Monatshefte für Mathematik

, Volume 174, Issue 2, pp 169–193 | Cite as

Extending structures for Lie algebras

Article

Abstract

Let \(\mathfrak{g }\) be a Lie algebra, \(E\) a vector space containing \(\mathfrak{g }\) as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on \(E\) such that \(\mathfrak{g }\) is a Lie subalgebra of \(E\). A general product, called the unified product, is introduced as a tool for our approach. Let \(V\) be a complement of \(\mathfrak{g }\) in \(E\): the unified product \(\mathfrak{g } \,\natural \, V\) is associated to a system \((\triangleleft , \, \triangleright , \, f, \{-, \, -\})\) consisting of two actions \(\triangleleft \) and \(\triangleright \), a generalized cocycle \(f\) and a twisted Jacobi bracket \(\{-, \, -\}\) on \(V\). There exists a Lie algebra structure \([-,-]\) on \(E\) containing \(\mathfrak{g }\) as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras \((E, [-,-]) \cong \mathfrak{g } \,\natural \, V\). All such Lie algebra structures on \(E\) are classified by two cohomological type objects which are explicitly constructed. The first one \(\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })\) will classify all Lie algebra structures on \(E\) up to an isomorphism that stabilizes \(\mathfrak{g }\) while the second object \(\mathcal{H }^{2} (V, \mathfrak{g })\) provides the classification from the view point of the extension problem. Several examples that compute both classifying objects \(\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })\) and \(\mathcal{H }^{2} (V, \mathfrak{g })\) are worked out in detail in the case of flag extending structures.

Keywords

The extension and the factorization problem Unified products Relative (non-abelian) cohomology for Lie algebras 

Mathematics Subject Classification (2010)

17B05 17B55 17B56 

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Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Faculty of EngineeringVrije Universiteit BrusselBrusselsBelgium
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharest 1Romania

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