Symmetries of M-theory and free Lie superalgebras

Abstract

We study systematically various extensions of the Poincaré superalgebra. The most general structure starting from a set of spinorial supercharges Q α is a free Lie superalgebra that we discuss in detail. We explain how this universal extension of the Poincaré 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.

Generalized cosmological term in $$D=4$$ D = 4 supergravity from a new AdS–Lorentz superalgebra

“…In this way, as it was shown in [23] (see also [24]), the most general structure one obtains is the infinite-dimensional free Lie algebra, called Maxwell ∞ in the quoted references, generated by the P a 's. The ("conventional") Maxwell algebra M of [13][14][15]17] (also denoted in the literature as B 4 ) is generated by the set {J ab , P a , Z ab } and its commutation relations read as follows: Here, the bosonic generators Z ab are tensorial abelian charges.…”

Non-relativistic gravity theory based on an enlargement of the extended Bargmann algebra

“…and three extra U (1) generators Y 1 , Y 2 and Y 3 . At the relativistic level, there has been a growing interest in exploring (super)gravity models based on the Maxwell algebra and its generalizations [68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85][86][87]. Interestingly a dual version of the three-dimensional Maxwell algebra, known as Hietarinta algebra [88], can be obtained by interchanging the role of the momentum generator P A with the Maxwell gravitational generator Z A .…”

Galilean free Lie algebras

“…The algebraic structure can be embedded in a free Lie algebra construction as shown in [5] such that different quotients of the free Lie algebra yield the known relativistic Maxwell algebra or related extensions [6][7][8][9]. A similar analysis was undertaken for the supersymmetric theory in [10][11][12][13][14][15].…”