie superalgebra, and we describe such gradings. The graded Lie superalgebra g defines an isotropy irreducible homogeneous supermanifold M = G/G0 where G, G0 are Lie supergroups respectively associated with the Lie superalgebras g and g0 := p≥0 g p .

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Section: Introductionmentioning

confidence: 90%

Homogeneous irreducible supermanifolds and graded Lie superalgebras

Alekseevsky

Santi

2016

Int Math Res Notices

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“…Пространство P(h) возникло как пространство значений некоторой компоненты тензора кривизны лоренцева многообразия. Позже оказалось, что этому пространству принадлежит также некоторая компонента тензора кривизны риманова супермногообразия [63].…”

Section: а с галаевunclassified

“…Группы голономии определены также для многообразий с конформными метриками, в частности, эти группы позволяют определить наличие метрик Эйнштейна в конформном классе [14]. Понятие группы голономии используется и для суперсвязностей на супермногообразиях [1], [63].…”

Section: а с галаевunclassified

Holonomy groups of Lorentzian manifolds

Galaev¹

2015

Uspekhi Mat. Nauk

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В работе дан обзор недавних результатов о классификации связных групп голономии лоренцевых многообразий. Получено упрощение конструкции лоренцевых метрик со всеми возможными связными группами голономии. В качестве применений рассмотрены уравнение Эйнштейна, лоренцевы многообразия с параллельными и рекуррентными спинорными полями, конформно плоские метрики Волкера и классификация 2-симметрических лоренцевых многообразий. Библиография: 123 названия. Ключевые слова: лоренцево многообразие, группа голономии, алгебра голономии, многообразие Волкера, уравнение Эйнштейна, рекуррентное спинорное поле, конформно плоское многообразие, 2-симметрическое лоренцево многообразие.

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“…Riemannian structures on supermanifolds are not so well studied as their classical counterparts. Relatively recent papers on the subject include [1][2][3][4]. One of the interesting aspects of the Riemannian supermanifolds is that we can have metrics that are either degree zero or degree one, referred to as even and odd Riemannian metrics, respectively.…”

Section: Introductionmentioning

confidence: 99%

Riemannian Structures on Z 2 n -Manifolds

Bruce

Grabowski

2020

Mathematics

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Very loosely, Z2n-manifolds are ‘manifolds’ with Z2n-graded coordinates and their sign rule is determined by the scalar product of their Z2n-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Z2n-manifold, i.e., a Z2n-manifold equipped with a Riemannian metric that may carry non-zero Z2n-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Z2n-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

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Irreducible holonomy algebras of Riemannian supermanifolds

Cited by 12 publications

References 38 publications

Homogeneous irreducible supermanifolds and graded Lie superalgebras

Homogeneous irreducible supermanifolds and graded Lie superalgebras

Holonomy groups of Lorentzian manifolds

Riemannian Structures on Z 2 n -Manifolds

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