  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.
