Tensor models and 3-ary algebras

TL;DR

This paper explores the algebraic structures of tensor models used in quantum gravity, revealing their symmetries are described by 3-ary algebras and connecting them to noncommutative spacetimes like Snyder's model.

Contribution

It demonstrates that tensor models' symmetries are represented by 3-ary algebras and links these to Lie triple systems and noncommutative spacetime symmetries.

Findings

Tensor models have algebraic expressions with symmetries represented by 3-ary algebras.

Coordinate algebras in fuzzy flat spacetimes form Lie triple systems.

Poincare transformations are generated by 3-ary algebras.

Abstract

Tensor models are the generalization of matrix models, and are studied as models of quantum gravity in general dimensions. In this paper, I discuss the algebraic structure in the fuzzy space interpretation of the tensor models which have a tensor with three indices as its only dynamical variable. The algebraic structure is studied mainly from the perspective of 3-ary algebras. It is shown that the tensor models have algebraic expressions, and that their symmetries are represented by 3-ary algebras. It is also shown that the 3-ary algebras of coordinates, which appear in the nonassociative fuzzy flat spacetimes corresponding to a certain class of configurations with Gaussian functions in the tensor models, form Lie triple systems, and the associated Lie algebras are shown to agree with those of the Snyder's noncommutative spacetimes. The Poincare transformations on the fuzzy flat spacetimes are shown to be generated by 3-ary algebras.