Ternary non -abelian symmetries on the multidimensional space entanglemen T
bic matrices opens up new possibilities for expanding a number of the basic principles of quantum physics as expanding the concepts of anticommutativity, particle identity, the principle of superposition, geometry of qutrites, unusual spinor quantum properties of invisible matter and maarcrions. The latter could play a role in solving the problem of quantization of a proton and neutron, and even an electron. In this article, we consider the entanglement geometry of three three -dimensional spaces in a 9 -dimensional Euclidean space based on ternary symmetries of the non -compact group TSU (3). Abelian and non -Abelian symmetry groups for describing qutrites are sequential components of an infinite series of groups and their corresponding n -ary algebras. The appearance on the group variety TSU (3) of new differential structures corresponds to an algebra with ternary commutation relations generalizing binary commutation Lie relations, and generators can be realized through cubic 3x3x3 matrices. Our entire analysis easily extends to the entire infinite network of n -ary groups and their corresponding algebras with matrix generators of dimension nxnx…xn.