An analytic method forS-Expansion involving resonance and reduction

Abstract

In this paper we describe an analytic method able to give the multiplication table(s) of the set(s) involved in an S-expansion process (with either resonance or 0 S -resonant-reduction) for reaching a target Lie (super)algebra from a starting one, after having properly chosen the partitions over subspaces of the considered (super)algebras.This analytic method gives us a simple set of expressions to find the subset decomposition of the set(s) involved in the process. Then, we use the information coming from both the initial (super)algebra and the target one for reaching the multiplication table(s) of the mentioned set(s). Finally, we check associativity with an auxiliary computational algorithm, in order to understand whether the obtained set(s) can describe semigroup(s) or just abelian set(s) connecting two (super)algebras.We also give some interesting examples of application, which check and corroborate our analytic procedure and also generalize some result already presented in the literature. * marcalderon@udec.cl †

Generalizing the $$\mathfrak {bms}_{3}$$ bms 3 and 2D-conformal algebras by expanding the Virasoro algebra

“…The S-expansion method [50] consists in combining the structure constants of a Lie algebra g with the elements of a semigroup S to obtain a new Lie algebra G = S × g. In this section we show that the centrally extended 2D-conformal algebra and the bms 3 algebra can be obtained explicitly as an S-expansion of the Virasoro algebra for suitable semigroups. For details regarding the notation and the S-expansion procedure we refer the reader to the original references [75][76][77][78][79][80][81][82].…”

Generalized Chern–Simons higher-spin gravity theories in three dimensions

“…A key ingredient in the construction of these algebras is the semigroup expansion method [79][80][81][82][83][84], which combines the structure a given Lie algebra with an abelian semigroup to form a Lie algebra of greater dimension. By applying this method to the sl(3, R) algebra we will construct spin-3 extensions of the Maxwell and the AdS-Lorentz algebras in three dimensions and the associated CS actions.…”

On the supersymmetric extension of asymptotic symmetries in three spacetime dimensions

“…is called the resonant subalgebra of G. Further studies of the S-expansion method have been developed by diverse authors and can be found in [58][59][60][61][62][63][64][65][66].…”