Marmo G. A Manifold view point of Lie Algebras (abstract)

Lie algebra brackets play a prominent role in the description of evolution (equations of motion) for any physical system, be it classical or quantum. On the other hand, the advent of general relativity has called for a description of physical systems in a coordinate independent manner. Special relativity has introduced the need for a composition law for velocities whose expression $\displaystyle{ (v+w)/[1+(vw/c^2)] }$ is alternative to the usual Galilean one $\displaystyle{ v+w }$, which appears to be a "contraction" of previous one when $\displaystyle{ c }$, the speed of light, goes to infinity. The composition law of special relativity is not Archimedean and is not associative in usual space-time (1+3). However in 1+1 space-time it defines an alternative "linear structure".