Algebra with ternary cyclic relations, representations and quark model

Abstract

We propose a unital associative algebra, motivated by a generalization of the Pauli’s exclusion principle proposed for the quark model. The generators of this algebra satisfy the following relations: The sum of squares of all generators is equal to zero (binary relation) and the sum of cyclic permutations of the factors in any triple product of generators is equal to zero (ternary relations). We study the structure of this algebra and calculate the dimensions of spaces spanned by homogeneous monomials. We establish a relation between our algebra and the irreducible representations of the rotation group. In particular, we show that the 10-dimensional space spanned by triple monomials is the space of a double irreducible unitary representation of the rotation group. We use ternary q- and qÌ -commutators, where q, qÌ are primitive 3rd order roots of unity, to split the 10-dimensional space spanned by triple monomials into a direct sum of two 5-dimensional subspaces. We endow these subspaces with a Hermitian scalar product by means of an orthonormal basis of triple monomials. In each subspace, there is an irreducible unitary representation so(3) → su(5). We calculate the matrix of this representation. The structure of this matrix indicates a possible connection between our algebra and the Georgi–Glashow model.

Keywords

• ternary algebras
• irreducible representations
• ternary extension of lie algebra
• ternary generalization of pauli’s exclusion principle
• quark model.