Hom-Lie algebras having non-invertible and equivariant twist maps are
studied. Central extensions of Hom-Lie algebras having these properties are
obtained and shown how the same properties are preserved. Conditions are given
so that the produced central extension has an invariant metric with respect to
its Hom-Lie product making its twist map self-adjoint when the original Hom-Lie
algebra has such a metric. This work is focused on algebras with these
properties and we call them quadratic Hom-Lie algebras. It is shown how a
quadratic Hom-Lie algebra gives rise to a quadratic Lie algebra and that the
Lie algebra associated to the given Hom-Lie central extension is a Lie algebra
central extension of it. It is also shown that if the 2-cocycle associated to
the central extension is not a coboundary, there exists a non-abelian and
non-associative algebra, the commutator of whose product is precisely the
Hom-Lie product of the Hom-Lie central extension. Moreover, the algebra whose
commutator realizes this Hom-Lie product is shown to be simple if the
associated Lie algebra is nilpotent. Non-trivial examples are provided