Metric Lie algebras and quadratic extensions

Details Metric Lie algebras and quadratic extensions Metric Lie algebras and quadratic extensions

2003-12-11

arxiv.org/abs/math/0312243v1

Mathematics

Differential Geometry

53 pages

Scientific paper

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a non-degenerate invariant symmetric bilinear form. We show that any metric Lie algebra without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module A in a canonical way. Identifying equivalence classes of quadratic extensions of l by A with a certain cohomology set H^2_Q(l,A) we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.

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