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.
Highlights
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
The free Lie algebra permit a plethora of quotients that connect to the Bargmann algebra and other non-relativistic symmetry algebras that have appeared recently in the literature [20,21,22,23]
It is not clear how to fit this extended algebra into a free Lie algebra construction similar to the one considered in the present paper
Summary
Going back to Le Bellac and Levy-Leblond one can study several limits of the relativistic Maxwell and Lorentz equations that lead to different forms of non-relativistic systems. One could consider a third case: 3) The pulse/shockwave Galilean Maxwell equations, where the limit is taken with the electric field and magnetic field large and of equal modulus. This last case has appeared in a recent study [18] of contractions of the Maxwell algebra. In the pulse case the algebra becomes (the scaling for Zij is with ω in this case, so that the electric and magnetic fields scale in the same way) [18] Comparing this to (2.2) and (2.3) we see that both the large electric and magnetic field generators have become boost invariant and that the magnetic field generator is not generated by the commutator of two translations anymore. We note that the pulse algebra (2.4) can be obtained from a further contraction of either (2.2) or (2.3) by scaling Zij appropriately such that the overall scaling of Zij and Zi match
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
