Lie algebraic Carroll/Galilei duality

We characterise Lie groups with bi-invariant bargmannian, galilean or carrollian structures. Localising at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian or galilean structures are actually determined by the same data: a metric Lie algebra with a skew-symmetric derivation. This is the same data defining a one-dimensional double extension of the metric Lie algebra and, indeed, bargmannian Lie algebras coincide with such double extensions, containing carrollian Lie algebras as an ideal and projecting to galilean Lie algebras. This sets up a canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras. This reformulation allows us to use the structure theory of metric Lie algebras to give a list of bargmannian, carrollian and galilean Lie algebras in the positive-semidefinite case. We also characterise Lie groups admitting a bi-invariant (ambient) leibnizian structure. Leibnizian Lie algebras extend the class of bargmannian Lie algebras and also set up a non-canonical correspondence between carrollian and galilean Lie algebras.