Extended kinematical 3D gravity theories

Abstract

In this work, we classify all extended and generalized kinematical Lie algebras that can be obtained by expanding the $$ \mathfrak{so} $$ so (2, 2) algebra. We show that the Lie algebra expansion method based on semigroups reproduces not only the original kinematical algebras but also a family of non- and ultra-relativistic algebras. Remarkably, the extended kinematical algebras obtained as sequential expansions of the AdS algebra are characterized by a non-degenerate bilinear invariant form, ensuring the construction of a well-defined Chern-Simons gravity action in three spacetime dimensions. Contrary to the contraction process, the degeneracy of the non-Lorentzian theories is avoided without extending the relativistic algebra but considering a bigger semigroup. Using the properties of the expansion procedure, we show that our construction also applies at the level of the Chern-Simons action.

Four dimensional topological supergravities from transgression field theory

“…It would be interesting to consider the semigroup expansion method [44] in order to derive a non-relativistic five-dimensional supergravity action. As it was shown in [45], some semigroups are useful to derive non-relativistic algebras with a non-degenerate invariant tensor. It would be worth exploring if such particularity can be extended in presence of supersymmetry.…”