A free Lie algebra approach to curvature corrections to flat space-time

We investigate a systematic approach to include curvature corrections to the isometry algebra of flat space-time order-by-order in the curvature scale. The Poincaré algebra is extended to a free Lie algebra, with generalised boosts and translations that no longer commute. The additional generators satisfy a level-ordering and encode the curvature corrections at that order. This eventually results in an infinite-dimensional algebra that we refer to as Poincaré∞, and we show that it contains among others an (A)dS quotient. We discuss a non-linear realisation of this infinite-dimensional algebra, and construct a particle action based on it. The latter yields a geodesic equation that includes (A)dS curvature corrections at every order.

Presentations for Subrings and Subalgebras of Finite CO-Rank

Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

Soluble Product of Parafree Lie Algebras and Its Residual Properties

The parafree Lie algebras are an extraordinary class of Lie algebras which shares many properties with a free Lie algebra. In this work, we turn our attention to soluble product of parafree Lie algebras. We show that soluble product of parafree Lie algebras is parafree. Furthermore, we investigate some residual properties of that product.

Automorphisms of the completion of relatively free Lie algebras

Let [Formula: see text] be a relatively free Lie algebra of finite rank [Formula: see text], with [Formula: see text], [Formula: see text] be the completion of [Formula: see text] with respect to the topology defined by the lower central series [Formula: see text] of [Formula: see text] and [Formula: see text], with [Formula: see text]. We prove that, with respect to the formal power series topology, the automorphism group [Formula: see text] of [Formula: see text] is dense in the automorphism group [Formula: see text] of [Formula: see text] if and only if [Formula: see text] is nilpotent. Furthermore, we show that there exists a dense subgroup of [Formula: see text] generated by [Formula: see text] and a finite set of IA-automorphisms if and only if [Formula: see text] is generated by [Formula: see text] and a finite set of IA-automorphisms independent upon [Formula: see text] for all [Formula: see text]. We apply our study to several varieties of Lie algebras.