A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and nondegenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariant metrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involved give rise to.

In this paper, we study and classify the 3-dimensional Hom-Lie algebras over C. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space g. The well known Lie brackets for 3-dimensional Lie algebras arise within the appropriate representatives. Then, for each product representative, we provide a complete set of canonical forms for the linear maps g → g that turn g into a Hom-Lie algebra. As by-products, Hom-Lie algebras for which the linear maps g → g are not automorphisms for their corresponding products, are exhibited. Also, interesting examples are given which, despite of having a fixed Lie-algebra product defined on g, break down into non-isomorphic families of Hom-Lie algebras. This is the case for the complex simple Lie algebra sl 2 , for which an exhaustive list of representatives of the isomorphism classes of Hom-Lie algebras that can be defined for its ordanary Lie algebra bracket is given. Similarly, interesting examples are given of representatives for which the skew-symmetric bilinear products can never be Lie algebra brackets on g.

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In this work, we show that the existence of invertible generalized derivations impose strong restrictions on the structure of a complex finite-dimensional Lie algebra. In particular, we recover the fact that a real Lie algebra admitting an abelian complex structure is necessarily solvable. On the other hand, we state a structure theorem for a Lie algebra [Formula: see text] admitting a periodic generalized derivation [Formula: see text].