Galilean free Lie algebras

Abstract

We construct free Lie algebras which, together with the algebra of spatial rotations, form infinite-dimensional extensions of finite-dimensional Galilei Maxwell algebras appearing as global spacetime symmetries of extended non-relativistic objects and non-relativistic gravity theories. We show how various extensions of the ordinary Galilei algebra can be obtained by truncations and contractions, in some cases via an affine Kac-Moody algebra. The infinite-dimensional Lie algebras could be useful in the construction of generalized Newton-Cartan theories gravity theories and the objects that couple to them.

Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity

“…In the case Λ = 0 this algebra corresponds to the one found in [9] and further studied in [8,18,12]. One can extend this algebra by considering N = 4 in the expansion prescription (2.9).…”

“…a , H (m) for m > 2. Taking the limit Λ → 0, we obtain as a contraction of (2.24), the infinite-dimensional extension of the Galilei algebra introduced in [10] and further studied in [18].…”

“…(where ∧ denotes the antisymmetric tensor product, coming from the antisymmetry of the Lie bracket). The free Lie algebra construction continues to all positive levels [48,18] but in order to reproduce the algebra (2.24) we need to take a quotient. This quotient leads to the subalgebra of g (1) r+1 at positive levels and the ideal that one quotients out is generated by the Serre relations at level ℓ = 2.…”

“…This generalises previous constructions to include a cosmological constant and generates an infinite family of algebras of Newton-Hooke [19,20] type. By taking the limit of the cosmological constant to zero one recovers the non-relativistic algebras introduced in [10] and further studied in [8,18,12].…”

“…The Lie algebra expansion method can also be related to a sequence of post-Newtonian limits as shown in [12], and has also been applied to derive diverse non-relativistic symmetries in the context of (super-)gravity [13][14][15][16][17]. Another method is based on a Galilean free Lie algebra [18] that can be thought as the most general extension of the Galilei algebra and, upon taking quotients, has a connection to Lie algebra expansions and Kac-Moody algebras. One interesting conceptual point made in [12] is that the sequence of Lie algebras naturally comes with a generalisation of Minkowski space with more coordinates and an extension of the Minkowski metric.…”