Physicists Build a Maxwellian Cube of Spacetime Symmetries for 3D Gravity

Every theory of space and time begins with a question about observers: how do two people moving at constant velocity relative to one another describe the same events? The mathematical answer is encoded in a kinematical Lie algebra, a compact set of generators describing rotations, boosts, and translations. In 1968, Henri Bacry and Jean-Marc Lévy-Leblond classified all such algebras under mild assumptions of homogeneity, isotropy, and parity invariance, and arranged them into an elegant diagram known as the kinematical cube. At one corner sits the anti-de Sitter algebra of a curved spacetime with negative cosmological constant; at another, the Poincaré algebra of flat spacetime; and along the remaining edges, the Galilean and Carrollian algebras that emerge when the speed of light is taken to infinity or zero. A new theoretical study published in The European Physical Journal C has now rebuilt this fifty-year-old structure on a more powerful foundation, and in doing so has opened a systematic route to entirely new families of gravitational theories in three dimensions.

The work, carried out by Patrick Concha, Nelson Gallegos, Evelyn Rodríguez, and Sebastián Salgado, replaces the contraction limits that generate the original cube with a technique called semigroup expansion. Where a contraction typically discards structure, an expansion does the opposite: it enlarges the algebra, adding new generators while preserving the ability to define consistent dynamics. The key insight is that a particular abelian semigroup, denoted S_E^(2), reproduces the familiar Inönü–Wigner contraction when applied in a specific way, meaning the original Bacry–Lévy-Leblond cube can be reinterpreted as a diagram of expansions rather than contractions. Once that reinterpretation is made, each arrow of the cube can be upgraded to a higher-order expansion, producing what the authors call a Maxwellian kinematical cube, and beyond it an infinite hierarchy of generalized kinematical algebras.

Why does the Maxwell algebra matter? Introduced originally to describe particles in a constant electromagnetic background, it extends the Poincaré algebra so that the translation generators no longer commute with each other; instead, their commutator produces a new set of generators associated with a constant field strength. In three-dimensional gravity, formulated as a Chern–Simons gauge theory, this extra generator comes with an additional gauge field that contributes non-trivially to conserved charges, asymptotic symmetries, and even the thermodynamics of black-hole and cosmological solutions. Recent analyses have shown that this gravitational Maxwell field modifies the first law of thermodynamics for asymptotically flat cosmological solutions, making the algebra far more than a mathematical curiosity.

The central technical obstacle in extending such structures to non-Lorentzian regimes is degeneracy. A Chern–Simons action is only well-defined if the underlying Lie algebra admits a non-degenerate invariant bilinear form, a symmetric pairing of generators that guarantees the field equations force the gauge curvature to vanish. In three dimensions, the plain Galilei algebra fails this test, and the consistent non-relativistic theory requires the extended Bargmann algebra, a double central extension of Galilei. On the ultra-relativistic side, the Carroll algebra happens to pass the test on its own, but its Maxwellian extension does not. The new paper shows that both the Maxwellian extended Bargmann algebra and the Maxwellian extended Carroll algebra, previously constructed by hand, arise naturally as vertices of the Maxwellian cube, each obtained through a resonant S_E^(2) expansion followed by a reduction that removes the semigroup’s zero element.

The construction is strikingly economical. Starting from the anti-de Sitter algebra so(2,2), the authors split its generators into two subspaces respecting a Z2 gradation, apply the semigroup multiplication law, and read off the commutation relations of the expanded algebra. The relativistic Maxwell algebra emerges first, and remarkably it turns out to be isomorphic to the extended AdS-Carroll algebra, an unexpected identification that the authors suggest justifies renaming that structure the Para-Maxwell algebra. Applying the same expansion machinery to the Maxwell algebra itself then yields the non-relativistic Maxwellian extended Bargmann algebra and the ultra-relativistic Maxwellian extended Carroll algebra, each inheriting a non-degenerate invariant tensor whose non-vanishing components the expansion method supplies automatically.

With the invariant tensors in hand, the authors write down explicit Chern–Simons gravity actions for each new algebra. In the Maxwellian extended Bargmann case, the action decomposes into sectors proportional to three coupling constants, and the first two sectors reproduce the most general Chern–Simons action for the ordinary extended Bargmann algebra, while the third carries the genuinely Maxwellian contribution. In the Carrollian case, the equations of motion retain the characteristic non-vanishing temporal torsion of Carroll gravity, but the Maxwellian extension introduces three additional gauge fields required to keep the invariant tensor non-degenerate. The authors also construct a Maxwellian extended static algebra, a structure not previously reported in the literature, and show that its full set of generators, including two abelian central charges, is the minimal content consistent with well-defined field equations.

The framework generalizes further. By replacing the semigroup S_E^(2) with an arbitrary higher-order semigroup S_E^(N), the authors generate an infinite hierarchy of kinematical cubes based on the so-called B_k algebras. The case N=1 reproduces the original Bacry–Lévy-Leblond cube, while N=2 gives exactly the Maxwellian extension. For N greater than or equal to 3, genuinely new algebras appear that are neither post-Newtonian nor post-Carrollian and cannot be obtained as extensions of the Galilei or Carroll algebras. Notably, the B_5 algebra at the relativistic level has previously been shown to recover standard General Relativity from Chern–Simons and Born–Infeld gravity actions, lending physical weight to the hierarchy. The authors derive the non-relativistic, Carrollian, and static counterparts of B_5, two of which had never been reported, and show that the Chern–Simons actions obey a recursive structure: each level of the hierarchy adds a single new term to the action of the level below.

The broader significance of the work extends beyond pure algebra. Non-relativistic symmetries have become central to holography, the correspondence relating gravitational theories to lower-dimensional field theories, and appear in effective descriptions of condensed matter systems such as the quantum Hall effect. Carrollian symmetries, meanwhile, have emerged in the physics of tensionless strings, black-hole horizons, and flat-space holography. Maxwell symmetry itself has recently appeared in gauge-theoretical descriptions of topological phases of matter, where Chern–Simons actions based on the Maxwell algebra reproduce relativistic Wen–Zee response terms characteristic of quantum Hall states. A unified framework that generates all of these symmetry structures, together with the invariant tensors needed to build consistent actions, provides a common language for what had been a scattered collection of special cases.

Important questions remain open. The additional gauge fields that appear in the non-Lorentzian Maxwellian theories go beyond the standard geometric data of clocks, rulers, and spin connections, and their physical interpretation is largely unexplored. The authors suggest they may encode post-Newtonian or post-Carrollian corrections, or perhaps gravito-magnetic-like effects, generalized torsion, or effective background fluxes. A complete second-order formulation, in which the gauge fields are solved in terms of geometric variables, is known only for specific cases such as the Bargmann and Newton–Hooke theories, where the Chern–Simons formulation connects to three-dimensional Hořava–Lifshitz gravity. For the full Maxwellian hierarchy, such formulations must be developed case by case. It also remains to be seen whether the new algebras arise as asymptotic symmetries of gravitational models, potentially yielding new extensions of the bms3 algebra that governs flat-space holography.

The authors outline several directions for future work, including the study of black-hole solutions and their thermodynamics in the generalized B_k Chern–Simons theories, where the extra gauge fields are expected to modify conserved charges and introduce new thermodynamic contributions. Extensions to supersymmetric and higher-spin theories are also on the horizon, since the semigroup expansion method has already proven effective for constructing non-Lorentzian supergravity theories with non-degenerate invariant tensors. What the present result delivers, however, is a conceptual shift: the venerable kinematical cube of Bacry and Lévy-Leblond is no longer an endpoint but the first rung of an infinite ladder, and every rung now comes equipped with the algebraic ingredients needed to write down a gravitational theory. For a field searching for structure beyond Einstein’s equations, that is a map worth having.

Subject of Research: Maxwell extensions of kinematical Lie algebras and their application to three-dimensional Chern–Simons gravity

Article Title: Maxwell kinematical algebras and 3D gravities

Article References: Concha, P., Gallegos, N., Rodríguez, E., & Salgado, S. (2026). Maxwell kinematical algebras and 3D gravities. The European Physical Journal C, 86(9), Article 1035. https://doi.org/10.1140/epjc/s10052-026-16284-w

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16284-w

Keywords: kinematical algebras, Maxwell algebra, Chern–Simons gravity, three-dimensional gravity, semigroup expansion, Bacry–Lévy-Leblond cube, Carroll symmetry, Bargmann algebra, non-Lorentzian gravity, Lie algebra expansion, holography, B_k algebras