Einstein-Hilbert action with cosmological term from Chern-Simons gravity

TL;DR

This paper introduces a modified Lie algebra expansion method using cyclic groups, leading to new invariant algebras and a Chern-Simons Lagrangian that relates to general relativity.

Contribution

It proposes a novel modification to the $S$-expansion method for Lie algebras, specifically for cyclic groups of even order, and applies it to derive new invariant tensors and a Chern-Simons Lagrangian.

Findings

Derived invariant tensors for $S_{H}$-expanded algebras.

Constructed a Chern-Simons Lagrangian invariant under the expanded algebra.

Explored the relationship between the new Lagrangian and general relativity.

Abstract

We propose a modification to the Lie algebra $S$-expansion method. The modification is carried out by imposing a condition on the $S$-expansion procedure, when the semigroup is given by a cyclic group of even order. The $S$-expanded algebras are called $S_{H}$-expanded algebras where $S=Z_{2n}$. The invariant tensors for $S_{H}$-expanded algebras are calculated and the dual formulation of $S_{H}$-expansion procedure is proposed. We consider the $S_{H}$-expansion of the five-dimensional $AdS$ algebra and its corresponding invariants tensors are found. Then a Chern-Simons Lagrangian invariant under the five-dimensional $AdS$ algebra $S_{H}$-expanded is constructed and its relationship to the general relativity is studied.