On the Structure of Ternary Clifford Algebras and Their Irreducible Representations

SO(3)-Irreducible Geometry in Complex Dimension Five and Ternary Generalization of Pauli Exclusion Principle

“…It is important to note that in the binary case, the relation ( 1) is equivalent to the relation (2): that is, one immediately follows from the other. This is generally not true in the ternary case because the relation ( 3) is not equivalent to the relation (4) or (5).…”

“…The relation (3) follows from the relation (4) or ( 5), but not vice-versa. In other words, (3) is a more general condition than (4) or (5). Thus, we have an approach for extending the concept of skew symmetry from binary to ternary multiplication that differs from the classical one (rearranging two elements changes the sign of a product).…”

“…Relations (3)-( 5) can be used to construct a ternary analogue of Grassmann and Clifford algebra: that is, we can consider an algebra over the field of complex numbers generated by a system of generators that obey one of the relations (3)- (5). The properties, structure, and possible applications of such algebras were studied in papers [5][6][7][8][9][10][11][12][13][14]. A generalization of the Dirac operator by means of the above-mentioned algebras can be found in [3,4,12,15,16].…”

“…A twofold irreducible tensor representation of the rotation group in T 3 splits into two irreducible tensor representations if we decompose the 10-dimensional representation space T 3 into a direct sum of two 5-dimensional subspaces in a way that is invariant under the action of the rotation group. A decomposition into two subspaces can be made with the help of the relations (4) and (5): that is, we define the subspace T 3 q ⊂ T 3 by imposing the additional condition…”

“…It is known that every representation of the rotation group can be made unitary if we endow a representation space with an appropriate Hermitian metric. We endow the space T 3 q with the Hermitian metric h(T, S) = T ijk S ijk (13) and show that the irreducible representation of the rotation group in T 3 q is an inclusion R : SO(3) → SU (5). We find the orthonormal basis E A , 1 ≤ A ≤ 5 (here, E A are complexvalued third-order covariant tensors satisfying ( 9) and ( 10)) for the Hermitian space T 3 q and identify the space C 5 with the Hermitian vector space of tensors T 3 q by putting…”