Lie Algebras with Ad-Invariant Metrics. a Survey
LIE ALGEBRAS WITH AD-INVARIANT METRICS A SURVEY GABRIELA P. OVANDO Abstract. This is a survey work on Lie algebras with ad-invariant met- rics. We summarize main features, notions and constructions, in the aim of bringing into consideration the main research on the topic. We also give some list of examples in low dimensions. To the memory of our friend Sergio 1. Introduction An ad-invariant metric on a Lie algebra g is a non-degenerate symmetric bilinear form h , i which satisfies (1) h[x, y], zi + hy, [x, z]i = 0 for all x, y, z ∈ g. Lie algebras equipped with ad-invariant metrics are known as: “metric”[22], “metrised” [10], “orthogonal” [28], “quadratic” [27], “regular quadratic”[16], ”symmetric self-dual” [17], ”self-dual” [36] Lie algebras. Lie algebras which can be endowed with an ad-invariant metric are also known as “quadrable” or “metrisable”. They became relevant some years ago when they were useful in the formulation of some physical problems, for instance in the so known Adler-Kostant-Symes scheme. More recently they appeared in conformal field theory precisely as the Lie algebras for which a Sugawara construction exists [17]. They also constitute the basis for the construction of bi-algebras and they arXiv:1512.03997v2 [math.DG] 14 Jun 2017 give rise to interesting pseudo-Riemannian geometry [13]. For instance in [34] a result originally due to Kostant [26] was revalidated for pseudo-Riemannian metrics: it states that there exists a Lie group acting by isometries on a pseudo- Riemannian naturally reductive space (in particular symmetric spaces) whose Lie algebra can be endowed with an ad-invariant metric.
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