  We show that each irreducible tensor representation of weight 2 of the
rotation group of three-dimensional space in the space of rank 3 covariant
tensors gives rise to an associative algebra with unity. We find the algebraic
relations that the generators of these algebras must satisfy. Part of these
relations has a form of binary relations and another part has a form of ternary
relations. The structure of ternary relations is based on the cyclic group Z_3
and the primitive cubic root of unity q=\exp(2\pi i/3). The subspace of each
algebra spanned by the triple products of generators is 5-dimensional and it is
the space of an irreducible tensor representation of weight 2 of the rotation
group SO(3). We define a Hermitian scalar product in this 5-dimensional
subspace and construct an orthonormal basis for it. Then we find the
representation matrix of an infinitesimal rotation. We show that constructed
algebras with binary and ternary relations can have applications in the quark
model and Grand Unification Theories.
