Symmetries of M-theory and free Lie superalgebras

We study systematically various extensions of the Poincar\'e superalgebra. The most general structure starting from a set of spinorial supercharges $Q_\alpha$ is a free Lie superalgebra that we discuss in detail. We explain how this universal extension of the Poincar\'e superalgebra gives rise to many other algebras as quotients, some of which have appeared previously in various places in the literature. In particular, we show how some quotients can be very neatly related to Borcherds superalgebras. The ideas put forward also offer some new angles on exotic branes and extended symmetry structures in M-theory.