Ternary Z3-graded Generalization of Heisenberg's Algebra

Many Z2-graded algebraic structures can be generalized to the case of Z3-grading. Several such structures are presented: the Grassmann algenra, algebra of exterior forms, lie algebras. In the latter, the antisymmetric binary product is replaced by ternary Z3-skew symmetric product. Ternary Heisenberg algebra is then introduced and its Bogolyubov symmetry group established. The second quantization of this structure is introduces, and the sixth-order Hamiltonian defined, along with its eigenstates. The Bohr-Sommerfeld quantization is applied to the stationary periodic solutions and the eigenvalues of energy are computed.