Two-step nilpotent Lie algebras with ad-invariant metrics and a special kind of skew-symmetric maps

Details Two-step nilpotent Lie algebras with ad-invariant metrics and a special kind of skew-symmetric maps Two-step nilpotent Lie algebras with ad-invariant metrics and a special kind of skew-symmetric maps

2006-11-26

arxiv.org/abs/math/0611789v1

J. Algebra Appl. 6 no. 6 (2007), 897 - 917.

Mathematics

Rings and Algebras

17 pages

Scientific paper

We prove that a 2-step nilpotent Lie algebras admitting an ad-invariant metric can be constructed from a vector space $\mathfrak v$ endowed with a inner product $<, >$ and an injective homomorphism $\rho: \mathfrak v \to \mathfrak{so}(\mathfrak v)$ satisfying $\rho(v)v=0$ for all $v\in \mathfrak v$. The corresponding simply connected pseudo-Riemannian Lie groups are flat and any isometry fixing the identity element does not depend on $\rho$. The description allows to construct examples starting with a compact semisimple Lie algebra and it is useful to show some applications.

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