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.
