Constructions of L∞ Algebras and Their Field Theory Realizations

We construct L∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these L∞ algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define L∞ algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term L∞ algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid.

Shift commutator algebras and multipliers

We determine the precise structure of all multipliers on the commutator algebra associated to the shift operator on a Hilbert space. The problem has its own interest by its connection with the theory of Toeplitz and Laurent operators.

The commutator algebra of covariant derivative as general framework for extended gravity. The Rastall theory case and the role of the torsion

In this short review, we discuss the approach of the commutator algebra of covariant derivative to analyze the gravitational theories, starting from the standard Einstein's general theory of relativity (GTR) and focusing on the Rastall theory. After that, we discuss the important role of the torsion in this mathematical framework.In the appendix of the paper, we analyze the importance of the nascent gravitational wave (GW) astronomy as a tool to discriminate among the GTR and alternative theories of gravity.

Commutator-based linearization of N = 1 nonlinear supersymmetry

We consider the linearization of N = 1 nonlinear supersymmetry (NLSUSY) based on a commutator algebra in Volkov–Akulov (VA) NLSUSY theory. We show explicitly that U(1) gauge and scalar supermultiplets in addition to a vector supermultiplet with general auxiliary fields in linear SUSY theories are obtained from a same set of bosonic and fermionic functionals (composites) which are expressed as simple products of the powers of a Nambu–Goldstone fermion and a fundamental determinant in the NLSUSY theory.

New Black Holes Solutions in a Modified Gravity

We present some basic concepts of a theory of modified gravity, inspired by the gauge theories, where the commutator algebra of covariant derivative gives us an added term with respect to the General Relativity, which represents the interaction of gravity with a substratum. New spherically symmetric solutions of this theory are obtained and can be viewed as solutions that reproduce the mass, the charge, the cosmological constant, and the Rindler acceleration, without coupling with the matter content, that is, in the vacuum.

Remarks on the Representation Theory of the Moyal Plane

We present an explicit construction of a unitary representation of the commutator algebra satisfied by position and momentum operators on the Moyal plane.