How to Find the Holonomy Algebra of a Lorentzian Manifold

GALAEV, Anton. How to Find the Holonomy Algebra of a Lorentzian Manifold. LETTERS IN MATHEMATICAL PHYSICS. DORDRECHT: SPRINGER, 2015, vol. 105, No 2, p. 199-219. ISSN 0377-9017. Available from: https://doi.org/10.1007/s11005-014-0741-y.

Basic information

Original name

How to Find the Holonomy Algebra of a Lorentzian Manifold

Authors

GALAEV, Anton

Edition

LETTERS IN MATHEMATICAL PHYSICS, DORDRECHT, SPRINGER, 2015, 0377-9017

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Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

Netherlands

Confidentiality degree

is not subject to a state or trade secret

Impact factor

Impact factor: 1.517

DOI

https://doi.org/10.1007/s11005-014-0741-y

UT WoS

000348355500003

Keywords in English

Lorentzian manifold; holonomy group; holonomy algebra; de Rham-Wu decomposition

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Abstract

In the original language

Manifolds with exceptional holonomy play an important role in string theory, supergravity and M-theory. It is explained how one can find the holonomy algebra of an arbitrary Riemannian or Lorentzian manifold. Using the de Rham and Wu decompositions, this problem is reduced to the case of locally indecomposable manifolds. In the case of locally indecomposable Riemannian manifolds, it is known that the holonomy algebra can be found from the analysis of special geometric structures on the manifold. If the holonomy algebra of a locally indecomposable Lorentzian manifold (M, g) of dimension n is different from , then it is contained in the similitude algebra . There are four types of such holonomy algebras. Criterion to find the type of is given, and special geometric structures corresponding to each type are described. To each there is a canonically associated subalgebra . An algorithm to find is provided.