We study and classify Lie algebras, homogeneous spacetimes and coadjoint
    orbits (“particles”) of Lie groups generated by spatial rotations, temporal and
    spatial translations and an additional scalar generator. As a first step we
    classify Lie algebras of this type in arbitrary dimension. Among them is the
    prototypical Lifshitz algebra, which motivates this work and the name “Lifshitz
    Lie algebras”. We classify homogeneous spacetimes of Lifshitz Lie groups.
    Depending on the interpretation of the additional scalar generator, these
    spacetimes fall into three classes:

    (1) ($d+2$)-dimensional Lifshitz spacetimes which have one additional
    holographic direction;

    (2) ($d+1$)-dimensional Lifshitz–Weyl spacetimes which can be seen as the
    boundary geometry of the spacetimes in (1) and where the scalar generator is
    interpreted as an anisotropic dilation; and

    (3) ($d+1$)-dimensional aristotelian spacetimes with one scalar charge,
    including exotic fracton-like symmetries that generalise multipole algebras.

    We also classify the possible central extensions of Lifshitz Lie algebras and
    we discuss the homogeneous symplectic manifolds of Lifshitz Lie groups in terms
    of coadjoint orbits.

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